Analytical and Numerical Methods For Solving Partial Differential Equations and Integral Equations Arising in Physical Models

Abstract and Applied Analysis
Analytical and Numerical Methods

for Solving Partial Differential
Equations and Integral Equations
Arising in Physical Models
Guest Editors: Santanu Saha Ray, Om P. Agrawal, R. K. Bera, Shantanu Das,
and T. Raja Sekhar
Analytical and Numerical Methods for Solving
Partial Differential Equations and Integral
Equations Arising in Physical Models

Partial Differential Equations and Integral
Equations Arising in Physical Models
Guest Editors: Santanu Saha Ray, Om P. Agrawal, R. K. Bera,

Shantanu Das, and T. Raja Sekhar
Copyright © 2014 Hindawi Publishing Corporation. All rights reserved.
This is a special issue published in “Abstract and Applied Analysis.” All articles are open access articles distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original
work is properly cited.
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Contents
Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations
Arising in Physical Models, Santanu Saha Ray, Om P. Agrawal, R. K. Bera, Shantanu Das, and T. Raja Sekhar
Volume 2013, Article ID 635235, 3 pages
Numerical Methods for Solving Fredholm Integral Equations of Second Kind, S. Saha Ray and P. K. Sahu
Classification of Exact Solutions for Generalized Form of 𝐾(𝑚, 𝑛) Equation, Hasan Bulut
Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional

Fractional-Order Legendre Functions, Fukang Yin, Junqiang Song, Yongwen Wu, and Lilun Zhang
Persistence Property and Estimate on Momentum Support for the Integrable Degasperis-Procesi
Equation, Zhengguang Guo and Liangbing Jin
Existence and Decay Estimate of Global Solutions to Systems of Nonlinear Wave Equations with
Damping and Source Terms, Yaojun Ye
A Class of Spectral Element Methods and Its A Priori/A Posteriori Error Estimates for 2nd-Order
Elliptic Eigenvalue Problems, Jiayu Han and Yidu Yang
Nonlinear Hydroelastic Waves beneath a Floating Ice Sheet in a Fluid of Finite Depth,
Ping Wang and Zunshui Cheng
Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse

Functions and Normalized Bernstein Polynomials, S. H. Behiry
Semi-Idealized Study on Estimation of Partly and Fully Space Varying Open Boundary Conditions for
Tidal Models, Jicai Zhang and Haibo Chen
Decoupling the Stationary Navier-Stokes-Darcy System with the Beavers-Joseph-Saffman Interface

Condition, Yong Cao, Yuchuan Chu, Xiaoming He, and Mingzhen Wei
A New Integro-Differential Equation for Rossby Solitary Waves with Topography Effect in Deep
Rotational Fluids, Hongwei Yang, Qingfeng Zhao, Baoshu Yin, and Huanhe Dong
A One Step Optimal Homotopy Analysis Method for Propagation of Harmonic Waves in Nonlinear
Generalized Magnetothermoelasticity with Two Relaxation Times under Influence of Rotation,
S. M. Abo-Dahab, Mohamed S. Mohamed, and T. A. Nofal
The Effect of Boundary Slip on the Transient Pulsatile Flow of a Modified Second-Grade Fluid,
N. Khajohnsaksumeth, B. Wiwatanapataphee, and Y. H. Wu
Analytical Solutions of Boundary Values Problem of 2D and 3D Poisson and Biharmonic Equations by
Homotopy Decomposition Method, Abdon Atangana and Adem Kılıçman
Pattern Dynamics in a Spatial Predator-Prey System with Allee Effect, Gui-Quan Sun, Li Li, Zhen Jin,
Zi-Ke Zhang, and Tao Zhou
The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu Transform
Method, Hasan Bulut, Haci Mehmet Baskonus, and Fethi Bin Muhammad Belgacem
Improved (𝐺󸀠 /𝐺)-Expansion Method for the Space and Time Fractional Foam Drainage and KdV
Equations, Ali Akgül, Adem Kılıçman, and Mustafa Inc
The Solution to the BCS Gap Equation for Superconductivity and Its Temperature Dependence,
Shuji Watanabe
Numerical Solution for IVP in Volterra Type Linear Integrodifferential Equations System,
F. Ghomanjani, A. Kılıçman, and S. Effati
Analytical and Multishaped Solitary Wave Solutions for Extended Reduced Ostrovsky Equation,
Ben-gong Zhang
New Exact Solutions for a Generalized Double Sinh-Gordon Equation,

Gabriel Magalakwe and Chaudry Masood Khalique
Optimal Homotopy Asymptotic Method for Solving the Linear Fredholm Integral Equations of the First
Kind, Mohammad Almousa and Ahmad Ismail
Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with
DTM, Reza Abazari and Adem Kılıçman
A Pressure-Stabilized Lagrange-Galerkin Method in a Parallel Domain Decomposition System,

Qinghe Yao and Qingyong Zhu
A Note on the Triple Laplace Transform and Its Applications to Some Kind of Third-Order Differential
Equation, Abdon Atangana
Approximate Solution of Tuberculosis Disease Population Dynamics Model,

Abdon Atangana and Necdet Bildik
A Rectangular Mixed Finite Element Method with a Continuous Flux for an Elliptic Equation
Modelling Darcy Flow, Xindong Li and Hongxing Rui
Hindawi Publishing Corporation
http://dx.doi.org/10.1155/2014/635235
Editorial
Partial Differential Equations and Integral Equations
Arising in Physical Models
Santanu Saha Ray,1 Om P. Agrawal,2 R. K. Bera,3 Shantanu Das,4 and T. Raja Sekhar5
1
Department of Mathematics, National Institute of Technology, Rourkela 769008, India
2
Department of Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901, USA
3
Department of Science, National Institute of Technical Teachers’ Training and Research, Kolkata 700106, India
4
Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India
5
Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India
Correspondence should be addressed to Santanu Saha Ray; santanusaharay@yahoo.com
Received 15 December 2013; Accepted 15 December 2013; Published 9 January 2014
Copyright © 2014 Santanu Saha Ray et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Mathematical modelling of real-life problems usually results integral equations in physical systems are included in the
in functional equations, like ordinary or partial differential main focus of the issue.
equations, integral and integrodifferential equations, and Accordingly, various papers on partial differential equa-
stochastic equations. Many mathematical formulations of tions and integral equations have been included in this special
physical phenomena contain integrodifferential equations; issue after completing a heedful, rigorous, and peer-review
these equations arise in many fields like fluid dynamics, process. In particular, the nonlinear hydroelastic waves prop-
biological models, and chemical kinetics. Partial differential agating beneath an infinite ice sheet floating on an inviscid
equations (PDEs) have become a useful tool for describing fluid of finite depth are investigated analytically in one of the
the natural phenomena of science and engineering models. In papers. In this paper, the approximate series solutions for the
addition, most physical phenomena of fluid dynamics, quan- velocity potential and the wave surface elevation are derived,
tum mechanics, electricity, ecological systems, and many respectively, by an analytic approximation technique named
other models are controlled within their domain of validity homotopy analysis method (HAM) and are presented for the
by PDEs. Therefore, it becomes increasingly important to be second-order components.
familiar with all traditional and recently developed methods In another paper, a domain decomposition method is
for solving PDEs and the implementations of these methods. proposed for the coupled stationary Navier-Stokes and Darcy
Leaving aside quantum mechanics, which remains to date an equations with the Beavers-Joseph-Saffman interface condi-
inherently linear theory, most real-world physical systems, tion in order to improve the efficiency of the finite element
including gas dynamics, fluid mechanics, elasticity, relativity, method. The physical interface conditions are directly uti-
ecology, neurology, and thermodynamics, are modelled by lized to construct the boundary conditions on the interface
nonlinear partial differential equations. and then decouple the Navier-Stokes and Darcy equations.
The aim of this special issue is to bring together the lead- Newton iteration is used to deal with the nonlinear systems.
ing researchers of dynamics, quantum mechanics, ecology, Another paper proposes a pressure-stabilized Lagrange-
and neurology area including applied mathematicians and Galerkin method in a parallel domain decomposition system
allow them to share their original research work. Analytical in which the new stabilization strategy is proved to be
and numerical methods with advanced mathematical and effective for large Reynolds number and Rayleigh number
real physical modelling, recent developments of PDEs, and simulations. The symmetry of the stiffness matrix enables
2 Abstract and Applied Analysis
the interface problems of the linear system to be solved by the importance of interdisciplinary effort for advancing the
the preconditioned conjugate method, and an incomplete study on numerical methods for solving integral equations.
balanced domain preconditioner is applied to the flow- Also one of the papers has used a numerical method like func-
thermal coupled problems. tion approximation to determine the numerical solution of
One of the papers is of use of Sumudu transform on system of linear Volterra integrodifferential equations using
fractional derivatives for solving some interesting nonho- Bezier curves. Two-dimensional Volterra integral equations
mogeneous fractional ordinary differential equations. Then have also been solved using more recent semianalytic method
spectral and spectral element methods have been discussed like the reduced differential transform method and also
with Legendre-Gauss-Lobatto nodal basis for general 2nd- compared with the differential transform method. One of
order elliptic eigenvalue problems. A priori and a posteriori the papers has presented a numerical method to achieve the
error estimates for spectral and spectral element methods approximate solutions in a generalized expansion form of
have been proposed. In the another paper, a generalized two-dimensional fractional-order Legendre functions (2D-
double sinh-Gordon equation has many more applications FLFs). The operational matrices of integration and derivative
in various fields such as fluid dynamics, integrable quantum for 2D-FLFs have been derived.
field theory, and kink dynamics has been solved by Exp- Then a mixed finite element method has been introduced
function method to obtain new exact solutions for this for an elliptic equation modelling of Darcy flow in porous
generalized double sinh-Gordon equation. A semianalytical media. In present mixed finite element, the approximate
method called the optimal homotopy asymptotic method has velocity is continuous and the conservation law holds locally.
been also applied for solving the linear Fredholm integral In order to assess the rotational potential vorticity-conserved
equations of the first kind in another paper. In one of equation with topography effect and dissipation effect, the
the papers, two strategies for inverting the open boundary multiple-scale method has been studied to describe the
conditions with adjoint method are compared by carrying out Rossby solitary waves in deep rotational fluids. A one step
semi-idealized numerical experiments. In the first strategy, optimal homotopy analysis method has been applied numer-
the open boundary curves are assumed to be partly space ically to harmonic wave propagation in a nonlinear ther-
varying and are generated by linearly interpolating the values moelasticity under influence of rotation, thermal relaxation
at feature points and, in the second strategy, the open times, and magnetic field. The problem has been solved in
boundary conditions are assumed to be fully space varying one-dimensional elastic half-space model subjected initially
and the values at every open boundary points are taken to a prescribed harmonic displacement and the temperature
as control variables. Another paper contains the use of a of the medium. In one of the papers, the analytical and
relatively new analytical method like homotopy decompo- multishaped solitary wave solutions have been presented
sition method to solve the 2D and 3D Poisson equations for extended reduced Ostrovsky equation. The exact soli-
and biharmonic equations. The method does not require tary (traveling) wave solutions are also expressed by three
the linearization or assumptions of weak nonlinearity, the types of functions which are hyperbolic function solution,
solutions are generated in the form of general solution, and trigonometric function solution, and rational solution. In
it is more realistic compared to the method of simplifying the order to classify the exact solutions, including solitons and
physical problems. elliptic solutions, of the generalized 𝐾(𝑚, 𝑛) equation by
One of the papers has shown that a strong solution of the the complete discrimination system a polynomial method
Degasperis-Procesi equation possesses persistence property has been obtained. To examine the possible approximate
in the sense that the solution with algebraically decaying solutions of both integer and noninteger systems of nonlinear
initial data and its spatial derivative must retain this property. differential equations which describe tuberculosis disease
In another paper, the fractional complex transformation population dynamics, the relatively new analytical technique
has been used to transform nonlinear partial differential like homotopy decomposition method has been proposed. In
equations to nonlinear ordinary differential equations. The one of the papers, a relatively new operator called the triple
improved (𝐺󸀠 /𝐺)-expansion method has suggested solving Laplace transform has been introduced and to make use of
the space and time fractional foam drainage and KdV equa- the operator some kind of third-order differential equation
tions. Integral equation has been one of the essential tools called Mboctara equations has been solved.
for various areas of applied mathematics. For solving non- Another paper investigates the effect of boundary slip
linear Fredholm integrodifferential equations, the method on the transient pulsatile fluid flow through a vessel with
based on hybrid functions approximate has been proposed body acceleration. To describe the non-Newtonian behavior,
in one of the papers. The properties of hybrid of block the modified second-grade fluid model has been analyzed
pulse functions and orthonormal Bernstein polynomials have in which the viscosity and the normal stresses have been
been presented and utilized to reduce the problem to the represented in terms of the shear rate. One of the papers
solution of nonlinear algebraic equations. Another paper proves the existence of global solutions for nonlinear wave
contains many numerical methods, namely, B-Spline wavelet equations with damping and source terms by constructing
method, Wavelet Galerkin method, and quadrature method, a stable set and also obtaining the asymptotic stability of
for solving Fredholm integral equations of second kind. A global solutions through the use of a difference inequality. In
peer-review of different numerical methods for solving both order to assess the spatial dynamical behavior of a predator-
linear and nonlinear Fredholm integral equations of second prey system with Allee effect, the bifurcation analyses have
kind has been presented. This paper has more emphasized on been used in which the exact Turing domain has been found
Abstract and Applied Analysis 3
in the parameters space. According to the operator theory,

the temperature dependence of the solution to the BCS gap
equation has been connected with superconductivity. When
the potential is a positive constant, the BCS gap equation
reduces to the simple gap equation. The solution to the BCS
gap equation has been indeed continuous with respect to both
the temperature and the energy under a certain condition
when the potential is not a constant. This study represents
that there is a unique nonnegative solution to the simple gap
equation, which is continuous and strictly decreasing and is
of class 𝐶2 with respect to the temperature.
At present, the use of partial differential equation and
integral equation in real physical systems is commonly
encountered in the fields of science and engineering. Analysis
and numerical approximate of such physical models are
required for efficient computational tools. The present issue
has addressed recent trends and developments regarding
the analytical and numerical methods that may be used in
the dynamical system. Eventually, it may be expected that
the present special issue would certainly helpful to explore
the researchers with their new arising problems and elevate
the efficiency and accuracy of the solution methods in use
nowadays.
Santanu Saha Ray
Om P. Agrawal
R. K. Bera
Shantanu Das
T. Raja Sekhar
http://dx.doi.org/10.1155/2013/426916
Research Article
Numerical Methods for Solving Fredholm Integral
Equations of Second Kind
S. Saha Ray and P. K. Sahu

Department of Mathematics, National Institute of Technology, Rourkela 769008, India
Correspondence should be addressed to S. Saha Ray; santanusaharay@yahoo.com
Received 3 September 2013; Accepted 3 October 2013
Academic Editor: Rasajit Bera
Copyright © 2013 S. S. Ray and P. K. Sahu. This is an open access article distributed under the Creative Commons Attribution
cited.
Integral equation has been one of the essential tools for various areas of applied mathematics. In this paper, we review different
numerical methods for solving both linear and nonlinear Fredholm integral equations of second kind. The goal is to categorize the
selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize
the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations.
1. Introduction functions only, then several approximate methods of solution

of integral equations can be developed.
Integral equations occur naturally in many fields of science A computational approach to solving integral equation
and engineering [1]. A computational approach to solve
is an essential work in scientific research. Some methods
integral equation is an essential work in scientific research.
for solving second kind Fredholm integral equation are
Integral equation is encountered in a variety of appli-
available in the open literature. The B-spline wavelet method,
cations in many fields including continuum mechanics,
potential theory, geophysics, electricity and magnetism, the method of moments based on B-spline wavelets by
kinetic theory of gases, hereditary phenomena in physics Maleknejad and Sahlan [2], and variational iteration method
and biology, renewal theory, quantum mechanics, radiation, (VIM) by He [3–5] have been applied to solve second kind
optimization, optimal control systems, communication the- Fredholm linear integral equations. The learned researchers
ory, mathematical economics, population genetics, queuing Maleknejad et al. proposed some numerical methods for
theory, medicine, mathematical problems of radiative equi- solving linear Fredholm integral equations system of second
librium, the particle transport problems of astrophysics and kind using Rationalized Haar functions method, Block-Pulse
reactor theory, acoustics, fluid mechanics, steady state heat functions, and Taylor series expansion method [6–8]. Haar
conduction, fracture mechanics, and radiative heat transfer wavelet method with operational matrices of integration [9]
problems. Fredholm integral equation is one of the most has been applied to solve system of linear Fredholm integral
important integral equations. equations of second kind. Quadrature method [10], B-spline
Integral equations can be viewed as equations which are wavelet method [11], wavelet Galerkin method [12], and
results of transformation of points in a given vector spaces also VIM [13] can be applied to solve nonlinear Fredholm
of integrable functions by the use of certain specific integral integral equation of second kind. Some iterative methods
operators to points in the same space. If, in particular, one like Homotopy perturbation method (HPM) [14–16] and
is concerned with function spaces spanned by polynomials Adomian decomposition method (ADM) [16–18] have been
for which the kernel of the corresponding transforming applied to solve nonlinear Fredholm integral equation of
integral operator is separable being comprised of polynomial second kind.
2. Fredholm Integral Equation 2.5. System of Nonlinear Fredholm Integral Equations. System
of nonlinear Fredholm integral equations of second kind is
The general form of linear Fredholm integral equation is defined as follows:
defined as follows: 𝑛 𝑛 𝑏
∑ 𝑔𝑖,𝑗 𝑦𝑗 (𝑥) = 𝑓𝑖 (𝑥) + ∑ ∫ 𝐾𝑖,𝑗 (𝑥, 𝑡) 𝐹𝑖,𝑗 (𝑡, 𝑦𝑗 (𝑡)) 𝑑𝑡,
𝑏
𝑗=1 𝑗=1 𝑎
𝑔 (𝑥) 𝑦 (𝑥) = 𝑓 (𝑥) + 𝜆 ∫ 𝐾 (𝑥, 𝑡) 𝑦 (𝑡) 𝑑𝑡, (1)
𝑎
𝑖 = 1, 2, . . . , 𝑛,
where 𝑎 and 𝑏 are both constants. 𝑓(𝑥), 𝑔(𝑥), and 𝐾(𝑥, 𝑡) (6)
are known functions while 𝑦(𝑥) is unknown function. 𝜆 where 𝑓𝑖 (𝑥) and 𝐾𝑖,𝑗 (𝑥, 𝑡) are known functions and 𝑦𝑗 (𝑥) are
(nonzero parameter) is called eigenvalue of the integral the unknown functions for 𝑖, 𝑗 = 1, 2, . . . , 𝑛.
equation. The function 𝐾(𝑥, 𝑡) is known as kernel of the
integral equation.
3. Numerical Methods for Linear Fredholm
Integral Equation of Second Kind
2.1. Fredholm Integral Equation of First Kind. The linear
integral equation is of form (by setting 𝑔(𝑥) = 0 in (1)) Consider the following Fredholm integral equation of second
kind defined in (3)
𝑏 𝑏
𝑓 (𝑥) + 𝜆 ∫ 𝐾 (𝑥, 𝑡) 𝑦 (𝑡) 𝑑𝑡 = 0. (2) 𝑦 (𝑥) = 𝑓 (𝑥) + ∫ 𝐾 (𝑥, 𝑡) 𝑦 (𝑡) 𝑑𝑡, 𝑎 ≤ 𝑥 ≤ 𝑏, (7)
𝑎 𝑎
Equation (2) is known as Fredholm integral equation of first where 𝐾(𝑥, 𝑡) and 𝑔(𝑥) are known functions and 𝑦(𝑥) is
kind. unknown function to be determined.
3.1. B-Spline Wavelet Method

2.2. Fredholm Integral Equation of Second Kind. The linear
integral equation is of form (by setting 𝑔(𝑥) = 1 in (1)) 3.1.1. B-Spline Scaling and Wavelet Functions on the Interval
[0, 1]. Semiorthogonal wavelets using B-spline are specially
𝑏 constructed for the bounded interval and this wavelet can
𝑦 (𝑥) = 𝑓 (𝑥) + ∫ 𝐾 (𝑥, 𝑡) 𝑦 (𝑡) 𝑑𝑡. (3) be represented in a closed form. This provides a compact
𝑎
support. Semiorthogonal wavelets form the basis in the space
Equation (3) is known as Fredholm integral equation of 𝐿2 (𝑅).
second kind. Using this basis, an arbitrary function in 𝐿2 (𝑅) can be
expressed as the wavelet series. For the finite interval [0, 1],
the wavelet series cannot be completely presented by using
2.3. System of Linear Fredholm Integral Equations. The gen- this basis. This is because supports of some basis are truncated
eral form of system of linear Fredholm integral equations of at the left or right end points of the interval. Hence, a special
second kind is defined as follows: basis has to be introduced into the wavelet expansion on the
finite interval. These functions are referred to as the boundary
𝑛 𝑛 𝑏 scaling functions and boundary wavelet functions.
∑ 𝑔𝑖,𝑗 𝑦𝑗 (𝑥) = 𝑓𝑖 (𝑥) + ∑ ∫ 𝐾𝑖,𝑗 (𝑥, 𝑡) 𝑦𝑗 (𝑡) 𝑑𝑡, Let 𝑚 and 𝑛 be two positive integers and let
𝑗=1 𝑗=1 𝑎 (4)
𝑎 = 𝑥−𝑚+1 = ⋅ ⋅ ⋅ = 𝑥0 < 𝑥1
𝑖 = 1, 2, . . . , 𝑛,
< ⋅ ⋅ ⋅ < 𝑥𝑛 = 𝑥𝑛+1 (8)
where 𝑓𝑖 (𝑥) and 𝐾𝑖,𝑗 (𝑥, 𝑡) are known functions and 𝑦𝑗 (𝑥) are = ⋅ ⋅ ⋅ = 𝑥𝑛+𝑚−1 = 𝑏
the unknown functions for 𝑖, 𝑗 = 1, 2, . . . , 𝑛.
be an equally spaced knots sequence. The functions
2.4. Nonlinear Fredholm-Hammerstein Integral Equation of 𝑥 − 𝑥𝑗
Second Kind. Nonlinear Fredholm-Hammerstein integral 𝐵𝑚,𝑗,𝑋 (𝑥) = 𝐵𝑚−1,𝑗,𝑋 (𝑥)
𝑥𝑗+𝑚−1 − 𝑥𝑗
equation of second kind is defined as follows:
𝑥𝑗+𝑚 − 𝑥
𝑏
+ 𝐵𝑚−1,𝑗+1,𝑋 (𝑥) ,
𝑥𝑗+𝑚 − 𝑥𝑗+1
𝑦 (𝑥) = 𝑓 (𝑥) + ∫ 𝐾 (𝑥, 𝑡) 𝐹 (𝑦 (𝑡)) 𝑑𝑡, (5) (9)
𝑎
𝑗 = −𝑚 + 1, . . . , 𝑛 − 1,
where 𝐾(𝑥, 𝑡) is the kernel of the integral equation, 𝑓(𝑥) 1, 𝑥 ∈ [𝑥𝑗 , 𝑥𝑗+1 ) ,
and 𝐾(𝑥, 𝑡) are known functions, and 𝑦(𝑥) is the unknown 𝐵1,𝑗,𝑋 (𝑥) = {
function that is to be determined. 0, otherwise,
are called cardinal B-spline functions of order 𝑚 ≥ 2 for 3.1.2. Function Approximation. A function 𝑓(𝑥) defined over
the knot sequence 𝑋 = {𝑥𝑖 }𝑛+𝑚−1 𝑖=−𝑚+1 and Supp𝐵𝑚,𝑗,𝑋 (𝑥) =
[0, 1] may be approximated by B-spline wavelets as [21, 22]
[𝑥𝑗 , 𝑥𝑗+𝑚 ] ∩ [𝑎, 𝑏].
2𝑗0 −1
By considering the interval [𝑎, 𝑏] = [0, 1], at any level 𝑗 ∈
Ζ+ , the discretization step is 2−𝑗 , and this generates 𝑛 = 2𝑗 𝑓 (𝑥) = ∑ 𝑐𝑗0 ,𝑘 𝜑𝑗0 ,𝑘 (𝑥)
𝑘=1−𝑚
number of segments in [0, 1] with knot sequence (15)
(𝑗) (𝑗) ∞ 2𝑗 −𝑚
{ 𝑥−𝑚+1 = ⋅⋅⋅ = 𝑥0 = 0, + ∑ ∑ 𝑑𝑗,𝑘 𝜓𝑗,𝑘 (𝑥) .
{
{ (𝑗)
(𝑗) 𝑘
𝑋 = { 𝑥𝑘 = , 𝑘 = 1, . . . , 𝑛 − 1, (10) 𝑗=𝑗0 𝑘=1−𝑚
{
{ (𝑗) 2𝑗 (𝑗)
{𝑥𝑛 = ⋅ ⋅ ⋅ = 𝑥𝑛+𝑚−1 = 1. If the infinite series in (15) is truncated at 𝑀, then (15) can be
written as [2]
Let 𝑗0 be the level for which 2𝑗0 ≥ 2 𝑚−1; for each level, 𝑗 ≥ 𝑗0 ,
the scaling functions of order 𝑚 can be defined as follows in 2𝑗0 −1
[2]: 𝑓 (𝑥) ≅ ∑ 𝑐𝑗0 ,𝑘 𝜑𝑗0 ,𝑘 (𝑥)
𝑘=1−𝑚
𝜑𝑚,𝑗,𝑖 (𝑥) (16)
𝑀 2𝑗 −𝑚
{ 𝐵 (2𝑗−𝑗0 𝑥) 𝑖 = −𝑚 + 1, . . . , −1, + ∑ ∑ 𝑑𝑗,𝑘 𝜓𝑗,𝑘 (𝑥) ,

{ 𝑚,𝑗0 ,𝑖 𝑗=𝑗0 𝑘=1−𝑚
= {𝐵𝑚,𝑗0 ,2𝑗 −𝑚−𝑖 (1 − 2𝑗−𝑗0 𝑥) 𝑖 = 2𝑗 − 𝑚 + 1, . . . , 2𝑗 − 1,
{ 𝑗−𝑗0 −𝑗0
{𝐵𝑚,𝑗0 ,0 (2 𝑥 − 2 𝑖) 𝑖 = 0, . . . , 2𝑗 − 𝑚. where 𝜑2,𝑘 and 𝜓𝑗,𝑘 are scaling and wavelets functions,
(11) respectively, and 𝐶 and Ψ are (2𝑀+1 + 𝑚 − 1) × 1 vectors given
And the two scale relations for the 𝑚-order semiorthogonal by
compactly supported B-wavelet functions are defined as
follows: 𝐶 = [𝑐𝑗0 ,1−𝑚 , . . . , 𝑐𝑗0 ,2𝑗0 −1 , 𝑑𝑗0 ,1−𝑚 , . . . ,
2𝑖+2𝑚−2
(17)
𝑇
𝜓𝑚,𝑗,𝑖−𝑚 = ∑ 𝑞𝑖,𝑘 𝐵𝑚,𝑗,𝑘−𝑚 , 𝑖 = 1, . . . , 𝑚 − 1, 𝑑𝑗0 ,2𝑗0 −𝑚 , . . . , 𝑑𝑀,1−𝑚 , . . . , 𝑑𝑀,2𝑀 −𝑚 ] ,
𝑘=𝑖
Ψ = [𝜑𝑗0 ,1−𝑚 , . . . , 𝜑𝑗0 ,2𝑗0 −1 , 𝜓𝑗0 ,1−𝑚 , . . . ,
2𝑖+2𝑚−2
(18)
𝜓𝑚,𝑗,𝑖−𝑚 = ∑ 𝑞𝑖,𝑘 𝐵𝑚,𝑗,𝑘−𝑚 , 𝑖 = 𝑚, . . . , 𝑛 − 𝑚 + 1, 𝑇
𝑘=2𝑖−𝑚
𝜓𝑗0 ,2𝑗0 −𝑚 , . . . , 𝜓𝑀,1−𝑚 , . . . , 𝜓𝑀,2𝑀 −𝑚 ] ,
𝑛+𝑖+𝑚−1 with
𝜓𝑚,𝑗,𝑖−𝑚 = ∑ 𝑞𝑖,𝑘 𝐵𝑚,𝑗,𝑘−𝑚 , 𝑖 = 𝑛 − 𝑚 + 2, . . . , 𝑛,
1
𝑘=2𝑖−𝑚
𝑐𝑗0 ,𝑘 = ∫ 𝑓 (𝑥) 𝜑̃𝑗0 ,𝑘 (𝑥) 𝑑𝑥, 𝑘 = 1 − 𝑚, . . . , 2𝑗0 − 1,
(12) 0
where 𝑞𝑖,𝑘 = 𝑞𝑘−2𝑖 . 1

(19)
Hence, there are 2(𝑚 − 1) boundary wavelets and (𝑛 − 𝑑𝑗,𝑘 = ∫ 𝑓 (𝑥) 𝜓̃𝑗,𝑘 (𝑥) 𝑑𝑥,
0
2𝑚 + 2) inner wavelets in the bounded interval [𝑎, 𝑏]. Finally,
by considering the level 𝑗 with 𝑗 ≥ 𝑗0 , the B-wavelet functions 𝑗 = 𝑗0 , . . . , 𝑀, 𝑘 = 1 − 𝑚, . . . , 2𝑀 − 𝑚,
in [0, 1] can be expressed as follows:
where 𝜑̃𝑗0 ,𝑘 (𝑥) and 𝜓̃𝑗,𝑘 (𝑥) are dual functions of 𝜑𝑗0 ,𝑘 and 𝜓𝑗,𝑘 ,
𝜓𝑚,𝑗,𝑖 (𝑥)
respectively. These can be obtained by linear combinations of
𝜓 (2𝑗−𝑗0 𝑥) 𝑖 = −𝑚 + 1, . . . , −1, 𝜑𝑗0 ,𝑘 , 𝑘 = 1 − 𝑚, . . . , 2𝑗0 − 1, and 𝜓𝑗,𝑘 , 𝑗 = 𝑗0 , . . . , 𝑀, 𝑘 =
{
{ 𝑚,𝑗0 ,𝑖
= {𝜓𝑚,2𝑗 −2 𝑚+1−𝑖,𝑖 (1 − 2𝑗−𝑗0 𝑥) 𝑖 = 2𝑗 −2 𝑚+2, . . . , 2𝑗 −𝑚, 1 − 𝑚, . . . , 2𝑀 − 𝑚, as follows. Let
{ 𝑗−𝑗0 −𝑗0
{𝜓𝑚,𝑗0 ,0 (2 𝑥 − 2 𝑖) 𝑖 = 0, . . . , 2𝑗 − 2 𝑚 + 1. Φ = [𝜑𝑗0 ,1−𝑚 , . . . , 𝜑𝑗0 ,2𝑗0 −1 ] ,
𝑇
(20)
(13) 𝑇
Ψ = [𝜓𝑗0 ,1−𝑚 , . . . , 𝜓𝑗0 ,2𝑗0 −𝑚 , . . . , 𝜓𝑀,1−𝑚 , . . . , 𝜓𝑀,2𝑀 −𝑚 ] .
The scaling functions 𝜑𝑚,𝑗,𝑖 (𝑥) occupy 𝑚 segments and the (21)
wavelet functions 𝜓𝑚,𝑗,𝑖 (𝑥) occupy 2𝑚 − 1 segments.
When the semiorthogonal wavelets are constructed from Using (11), (20), (12)-(13), and (21), we get
B-spline of order 𝑚, the lowest octave level 𝑗 = 𝑗0 is
determined in [19, 20] by 1
∫ ΦΦ𝑇𝑑𝑥 = 𝑃1 ,
2𝑗0 ≥ 2 𝑚 − 1, (14) 0
(22)
1
so as to have a minimum of one complete wavelet on the 𝑇
∫ Ψ Ψ 𝑑𝑥 = 𝑃2 .
interval [0, 1]. 0
Suppose that Φ ̃ are the dual functions of Φ and Ψ,

̃ and Ψ and 𝑃 is a (2𝑀+1 + 𝑚 − 1) × (2𝑀+1 + 𝑚 − 1) square matrix given
respectively; then by
1 1
̃ 𝑇 𝑑𝑥 = 𝐼1 , 𝑃 0
∫ ΦΦ 𝑃 = ∫ Ψ (𝑥) Ψ𝑇 (𝑥) 𝑑𝑥 = ( 1 ). (33)
0 0 0 𝑃2
(23)
1
̃ Ψ𝑇 𝑑𝑥 = 𝐼 ,
∫ Ψ Consequently, from (31), we get 𝐶𝑇 = 𝐶1𝑇 (𝑃 − Θ)−1 . Hence,
2
0
we can calculate the solution for 𝑦(𝑥) = 𝐶𝑇 Ψ(𝑥).
̃ = 𝑃1 −1 Φ,
Φ
(24) 3.2. Method of Moments
̃ = 𝑃 −1 Ψ.
Ψ 2
3.2.1. Multiresolution Analysis (MRA) and Wavelets [2]. A set
3.1.3. Application of B-Spline Wavelet Method. In this section, of subspaces {𝑉𝑗 }𝑗∈𝑍 is said to be MRA of 𝐿2 (𝑅) if it possesses
linear Fredholm integral equation of the second kind of form the following properties:
(7) has been solved by using B-spline wavelets. For this, we
use (16) to approximate 𝑦(𝑥) as 𝑉𝑗 ⊂ 𝑉𝑗+1 , ∀𝑗 ∈ Ζ, (34)
𝑦 (𝑥) = 𝐶𝑇 Ψ (𝑥) , (25) ⋃ 𝑉𝑗 is dense in 𝐿2 (𝑅) , (35)

𝑗∈Ζ
𝑀+1
where Ψ(𝑥) is defined in (18) and 𝐶 is (2 + 𝑚 − 1) × 1
unknown vector defined similarly as in (17). We also expand ⋂ 𝑉𝑗 = 𝜙, (36)
̃ defined in (24)
𝑓(𝑥) and 𝐾(𝑥, 𝑡) by B-spline dual wavelets Ψ 𝑗∈Ζ
as
𝑓 (𝑥) ∈ 𝑉𝑗 ⇐⇒ 𝑓 (2𝑥) ∈ 𝑉𝑗+1 , ∀𝑗 ∈ Ζ, (37)
̃ (𝑥) ,
𝑓 (𝑥) = 𝐶1 𝑇 Ψ
(26) where 𝑍 denotes the set of integers. Properties (34)–(36) state
𝐾 (𝑥, 𝑡) = Ψ ̃ (𝑥) ,
̃𝑇 (𝑡) ΘΨ that {𝑉𝑗 }𝑗∈𝑍 is a nested sequence of subspaces that effectively
where covers 𝐿2 (𝑅). That is, every square integrable function can be
approximated as closely as desired by a function that belongs
1 1 to at least one of the subspaces 𝑉𝑗 . A function 𝜑 ∈ 𝐿2 (𝑅) is
Θ𝑖,𝑗 = ∫ [∫ 𝐾 (𝑥, 𝑡) Ψ𝑖 (𝑡) 𝑑𝑡] Ψ𝑗 (𝑥) 𝑑𝑥. (27) called a scaling function if it generates the nested sequence of
0 0
subspaces 𝑉𝑗 and satisfies the dilation equation; namely,
From (26) and (25), we get
1 1 𝜑 (𝑥) = ∑ 𝑝𝑘 𝜑 (𝑎𝑥 − 𝑘) , (38)
∫ 𝐾 (𝑥, 𝑡) 𝑦 (𝑡) 𝑑𝑡 = ∫ 𝐶 Ψ (𝑡) Ψ ̃ (𝑥) 𝑑𝑡
̃𝑇 (𝑡) ΘΨ𝑇 𝑘
0 0 (28)
̃ (𝑥)
𝑇 with 𝑝𝑘 ∈ 𝑙2 and 𝑎 being any rational number.
= 𝐶 ΘΨ For each scale 𝑗, since 𝑉𝑗 ⊂ 𝑉𝑗+1 , there exists a unique
since orthogonal complementary subspace 𝑊𝑗 of 𝑉𝑗 in 𝑉𝑗+1 . This
subspace 𝑊𝑗 is called wavelet subspace and is generated by
1
̃𝑇 (𝑡) 𝑑𝑡 = 𝐼. 𝜓𝑗,𝑘 = 𝜓(2𝑗 𝑥 − 𝑘), where 𝜓 ∈ 𝐿2 is called the wavelet. From
∫ Ψ (𝑡) Ψ (29)
0 the above discussion, these results follow easily:
By applying (25)–(28) in (7) we have 𝑉𝑗1 ∩ 𝑉𝑗2 = 𝑉𝑗2 , 𝑗1 > 𝑗2 ,

̃ (𝑥) = 𝐶𝑇 Ψ
𝐶𝑇 Ψ (𝑥) − 𝐶𝑇 ΘΨ ̃ (𝑥) . (30) 𝑊𝑗1 ∩ 𝑊𝑗2 = 0, 𝑗1 ≠𝑗2 , (39)
1
By multiplying both sides of (30) with Ψ𝑇 (𝑥) from the right 𝑉𝑗1 ∩ 𝑊𝑗2 = 0, 𝑗1 ≤ 𝑗2 .
and integrating both sides with respect to 𝑥 from 0 to 1, we
get Some of the important properties relevant to the present
analysis are given below [2, 19].
𝐶𝑇 𝑃 − 𝐶𝑇 Θ = 𝐶1𝑇 , (31)
(1) Vanishing Moment. A wavelet is said to have a vanishing
since moment of order 𝑚 if
1 ∞
̃ (𝑥) Ψ𝑇 (𝑥) 𝑑𝑥 = 𝐼,
∫ Ψ (32) ∫ 𝑥𝑝 𝜓 (𝑥) 𝑑𝑥 = 0; 𝑝 = 0, . . . , 𝑚 − 1. (40)
0 −∞
All wavelets must satisfy the previously mentioned condition ⟨𝜓, 𝜓⟩ − ⟨𝐾𝜓, 𝜓⟩
for 𝑝 = 0. 1
= (∫ 𝜓𝑠,𝑙 (𝑥) 𝜓𝑘,𝑗 (𝑥) 𝑑𝑥
(2) Semiorthogonality. The wavelets 𝜓𝑗,𝑘 form a semiorthogo- 0
nal basis if 1 1
⟨𝜓𝑗,𝑘 , 𝜓𝑠,𝑖 ⟩ = 0; 𝑗 ≠𝑠; ∀𝑗, 𝑘, 𝑠, 𝑖 ∈ Ζ. (41) − ∫ 𝜓𝑠,𝑙 (𝑥) ∫ 𝐾(𝑥, 𝑡)𝜓𝑘,𝑗 (𝑡)𝑑𝑡 𝑑𝑥) ,
0 0 𝑙,𝑠,𝑘,𝑗
1
3.2.2. Method of Moments for the Solution of Fredholm Integral
𝐹1 = ∫ 𝑓 (𝑥) 𝜑𝑗0 ,𝑟 (𝑥) 𝑑𝑥,
Equation. In this section, we solve the integral equation of 0
form (7) in interval [0, 1] by using linear B-spline wavelets 1
[2]. The unknown function in (7) can be expanded in terms 𝐹2 = ∫ 𝑓 (𝑥) 𝜓𝑠,𝑙 (𝑥) 𝑑𝑥,
of the scaling and wavelet functions as follows: 0
2𝑗0 −1
(44)
𝑦 (𝑥) ≈ ∑ 𝑐𝑘 𝜑𝑗0 ,𝑘 (𝑥)
𝑘=−1 and the subscripts 𝑖, 𝑟, 𝑘, 𝑗, 𝑙, and 𝑠 assume values as given
below:
𝑀 2𝑗 −2 (42)
+ ∑ ∑ 𝑑𝑗,𝑘 𝜓𝑗,𝑘 (𝑥) 𝑖, 𝑟 = −1, . . . , 2𝑗0 − 1,
𝑗=𝑗0 𝑘=−1
𝑙, 𝑘 = 𝑗0 , . . . , 𝑀, (45)
= 𝐶𝑇 Ψ (𝑥) .
By substituting this expression into (7) and employing the 𝑠, 𝑗 = −1, . . . , 2𝑀 − 2.
Galerkin method, the following set of linear system of order
(2𝑀 + 1) is generated. The scaling and wavelet functions are In fact, the entries with significant magnitude are in the
used as testing and weighting functions: ⟨𝐾𝜑, 𝜑⟩ − ⟨𝜑, 𝜑⟩ and ⟨𝐾𝜓, 𝜓⟩ − ⟨𝜓, 𝜓⟩ submatrices which are
of order (2𝑗0 + 1) and (2𝑀+1 + 1), respectively.
⟨𝜑, 𝜑⟩ − ⟨𝐾𝜑, 𝜑⟩ ⟨𝜓, 𝜑⟩ − ⟨𝐾𝜓, 𝜑⟩ 𝐶 𝐹
( ) ( ) = ( 1) ,
⟨𝜑, 𝜓⟩ − ⟨𝐾𝜑, 𝜓⟩ ⟨𝜓, 𝜓⟩ − ⟨𝐾𝜓, 𝜓⟩ 𝐷 𝐹2
(43) 3.3. Variational Iteration Method [3–5]. In this section, Fred-
holm integral equation of second kind given in (7) has been
where considered for solving (7) by variational iteration method.
𝑇 First, we have to take the partial derivative of (7) with respect
𝐶 = [𝑐−1 , 𝑐0 , . . . , 𝑐3 ] ,
to 𝑥 yielding
𝐷 = [𝑑2,−1 , . . . , 𝑑2,2 , 𝑑3,−1 , . . . , 𝑑3,6 , . . . ,
1
𝑑𝑀,−1 , . . . , 𝑑𝑀,2𝑀 −2 ] ,
𝑇 𝑌󸀠 (𝑥) = 𝑓󸀠 (𝑥) + ∫ 𝐾󸀠 (𝑥, 𝑡) 𝑦 (𝑡) 𝑑𝑡. (46)
0
⟨𝜑, 𝜑⟩ − ⟨𝐾𝜑, 𝜑⟩ We apply variation iteration method for (46). According to

1 this method, correction functional can be defined as
= (∫ 𝜑𝑗0 ,𝑟 (𝑥) 𝜑𝑗0 ,𝑖 (𝑥) 𝑑𝑥
0
𝑦𝑛+1 (𝑥)
1 1
− ∫ 𝜑𝑗0 ,𝑟 (𝑥) ∫ 𝐾(𝑥, 𝑡)𝜑𝑗0 ,𝑖 (𝑡)𝑑𝑡 𝑑𝑥) , = 𝑦𝑛 (𝑥)
0 0 𝑖,𝑟
𝑥 𝑏
⟨𝜓, 𝜑⟩ − ⟨𝐾𝜓, 𝜑⟩ + ∫ 𝜆 (𝜉) (𝑦𝑛󸀠 (𝜉) − 𝑓󸀠 (𝜉) − ∫ 𝐾󸀠 (𝜉, 𝑡) 𝑦̃𝑛 (𝑡) 𝑑𝑡) 𝑑𝜉,
0 𝑎
1 (47)
= (∫ 𝜑𝑗0 ,𝑟 (𝑥) 𝜓𝑘,𝑗 (𝑥) 𝑑𝑥
0
where 𝜆(𝜉) is a general Lagrange multiplier which can be
1 1
identified optimally by the variational theory, the subscript
− ∫ 𝜑𝑗0 ,𝑟 (𝑥) ∫ 𝐾(𝑥, 𝑡)𝜓𝑘,𝑗 (𝑡)𝑑𝑡 𝑑𝑥) ,
0 0 𝑟,𝑘,𝑗 𝑛 denotes the 𝑛th order approximation, and 𝑦̃𝑛 is considered
as a restricted variation; that is, 𝛿𝑦̃𝑛 = 0. The successive
⟨𝜑, 𝜓⟩ − ⟨𝐾𝜑, 𝜓⟩ approximations 𝑦𝑛 (𝑥), 𝑛 ≥ 1 for the solution 𝑦(𝑥) can be
1
readily obtained after determining the Lagrange multiplier
= (∫ 𝜓𝑠,𝑙 (𝑥) 𝜑𝑗0 ,𝑖 (𝑥) 𝑑𝑥 and selecting an appropriate initial function 𝑦0 (𝑥). Conse-
0 quently the approximate solution may be obtained by using
1 1
− ∫ 𝜓𝑠,𝑙 (𝑥) ∫ 𝐾(𝑥, 𝑡)𝜑𝑗0 ,𝑖 (𝑡)𝑑𝑡 𝑑𝑥) , 𝑦 (𝑥) = lim 𝑦𝑛 (𝑥) . (48)
0 0 𝑛→∞
𝑖,𝑙,𝑠
To make the above correction functional stationary, we have 4.1. Application of Haar Wavelet Method [9]. In this section,
an efficient algorithm for solving Fredholm integral equations
𝛿𝑦𝑛+1 (𝑥) = 𝛿𝑦𝑛 (𝑥) with Haar wavelets is analyzed. The present algorithm takes
𝑥 the following essential strategy. The Haar wavelet is first
+ 𝛿 ∫ 𝜆 (𝜉) (𝑦𝑛󸀠 (𝜉) − 𝑓󸀠 (𝜉) used to decompose integral equations into algebraic systems
0 of linear equations, which are then solved by collocation
𝑏
methods.
− ∫ 𝐾󸀠 (𝜉, 𝑡) 𝑦̃𝑛 (𝑡) 𝑑𝑡) 𝑑𝜉
𝑎 4.1.1. Haar Wavelets. The compact set of scale functions is
𝑥 chosen as
= 𝛿𝑦𝑛 (𝑥) + ∫ 𝜆 (𝜉) 𝛿 (𝑦𝑛󸀠 (𝜉)) 𝑑𝜉
0 1, 0 ≤ 𝑥 < 1,
ℎ0 = { (56)
󵄨
𝑥 0, others.
= 𝛿𝑦𝑛 (𝑥) + 𝜆𝛿𝑦𝑛 󵄨󵄨󵄨𝜉=𝑥 − ∫ 𝜆󸀠 (𝜉) 𝛿𝑦𝑛 (𝜉) 𝑑𝜉.
0 The mother wavelet function is defined as
(49)
1
{
{ 1, 0≤𝑥< ,
Under stationary condition, {
{ 2
ℎ1 (𝑥) = {−1, 1 (57)
{ ≤ 𝑥 < 1,
𝛿𝑦𝑛+1 = 0 (50) {
{ 2
{0, others.
implies the following Euler Lagrange equation:
The family of wavelet functions generated by translation and
𝜆󸀠 (𝜉) = 0, (51) dilation of ℎ1 (𝑥) are given by
with the following natural boundary condition: ℎ𝑛 (𝑥) = ℎ1 (2𝑗 𝑥 − 𝑘) , (58)
󵄨
1 + 𝜆(𝜉)󵄨󵄨󵄨𝜉=𝑥 = 0. (52) where 𝑛 = 2𝑗 + 𝑘, 𝑗 ≥ 0, 0 ≤ 𝑘 < 2𝑗 .
Mutual orthogonalities of all Haar wavelets can be
Solving (51), along with boundary condition (52), we get the expressed as
general Lagrange multiplier
1
2−𝑗 , 𝑚 = 𝑛 = 2𝑗 + 𝑘,
𝜆 = −1. (53) ∫ ℎ𝑚 (𝑥) ℎ𝑛 (𝑥) 𝑑𝑥 = 2−𝑗 𝛿𝑚𝑛 = {
0 0, 𝑚 ≠𝑛.
Substituting the identified Lagrange multiplier into (47) (59)
results in the following iterative scheme:
4.1.2. Function Approximation. An arbitrary function 𝑦(𝑥) ∈
𝑦𝑛+1 (𝑥) = 𝑦𝑛 (𝑥)
𝐿2 [0, 1) can be expanded into the following Haar series:
𝑥 𝑏
+∞
− ∫ (𝑦𝑛󸀠 (𝜉) − 𝑓󸀠 (𝜉) − ∫ 𝐾󸀠 (𝜉, 𝑡) 𝑦̃𝑛 (𝑡) 𝑑𝑡) 𝑑𝜉,
0 𝑎 𝑦 (𝑥) = ∑ 𝑐𝑛 ℎ𝑛 (𝑥) , (60)
𝑛=0
𝑛 ≥ 0. where the coefficients 𝑐𝑛 are given by
(54)
1
By starting with initial approximate function 𝑦0 (𝑥) = 𝑓(𝑥) 𝑐𝑛 = 2𝑗 ∫ 𝑦 (𝑥) ℎ𝑛 (𝑥) 𝑑𝑥,
0 (61)
(say), we can determine the approximate solution 𝑦(𝑥) of (7).
𝑗 𝑗
𝑛 = 2 + 𝑘, 𝑗 ≥ 0, 0≤𝑘<2.
4. Numerical Methods for System of 1
In particular, 𝑐0 = ∫0 𝑦(𝑥)𝑑𝑥.
Linear Fredholm Integral Equations of
The previously mentioned expression in (60) can be
Second Kind approximately represented with finite terms as follows:
Consider the system of linear Fredholm integral equations of 𝑚−1
𝑇
second kind of the following form: 𝑦 (𝑥) ≈ ∑ 𝑐𝑛 ℎ𝑛 (𝑥) = 𝐶(𝑚) ℎ(𝑚) (𝑥) , (62)
𝑛=0
𝑛 𝑛 1
∑ 𝑦𝑗 (𝑥) = 𝑓𝑖 (𝑥) + ∑ ∫ 𝐾𝑖,𝑗 (𝑥, 𝑡) 𝑦𝑗 (𝑡) 𝑑𝑡, 𝑇
where the coefficient vector 𝐶(𝑚) and the Haar function vector
𝑗=1 𝑗=1 0 (55) ℎ(𝑚) (𝑥) are, respectively, defined as
𝑖 = 1, 2, . . . , 𝑛, 𝑇
𝐶(𝑚) = [𝑐0 , 𝑐1 , . . . , 𝑐𝑚−1 ] , 𝑚 = 2𝑗 ,
where 𝑓𝑖 (𝑥) and 𝐾𝑖,𝑗 (𝑥, 𝑡) are known functions and 𝑦𝑗 (𝑥) are (63)
𝑇
the unknown functions for 𝑖, 𝑗 = 1, 2, . . . , 𝑛. ℎ(𝑚) (𝑥) = [ℎ0 (𝑥) , ℎ1 (𝑥) , . . . , ℎ𝑚−1 (𝑥)] , 𝑚 = 2𝑗 .
The Haar expansion for function 𝐾(𝑥, 𝑡) of order 𝑚 is defined By recursion of the above formula, we obtain
as follows: 1 2 −1
𝑃(2) = [ ],
4 1 0
𝑚−1 𝑚−1
𝐾 (𝑥, 𝑡) ≈ ∑ ∑ 𝑎𝑢V ℎV (𝑥) ℎ𝑢 (𝑡) , (64) 8 −4 −2 −2
𝑢=0 V=0 1 [
[4 0 −2 2]
],
𝑃(4) =
16 [1 1 0 0]
1 [1 −1 0 0]
where 𝑎𝑢V = 2𝑖+𝑞 ∬0 𝐾(𝑥, 𝑡)ℎV (𝑥)ℎ𝑢 (𝑡)𝑑𝑥 𝑑𝑡, 𝑢 = 2𝑖 + 𝑗, V =
32 −16 −8 −8 −4 −4 −4 −4 (71)
2𝑞 + 𝑟, 𝑖, 𝑞 ≥ 0.
From (62) and (64), we obtain [16 0 −8 8 −4 −4 4 4]
[ ]
[4 4 0 0 −4 4 0 0]
1 [[4 4 0 0 0 0 −4 4]
]
𝑃(8) = [ .
𝑇
𝐾 (𝑥, 𝑡) ≈ ℎ(𝑚) (𝑡) 𝐾ℎ(𝑚) (𝑥) , (65) 64 [ 1 1 2 0 0 0 0 0]
]
[ 1 1 −2 0 0 0 0 0]]
[
[ 1 −1 0 2 0 0 0 0]
where [ 1 −1 0 −2 0 0 0 0]
Therefore, we get
𝑇
𝐾 = (𝑎𝑢V )𝑚×𝑚 . (66)
−1 1 𝑇
𝐻(𝑚) = ( ) 𝐻(𝑚)
4.1.3. Operational Matrices of Integration. We define 𝑚
22 , . . . , 22 , . . . , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
× diag (1, 1, 2, 2, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2𝛼−1 , . . . , 2𝛼−1 ) ,
1 3 2𝑚 − 1 22 2𝛼−1
𝐻(𝑚) = [ℎ(𝑚) ( ) , ℎ(𝑚) ( ) , . . . , ℎ(𝑚) ( )] ,
2𝑚 2𝑚 2𝑚
(67)
(72)
where 𝐻(1) = [1], 𝐻(2) = [ 11 −1

1 ]. where 𝑚 = 2𝛼 and 𝛼 is a positive integer.
Then, for 𝑚 = 4, the corresponding matrix can be The inner product of two Haar functions can be repre-
represented as sented as
1
𝑇
∫ ℎ(𝑚) (𝑡) ℎ(𝑚) (𝑡) 𝑑𝑡 = 𝐷, (73)
0
1 3 7
𝐻(4) = [ℎ(4) ( ) , ℎ(4) ( ) , . . . , ℎ(4) ( )] where
8 8 8
1 1 1 1 (68) 1/22 , . . . , 1/22 , . . . ,
𝐷 = diag (1, 1, 1/2, 1/2, ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
[1 1 −1 −1]
= [
[1
]. 22
−1 0 0] (74)
[0 0 1 −1]
1/2 𝛼−1 𝛼−1
, . . . , 1/2 ) .
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2𝛼−1
The integration of the Haar function vector ℎ(𝑚) (𝑡) is

4.1.4. Haar Wavelet Solution for Fredholm Integral Equations
System [9]. Consider the following Fredholm integral equa-
𝑥 tions system defined in (55):
∫ ℎ(𝑚) (𝑡) 𝑑𝑡 = 𝑃(𝑚) ℎ(𝑚) (𝑥) , 𝑚 𝑚
0 1
(69)
∑ 𝑦𝑗 (𝑥) = 𝑓𝑖 (𝑥) + ∑ ∫ 𝐾𝑖,𝑗 (𝑥, 𝑡) 𝑦𝑗 (𝑡) 𝑑𝑡,
𝑥 ∈ [0, 1) , 𝑗=1 𝑗=1 0 (75)
𝑖 = 1, 2, . . . , 𝑚.
where 𝑃(𝑚) is the operational matrix of order 𝑚, and
The Haar series of 𝑦𝑗 (𝑥) and 𝐾𝑖,𝑗 (𝑥, 𝑡), 𝑖 = 1, 2, . . . , 𝑚; 𝑗 =
1, 2, . . . , 𝑚 are, respectively, expanded as
1 𝑦𝑗 (𝑥) ≈ 𝑌𝑗𝑇ℎ(𝑚) (𝑥) , 𝑗 = 1, 2, . . . , 𝑚,
𝑃(1) = [ ] ,
2
𝑇
(70) 𝐾𝑖,𝑗 (𝑥, 𝑡) ≈ ℎ(𝑚) (𝑡) 𝐾𝑖,𝑗 ℎ(𝑚) (𝑥) , (76)
1 2𝑚𝑃(𝑚/2) −𝐻(𝑚/2)
𝑃(𝑚) = [ −1 ].
2 𝑚 𝐻(𝑚/2) 0 𝑖, 𝑗 = 1, 2, . . . , 𝑚.
1
Substituting (76) into (75), we get ∫0 ∑𝑛𝑗=1 𝐾𝑖,𝑗 (𝑥, 𝑡)𝐸(𝑡)𝑑𝑡, then, substituting (81) for 𝑦𝑗 (𝑡) into
𝑚
the integral in (80), we have
∑ 𝑌𝑗𝑇 ℎ(𝑚) (𝑥) 𝑦𝑖 (𝑥) ≈ 𝑓𝑖 (𝑥)
𝑗=1
𝑛 1 𝑚
𝑚 1
1
+ ∑ ∫ 𝐾𝑖,𝑗 (𝑥, 𝑡) ∑ (𝑡 − 𝑥)𝑟 𝑦𝑗(𝑟) (𝑥) 𝑑𝑡,
= 𝑓𝑖 (𝑥) + ∑ ∫ 𝑌𝑗𝑇 ℎ(𝑚) 𝑇
(𝑡) ℎ(𝑚) (𝑡) 𝐾𝑖,𝑗 ℎ(𝑚) (𝑥) 𝑑𝑡, 𝑗=1 0 𝑟=0 𝑟!
𝑗=1 0
𝑖 = 1, 2, . . . , 𝑛,
𝑖 = 1, 2, . . . , 𝑚. (82)
(77) 𝑦𝑖 (𝑥) ≈ 𝑓𝑖 (𝑥)
𝑛 𝑚 1
From (77) and (73), we get 1 (𝑟)
+∑∑ 𝑦𝑗 (𝑥) ∫ 𝐾𝑖,𝑗 (𝑥, 𝑡) (𝑡 − 𝑥)𝑟 𝑑𝑡,
𝑚 𝑚 𝑗=1 𝑟=0 𝑟! 0
∑ 𝑌𝑗𝑇 ℎ(𝑚) (𝑥) = 𝑓𝑖 (𝑥) + ∑ 𝑌𝑗𝑇 𝐷𝐾𝑖,𝑗 ℎ(𝑚) (𝑥) ,
𝑗=1 𝑗=1 (78) 𝑖 = 1, 2, . . . , 𝑛,
𝑛 𝑚 1
𝑖 = 1, 2, . . . , 𝑚. 1 (𝑟)
𝑦𝑖 (𝑥) − ∑ ∑ 𝑦𝑗 (𝑥) [∫ 𝐾𝑖,𝑗 (𝑥, 𝑡) (𝑡 − 𝑥)𝑟 𝑑𝑡]
𝑗=1 𝑟=0 𝑟! 0
Interpolating 𝑚 collocation points, that is, {𝑥𝑖 }𝑚
𝑖=1 , in the
interval [0, 1] leads to the following algebraic system of ≈ 𝑓𝑖 (𝑥) , 𝑖 = 1, 2, . . . , 𝑛.
equations: (83)
𝑚 𝑚
Equation (83) becomes a linear system of ordinary differential
∑ 𝑌𝑗𝑇 ℎ(𝑚) (𝑥𝑖 ) = 𝑓𝑖 (𝑥𝑖 ) + ∑ 𝑌𝑗𝑇 𝐷𝐾𝑖,𝑗 ℎ(𝑚) (𝑥𝑖 ) ,
𝑗=1 𝑗=1
equations that we have to solve. For solving the linear
(79)
system of ordinary differential equations (83), we require an
𝑖 = 1, 2, . . . , 𝑚. appropriate number of boundary conditions.
In order to construct boundary conditions, we first
Hence, 𝑌𝑗 , 𝑗 = 1, 2, . . . , 𝑚 can be computed by solving the differentiate 𝑠 times both sides of (80) with respect to 𝑥; that
above algebraic system of equations and consequently the is,
solutions 𝑦𝑗 (𝑥) ≈ 𝑌𝑗𝑇 ℎ(𝑚) (𝑥), 𝑗 = 1, 2, . . . , 𝑚. 𝑛 1
𝑦𝑖(𝑠) (𝑥) = 𝑓𝑖(𝑠) (𝑥) + ∑ ∫ 𝐾𝑖,𝑗
(𝑠)
(𝑥, 𝑡) 𝑦𝑗 (𝑡) 𝑑𝑡,
𝑗=1 0 (84)
4.2. Taylor Series Expansion Method. In this section, we
present Taylor series expansion method for solving Fredholm 𝑖 = 1, 2, . . . , 𝑛, 𝑠 = 1, 2, . . . , 𝑚,
integral equations system of second kind [7]. This method
(𝑠)
reduces the system of integral equations to a linear system where 𝐾𝑖,𝑗 (𝑥, 𝑡) = 𝜕(𝑠) 𝐾𝑖,𝑗 (𝑥, 𝑡)/𝜕𝑥(𝑠) , 𝑠 = 1, 2, . . . , 𝑚.
of ordinary differential equation. After including boundary Applying the mean value theorem for integral in (84), we
conditions, this system reduces to a system of equations that have
can be solved easily by any usual methods.
Consider the second kind Fredholm integral equations 𝑛 1
system defined in (55) as follows: 𝑦𝑖(𝑠) (𝑥) − [ ∑ ∫ 𝐾𝑖,𝑗
(𝑠)
(𝑥, 𝑡) 𝑑𝑡] 𝑦𝑗 (𝑥) ≈ 𝑓𝑖(𝑠) (𝑥) ,
0 (85)
𝑛 1
[𝑗=1 ]
𝑦𝑖 (𝑥) = 𝑓𝑖 (𝑥) + ∑ ∫ 𝐾𝑖,𝑗 (𝑥, 𝑡) 𝑦𝑗 (𝑡) 𝑑𝑡, 𝑖 = 1, 2, . . . , 𝑛, 𝑠 = 1, 2, . . . , 𝑚.
𝑗=1 0 (80)
Now (83) combined with (85) becomes a linear system
𝑖 = 1, 2, . . . , 𝑛, 0 ≤ 𝑥 ≤ 1. of algebraic equations that can be solved analytically or
numerically.
A Taylor series expansion can be made for the solution of
𝑦𝑗 (𝑡) in the integral equation (80):
4.3. Block-Pulse Functions for the Solution of Fredholm Integral
Equation. In this section, Block-Pulse functions (BPF) have
𝑦𝑗 (𝑡) = 𝑦𝑗 (𝑥) + 𝑦𝑗󸀠 (𝑥) (𝑡 − 𝑥) + ⋅ ⋅ ⋅ been utilized for the solution of system of Fredholm integral
(81) equations [6].
1 (𝑚)
+ 𝑦 (𝑥) (𝑡 − 𝑥)𝑚 + 𝐸 (𝑡) , An 𝑚-set of BPF is defined as follows:
𝑚! 𝑗
where 𝐸(𝑡) denotes the error between 𝑦𝑗 (𝑡) and its Taylor {1, (𝑖 − 1) 𝑇 ≤ 𝑡 < 𝑖 𝑇 ,
Φ𝑖 (𝑡) = { 𝑚 𝑚 (86)
series expansion in (81). 0, otherwise
If we use the first 𝑚 term of Taylor series {
expansion and neglect the term containing 𝐸(𝑡), that is, with 𝑡 ∈ [0, 𝑇), 𝑇/𝑚 = ℎ and 𝑖 = 1, 2, . . . , 𝑚.
4.3.1. Properties of BPF Now let 𝐾(𝑡, 𝑠) be two-variable function defined on 𝑡 ∈ [0, 𝑇)
and 𝑠 ∈ [0, 1); then 𝐾(𝑡, 𝑠) can be expanded to BPF as
(1) Disjointness. One has
𝐾 (𝑡, 𝑠) = Φ𝑇 (𝑡) 𝐾Ψ (𝑠) , (94)
Φ (𝑡) , 𝑖 = 𝑗;
Φ𝑖 (𝑡) Φ𝑗 (𝑡) = { 𝑖 (87)
0, 𝑖 ≠𝑗, where Φ(𝑡) and Ψ(𝑠) are 𝑚1 and 𝑚2 dimensional Block-Pulse
function vectors and 𝑘 is a 𝑚1 × 𝑚2 Block-Pulse coefficient
, 𝑖, 𝑗 = 1, 2, . . . , 𝑚. This property is obtained from definition matrix.
of BPF. There are two different cases of multiplication of two BPF.
The first case is
(2) Orthogonality. One has
𝑇
Φ1 (𝑡) 0 ⋅⋅⋅ 0
ℎ, 𝑖 = 𝑗; 0 Φ (𝑡) ⋅⋅⋅ 0
∫ Φ𝑖 (𝑡) Φ𝑗 (𝑡) 𝑑𝑡 = { (88) 2
0 0, 𝑖 ≠𝑗, Φ (𝑡) Φ𝑇 (𝑡) = ( .. .. .. ). (95)
. . d .
𝑡 ∈ [0, 𝑇), 𝑖, 𝑗 = 1, 2, . . . , 𝑚. This property is obtained from 0 0 ⋅ ⋅ ⋅ Φ𝑚 (𝑡)
the disjointness property.
It is obtained from disjointness property of BPF. It is a
(3) Completeness. For every 𝑓 ∈ 𝐿2 , {Φ} is complete; if ∫ Φ𝑓 = diagonal matrix with 𝑚 Block-Pulse functions.
0 then 𝑓 = 0 almost everywhere. Because of completeness of The second case is
{Φ}, we have
Φ𝑇 (𝑡) Φ (𝑡) = 1 (96)
𝑇 ∞
󵄩 󵄩2
2
∫ 𝑓 (𝑡) 𝑑𝑡 = ∑ 𝑓𝑖2 󵄩󵄩󵄩Φ𝑖 (𝑡)󵄩󵄩󵄩 (89)
0 𝑖=1
because ∑𝑚 2 𝑚
𝑖=1 (Φ𝑖 (𝑡)) = ∑𝑖=1 Φ𝑖 (𝑡) = 1.
for every real bounded function 𝑓(𝑡) which is square inte- Operational Matrix of Integration. BPF integration property
grable in the interval 𝑡 ∈ [0, 𝑇) and 𝑓𝑖 = (1/ℎ)𝑓(𝑡)Φ𝑖 (𝑡)𝑑𝑡. can be expressed by an operational equation as
𝑇
4.3.2. Function Approximation. The orthogonality property ∫ Φ (𝑡) 𝑑𝑡 = 𝑃Φ (𝑡) , (97)
of BPF is the basis of expanding functions into their Block- 0
Pulse series. For every 𝑓(𝑡) ∈ 𝐿2 (𝑅),
where
𝑚
𝑇
𝑓 (𝑡) = ∑ 𝑓𝑖 Φ𝑖 (𝑡) , (90) Φ (𝑡) = [Φ1 (𝑡) , Φ2 (𝑡) , . . . , Φ𝑚 (𝑡)] . (98)
𝑖=1
A general formula for 𝑃𝑚×𝑚 can be written as
where 𝑓𝑖 is the coefficient of Block-Pulse function, with
respect to 𝑖th Block-Pulse function Φ𝑖 (𝑡). 1 2 ⋅⋅⋅ 2 2
The criterion of this approximation is that mean square 0 1 ⋅⋅⋅ 2 2
error between 𝑓(𝑡) and its expansion is minimum 1
𝑃 = (0 0 1
⋅ ⋅ ⋅ 2) . (99)
2 2 .. .. ...
1 𝑇 𝑚
. .d .. .
𝜀 = ∫ (𝑓(𝑡) − ∑ 𝑓𝑗 Φ𝑗 (𝑡)) 𝑑𝑡 (91) 0 0 0 ⋅⋅⋅ 1
𝑇 0 𝑗=1
so that we can evaluate Block-Pulse coefficients. By using this matrix, we can express the integral of a function
𝑓(𝑡) into its Block-Pulse series
𝑚
𝜕𝜀 2 𝑇
Now = − ∫ (𝑓 (𝑡) − ∑ 𝑓𝑗 Φ𝑗 (𝑡)) Φ𝑖 (𝑡) 𝑑𝑡 = 0, 𝑡 𝑡
𝜕𝑓𝑖 𝑇 0 𝑗=1 ∫ 𝑓 (𝑡) 𝑑𝑡 = ∫ 𝐹𝑇 Φ (𝑡) 𝑑𝑡 = 𝐹𝑇 𝑃Φ (𝑡) . (100)
0 0
1 𝑇
󳨐⇒ 𝑓𝑖 = ∫ 𝑓 (𝑡) Φ𝑖 (𝑡) 𝑑𝑡 (using orthogonal property) . 4.3.3. Solution for Linear Integral Equations System. Consider
ℎ 0
(92) the integral equations system from (55) as follows:
𝑛 𝑛 𝛽
In the matrix form, we obtain the following from (90) as
follow: ∑ 𝑦𝑗 (𝑥) = 𝑓𝑖 (𝑥) + ∑ ∫ 𝐾𝑖,𝑗 (𝑥, 𝑡) 𝑦𝑗 (𝑡) 𝑑𝑡,
𝑗=1 𝑗=1 𝛼 (101)
𝑚
𝑇 𝑇
𝑓 (𝑡) = ∑ 𝑓𝑖 Φ𝑖 (𝑡) = 𝐹 Φ (𝑡) = Φ 𝐹 𝑖 = 1, 2, . . . , 𝑛.
𝑖=1
𝑇 (93) Block-Pulse coefficients of 𝑦𝑗 (𝑥), 𝑗 = 1, 2, . . . , 𝑛 in the interval

where 𝐹 = [𝑓1 , 𝑓2 , . . . , 𝑓𝑚 ] , 𝑥 ∈ [𝛼, 𝛽) can be determined from the known functions
𝑇 𝑓𝑖 (𝑥), 𝑖 = 1, 2, . . . , 𝑛 and the kernels 𝐾𝑖,𝑗 (𝑥, 𝑡), 𝑖, 𝑗 = 1, 2, . . . 𝑛.
Φ (𝑡) = [Φ1 (𝑡) , Φ2 (𝑡) , . . . , Φ𝑚 (𝑡)] . Usually, we consider 𝛼 = 0 to facilitie the use of Block-Pulse
functions. In case 𝛼 ≠0, we set 𝑋 = ((𝑥 − 𝛼)/(𝛽 − 𝛼))𝑇, where 5. Numerical Methods for Nonlinear
𝑇 = 𝑚ℎ. Fredholm-Hammerstein Integral Equation
We approximate 𝑓𝑖 (𝑥), 𝑦𝑗 (𝑥), 𝐾𝑖,𝑗 (𝑥, 𝑡) by its BPF as
follows: We consider the second kind nonlinear Fredholm integral
equation of the following form:
𝑓𝑖 (𝑥) ≈ 𝐹𝑖𝑇 Φ (𝑥) , 1
𝑢 (𝑥) = 𝑓 (𝑥) + ∫ 𝐾 (𝑥, 𝑡) 𝐹 (𝑡, 𝑢 (𝑡)) 𝑑𝑡,
𝑦𝑗 (𝑥) ≈ 𝑌𝑗𝑇 Φ (𝑥) , (102) 0 (109)
0 ≤ 𝑥 ≤ 1,
𝐾𝑖,𝑗 (𝑥, 𝑡) ≈ Φ𝑇 (𝑡) 𝐾𝑖,𝑗 Φ (𝑥) ,
where 𝐾(𝑥, 𝑡) is the kernel of the integral equation, 𝑓(𝑥)
where 𝐹𝑖 , 𝑌𝑗 , and 𝐾𝑖,𝑗 are defined in Section 4.3.2, and and 𝐾(𝑥, 𝑡) are known functions, and 𝑢(𝑥) is the unknown
substituting (102) into (101), we have function that is to be determined.
𝑛
5.1. B-Spline Wavelet Method. In this section, nonlinear
∑ 𝑌𝑗𝑇 Φ (𝑥) = 𝐹𝑖𝑇 Φ (𝑥)
Fredholm integral equation of second kind of the form given
𝑗=1
in (109) has been solved by using B-spline wavelets [11].
𝑛 𝑚ℎ B-spline scaling and wavelet functions in the interval
(103)
+∑∫ 𝑌𝑗𝑇Φ (𝑡) Φ𝑇 (𝑡) 𝐾𝑖,𝑗 Φ (𝑥) 𝑑𝑡, [0, 1] and function approximation have been defined in
𝑗=1 0 Sections 3.1.1 and 3.1.2, respectively.
First, we assume that
𝑖 = 1, 2, . . . , 𝑛,
𝑛 𝑛 𝑦 (𝑥) = 𝐹 (𝑥, 𝑢 (𝑥)) ,
∑ 𝑌𝑗𝑇 Φ (𝑥) = 𝐹𝑖𝑇 Φ (𝑥) + ∑ 𝑌𝑗𝑇 ℎ𝐼𝐾𝑖,𝑗 Φ (𝑥) , (110)
0 ≤ 𝑥 ≤ 1.
𝑗=1 𝑗=1 (104)
𝑖 = 1, 2, . . . , 𝑛, Now, from (16), we can approximate the functions 𝑢(𝑥) and
𝑦(𝑥) as
since
𝑢 (𝑥) = 𝐴𝑇 Ψ (𝑥) ,
𝑚ℎ
(111)
∫ Φ (𝑡) Φ𝑇 (𝑡) 𝑑𝑡 = ℎ𝐼. (105) 𝑦 (𝑥) = 𝐵𝑇 Ψ (𝑥) ,
0
where 𝐴 and 𝐵 are (2𝑀+1 + 𝑚 − 1) × 1 column vectors similar
From (104), we get to 𝐶 defined in (17).
Again, by using dual of the wavelet functions, we can
𝑛 approximate the functions 𝑓(𝑥) and 𝐾(𝑥, 𝑡) as follows:
𝑇
∑ (𝐼 − ℎ𝐾𝑖,𝑗 ) 𝑌𝑗 = 𝐹𝑖 , 𝑖 = 1, 2, . . . , 𝑛. (106)
𝑗=1 ̃ (𝑥) ,
𝐹 (𝑥) = 𝐷𝑇 Ψ
(112)
𝑇
Set 𝐴 𝑖,𝑗 = 𝐼 − ℎ𝐾𝑖,𝑗 ; then we have from (106) ̃𝑇 (𝑡) ΘΨ
𝐾 (𝑥, 𝑡) = Ψ ̃ (𝑥) ,
𝑛 where
∑ 𝐴 𝑖,𝑗 𝑌𝑗 = 𝐹𝑖 , 𝑖 = 1, 2, . . . , 𝑛 (107) 1 1
𝑗=1 Θ(𝑖,𝑗) = ∫ [∫ 𝐾 (𝑥, 𝑡) Ψ𝑖 (𝑡) 𝑑𝑡] Ψ𝑗 (𝑥) 𝑑𝑥. (113)
0 0
which is a linear system From (110)–(112), we get

1
𝐴 11 𝐴 12 . . . 𝐴 1𝑛 𝑌1 𝐹1 ∫ 𝐾 (𝑥, 𝑡) 𝐹 (𝑡, 𝑢 (𝑡)) 𝑑𝑡
𝐴 21 𝐴 22 . . . 𝐴 2𝑛 𝑌2 𝐹2 0
( . . . . )( . ) ( . )
= . 1
. . . . . . ̃𝑇 (𝑡) ΘΨ
= ∫ 𝐵𝑇 Ψ (𝑡) Ψ ̃ (𝑥) 𝑑𝑡
. . . . . . 0
(114)
(𝐴 𝑛1 𝐴 𝑛2 . . . 𝐴 𝑛𝑛 ) (𝑌𝑛 ) (𝐹𝑛 ) 1
(108)
𝑇 ̃ (𝑥)
̃𝑇
= 𝐵 [∫ Ψ (𝑡) Ψ (𝑡) 𝑑𝑡] ΘΨ
0
1
After solving the above system we can find 𝑌𝑗 , 𝑗 = 1, 2, . . . , 𝑛 ̃ (𝑥) , ̃𝑇 (𝑡) 𝑑𝑡 = 𝐼.
= 𝐵𝑇 ΘΨ since ∫ Ψ (𝑡) Ψ
and hence obtain the solutions 𝑦𝑗 = Φ𝑇 𝑌𝑗 , 𝑗 = 1, 2, . . . , 𝑛. 0
(𝑛−1)/2
Applying (110)–(114) in (109), we get 2ℎ
+ ∑ 𝑓 (𝑥2𝑖 )
3 𝑖=1
̃ (𝑥) + 𝐵𝑇 ΘΨ
𝐴𝑇 Ψ (𝑥) = 𝐷𝑇 Ψ ̃ (𝑥) . (115)
ℎ
+ 𝑓 (𝑏)
3
Multiplying (115) by Ψ𝑇 (𝑥) both sides from the right and
integrating both sides with respect to 𝑥 from 0 to 1, we have (𝑏 − 𝑎) 4 (4)
− ℎ 𝑓 (𝜂) .
180
(119)
𝐴𝑇 𝑃 = 𝐷𝑇 + 𝐵𝑇 Θ,
(116)
5.2.2. Modified Trapezoid Rule. One has
𝐴𝑇 𝑃 − 𝐷𝑇 − 𝐵𝑇 Θ = 0,
𝑏 𝑛 𝑥𝑖
∫ 𝑓 (𝑥) 𝑑𝑥 = ∑ ∫ 𝑓 (𝑥) 𝑑𝑥
where 𝑃 is a (2𝑀+1 + 𝑚 − 1) × (2𝑀+1 + 𝑚 − 1) square matrix 𝑎 𝑖=1 𝑥𝑖−1
given by 𝑛−1
ℎ
= 𝑓 (𝑎) + ℎ ∑ 𝑓 (𝑥𝑖 )
1
𝑃1 2 𝑖=1 (120)
𝑃 = ∫ Ψ (𝑥) Ψ𝑇 (𝑥) 𝑑𝑥 = [ ],
0 𝑃2 ℎ
(117) + 𝑓 (𝑏)
1 2
̃ (𝑥) Ψ𝑇 (𝑥) 𝑑𝑥 = 𝐼.
∫ Ψ
ℎ2 󸀠
0
[𝑓 (𝑎) − 𝑓󸀠 (𝑏)] .
+
12
Equation (116) gives a system of (2𝑀+1 + 𝑚 − 1) algebraic Consider the nonlinear Fredholm integral equation of second
equations with 2(2𝑀+1 +𝑚−1) unknowns for 𝐴 and 𝐵 vectors kind defined in (109) as follows:
given in (111). 𝑏
To find the solution 𝑢(𝑥) in (111), we first utilize the 𝑢 (𝑥) = 𝑓 (𝑥) + ∫ 𝐾 (𝑥, 𝑡) 𝐹 (𝑢 (𝑡)) 𝑑𝑡,
following equation: 𝑎 (121)
𝑎 ≤ 𝑥 ≤ 𝑏.
𝐹 (𝑥, 𝐴𝑇 Ψ (𝑥)) = 𝐵𝑇 Ψ (𝑥) , (118) For solving (121), we approximate the right-hand integral of
(121) with Simpson’s rule and modified trapezoid rule; then
with the collocation points 𝑥𝑖 = (𝑖 − 1)/(2𝑀+1 + 𝑚 − 2), where we get the following.
𝑖 = 1, 2, . . . , 2𝑀+1 + 𝑚 − 1. 5.2.3. Simpson’s Rule. One has
Equation (118) gives a system of (2𝑀+1 + 𝑚 − 1) algebraic
equations with 2(2𝑀+1 +𝑚−1) unknowns, for 𝐴 and 𝐵 vectors 𝑢 (𝑥) = 𝑓 (𝑥)
given in (111).
Combining (116) and (118), we have a total of 2(2𝑀+1 + ℎ[
+ 𝐾 (𝑥, 𝑡0 ) 𝐹 (𝑢0 )
𝑚 − 1) system of algebraic equations with 2(2𝑀+1 + 𝑚 − 3
[
1) unknowns for 𝐴 and 𝐵. Solving those equations for the
𝑛/2
unknown coefficients in the vectors 𝐴 and 𝐵, we can obtain
the solution 𝑢(𝑥) = 𝐴𝑇 Ψ(𝑥). + 4 ∑ 𝐾 (𝑥, 𝑡2𝑗−1 ) 𝐹 (𝑢2𝑗−1 )
𝑗=1 (122)
(𝑛/2)−1
5.2. Quadrature Method Applied to Fredholm Integral Equa-
+ 2 ∑ 𝐾 (𝑥, 𝑡2𝑗 ) 𝐹 (𝑢2𝑗 )
tion. In this section, Quadrature method has been applied 𝑗=1
to solve nonlinear Fredholm-Hammerstein integral equation
[10].
The quadrature methods like Simpson rule and modified + 𝐾 (𝑥, 𝑡𝑛 ) 𝐹 (𝑢𝑛 ) ] .
trapezoid method are applied for solving a definite integral as ]
follows.
Hence, for 𝑥 = 𝑥0 , 𝑥1 , . . . , 𝑥𝑛 and 𝑡 = 𝑡0 , 𝑡1 , . . . , 𝑡𝑛 in (122), we
have
5.2.1. Simpson’s Rule. One has 𝑢 (𝑥𝑖 ) = 𝑓 (𝑥𝑖 )
𝑏 𝑛−1 𝑥𝑖+1 ℎ[
∫ 𝑓 (𝑥) 𝑑𝑥 = ∑ ∫ 𝑓 (𝑥) 𝑑𝑥 + 𝐾 (𝑥𝑖 , 𝑡0 ) 𝐹 (𝑢0 )
𝑎 𝑥𝑖−1 3
𝑖=1 [
𝑛/2 𝑛/2
ℎ 4ℎ + 4 ∑ 𝐾 (𝑥𝑖 , 𝑡2𝑗−1 ) 𝐹 (𝑢2𝑗−1 )
= 𝑓 (𝑎) + ∑ 𝑓 (𝑥2𝑖−1 )
3 3 𝑖=1 𝑗=1
(𝑛/2)−1
This is a system of (𝑛+1) equations and (𝑛+3) unknowns.
+ 2 ∑ 𝐾 (𝑥𝑖 , 𝑡2𝑗 ) 𝐹 (𝑢2𝑗 ) By taking derivative from (121) and setting 𝐻(𝑥, 𝑡) =
𝑗=1
𝜕𝐾(𝑥, 𝑡)/𝜕𝑥, we obtain
+ 𝐾 (𝑥𝑖 , 𝑡𝑛 ) 𝐹 (𝑢𝑛 ) ] .
𝑏
]
(123) 𝑢󸀠 (𝑥) = 𝑓󸀠 (𝑥) + ∫ 𝐻 (𝑥, 𝑡) 𝐹 (𝑢 (𝑡)) 𝑑𝑡,
𝑎 (126)
𝑎 ≤ 𝑥 ≤ 𝑏.
Equation (123) is a nonlinear system of equations and, by
solving (123), we obtain the unknowns 𝑢(𝑥𝑖 ) for 𝑖 = 0, 1, . . . , 𝑛.
If 𝑢 is a solution of (121), then it is also solution of (126). By

5.2.4. Modified Trapezoid Rule. One has
using trapezoid rule for (126) and replacing 𝑥 = 𝑥𝑖 , we get
𝑢 (𝑥) = 𝑓 (𝑥)
ℎ 𝑢󸀠 (𝑥𝑖 ) = 𝑓󸀠 (𝑥𝑖 )
+ 𝐾 (𝑥, 𝑡0 ) 𝐹 (𝑢0 )
2 ℎ
𝑛−1 + 𝐻 (𝑥𝑖 , 𝑡0 ) 𝐹 (𝑢0 )
2
+ ℎ ∑ 𝐾 (𝑥, 𝑡𝑗 ) 𝐹 (𝑢𝑗 )
𝑗=1 𝑛−1 (127)
+ ℎ ∑ 𝐻 (𝑥𝑖 , 𝑡𝑗 ) 𝐹 (𝑢𝑗 )
ℎ 𝑗=1
+ 𝐾 (𝑥, 𝑡𝑛 ) 𝐹 (𝑢𝑛 ) (124)
2 ℎ
2 + 𝐻 (𝑥𝑖 , 𝑡𝑛 ) 𝐹 (𝑢𝑛 ) ,
ℎ 2
+ [𝐽 (𝑥, 𝑡0 ) 𝐹 (𝑢0 )
12
+ 𝐾 (𝑥, 𝑡0 ) 𝑢0󸀠 𝐹󸀠 (𝑢0 )
for 𝑖 = 0, 1, . . . , 𝑛. In case of 𝑖 = 0, 𝑛 from system (127), we
− 𝐽 (𝑥, 𝑡𝑛 ) 𝐹 (𝑢𝑛 ) obtain two equations.
Now (127) combined with (125) generates the nonlinear
−𝐾 (𝑥, 𝑡𝑛 ) 𝑢𝑛󸀠 𝐹󸀠 (𝑢𝑛 )] , system of equations as follows:
where 𝐽(𝑥, 𝑡) = 𝜕𝐾(𝑥, 𝑡)/𝜕𝑡.

For 𝑥 = 𝑥0 , 𝑥1 , . . . , 𝑥𝑛 and 𝑡 = 𝑡0 , 𝑡1 , . . . , 𝑡𝑛 in (124), we ℎ ℎ2
𝑢 (𝑥𝑖 ) = ( 𝐾 (𝑥𝑖 , 𝑡0 ) + 𝐽 (𝑥𝑖 , 𝑡0 )) 𝐹 (𝑢0 )
have 2 12
𝑛−1
𝑢 (𝑥𝑖 ) = 𝑓 (𝑥𝑖 ) + ℎ ∑ 𝐾 (𝑥𝑖 , 𝑡𝑗 ) 𝐹 (𝑢𝑗 )
𝑗=1
ℎ
+ 𝐾 (𝑥𝑖 , 𝑡0 ) 𝐹 (𝑢0 )
2 ℎ ℎ2
+ ( 𝐾 (𝑥𝑖 , 𝑡𝑛 ) − 𝐽 (𝑥𝑖 , 𝑡𝑛 )) 𝐹 (𝑢𝑛 )
𝑛−1 2 12
+ ℎ ∑ 𝐾 (𝑥𝑖 , 𝑡𝑗 ) 𝐹 (𝑢𝑗 )
𝑗=1 ℎ2
+ (𝐾 (𝑥𝑖 , 𝑡0 ) 𝑢0󸀠 𝐹󸀠 (𝑢0 )
ℎ 12
+ 𝐾 (𝑥𝑖 , 𝑡𝑛 ) 𝐹 (𝑢𝑛 ) (125)
2 −𝐾 (𝑥𝑖 , 𝑡𝑛 ) 𝑢𝑛󸀠 𝐹󸀠 (𝑢𝑛 )) ,
ℎ2 𝑢󸀠 (𝑥0 ) = 𝑓󸀠 (𝑥0 )
+ [𝐽 (𝑥𝑖 , 𝑡0 ) 𝐹 (𝑢0 )
12
ℎ
+ 𝐾 (𝑥𝑖 , 𝑡0 ) 𝑢0󸀠 𝐹󸀠 (𝑢0 ) + 𝐻 (𝑥0 , 𝑡0 ) 𝐹 (𝑢0 )
2
− 𝐽 (𝑥𝑖 , 𝑡𝑛 ) 𝐹 (𝑢𝑛 ) 𝑛−1
+ ℎ ∑ 𝐻 (𝑥0 , 𝑡𝑗 ) 𝐹 (𝑢𝑗 )
− 𝐾 (𝑥𝑖 , 𝑡𝑛 ) 𝑢𝑛󸀠 𝐹󸀠 (𝑢𝑛 )] , 𝑗=1
ℎ
for 𝑖 = 0, 1, . . . , 𝑛. + 𝐻 (𝑥0 , 𝑡𝑛 ) 𝐹 (𝑢𝑛 ) ,
2
𝑢󸀠 (𝑥𝑛 ) = 𝑓󸀠 (𝑥𝑛 ) where

ℎ
+ 𝐻 (𝑥𝑛 , 𝑡0 ) 𝐹 (𝑢0 ) 𝑐𝑛,𝑚 = ⟨𝑓 (𝑥) , 𝜓𝑛,𝑚 (𝑥)⟩ . (133)
2
𝑛−1 (128) If the infinite series in (132) is truncated, then (132) can be
+ ℎ ∑ 𝐻 (𝑥𝑛 , 𝑡𝑗 ) 𝐹 (𝑢𝑗 ) written as
𝑗=1
2𝑘−1 𝑀−1
ℎ 𝑓 (𝑥) ≈ ∑ ∑ 𝑐𝑛,𝑚 𝜓𝑛,𝑚 (𝑥) = 𝐶𝑇 Ψ (𝑥) , (134)
+ 𝐻 (𝑥𝑛 , 𝑡𝑛 ) 𝐹 (𝑢𝑛 ) .
2 𝑛=1 𝑚=0
By solving this system with (𝑛 + 3) nonlinear equations and

(𝑛 + 3) unknowns, we can obtain the solution of (109). where 𝐶 and Ψ(𝑥) are 2𝑘−1 𝑀 × 1 matrices given by
5.3. Wavelet Galerkin Method. In this section, the continuous 𝐶 = [𝑐1,0 , 𝑐1,1 , . . . , 𝑐1,𝑀−1 , 𝑐2,0 , . . . ,
Legendre wavelets [12], constructed on the interval [0, 1], are (135)
𝑇
applied to solve the nonlinear Fredholm integral equation of 𝑐2,𝑀−1 , . . . , 𝑐2𝑘−1 ,0 , . . . , 𝑐2𝑘−1 ,𝑀−1 ] ,
the second kind. The nonlinear part of the integral equation is Ψ (𝑥) = [𝜓1,0 (𝑥) , . . . , 𝜓1,𝑀−1 (𝑥) ,
approximated by Legendre wavelets, and the nonlinear inte-
gral equation is reduced to a system of nonlinear equations. 𝜓2,0 (𝑥) , . . . , 𝜓2,𝑀−1 (𝑥) , . . . , (136)
We have the following family of continuous wavelets with
𝑇
dilation parameter 𝑎 and the translation parameter 𝑏 𝜓2𝑘−1 ,0 (𝑥) , . . . , 𝜓2𝑘−1 ,𝑀−1 (𝑥)] .
𝑡−𝑏
𝜓𝑎,𝑏 (𝑡) = |𝑎|−1/2 𝜓 ( ), Similarly, a function 𝑘(𝑥, 𝑡) ∈ 𝐿2 ([0, 1] × [0, 1]) can be
𝑎 (129) approximated as
𝑎, 𝑏 ∈ 𝑅, 𝑎 ≠0.
𝑘 (𝑥, 𝑡) ≈ Ψ𝑇 (𝑡) 𝐾Ψ (𝑥) , (137)
Legendre wavelets 𝜓𝑚,𝑛 (𝑡) = 𝜓(𝑘, 𝑛̂, 𝑚, 𝑡) have four argu-
ments; 𝑘 = 2, 3, . . ., 𝑛̂ = 2𝑛 − 1, 𝑛 = 1, 2, . . . , 2𝑘−1 , 𝑚 is the where 𝐾 is (2𝑘−1 𝑀 × 2𝑘−1 𝑀) matrix, with
order for Legendre polynomials and 𝑡 is the normalized time.
Legendre wavelets are defined on [0, 1) by 𝐾𝑖,𝑗 = ⟨𝜓𝑖 (𝑡) , ⟨𝑘 (𝑥, 𝑡) , 𝜓𝑗 (𝑥)⟩⟩ . (138)
𝜓𝑚,𝑛 (𝑡)
Also, the integer power of a function can be approximated as
1/2
{(𝑚 + 1 ) 𝑛̂ − 1 𝑛̂ + 1
2𝑘/2 𝐿 𝑚 (2𝑘 𝑡 − 𝑛̂) , ≤𝑡< 𝑘 , 𝑝 𝑝 𝑇
={ 2 2𝑘 2 [𝑦 (𝑥)] = [𝑌𝑇Ψ (𝑥)] = 𝑌𝑝∗ Ψ (𝑥) , (139)
{0, otherwise,
(130) where 𝑌𝑝∗ is a column vector, whose elements are nonlinear
combinations of the elements of the vector 𝑌. 𝑌𝑝∗ is called the
where 𝐿 𝑚 (𝑡) are the well-known Legendre polynomials of
order m, which are orthogonal with respect to the weight operational vector of the 𝑝th power of the function 𝑦(𝑥).
function 𝑤(𝑡) = 1 and satisfy the following recursive formula:
5.3.2. The Operational Matrices. The integration of the vector
𝐿 0 (𝑡) = 1, Ψ(𝑥) defined in (136) can be obtained as
𝐿 1 (𝑡) = 𝑡, 𝑡
∫ Ψ (𝑡󸀠 ) 𝑑𝑡󸀠 = 𝑃Ψ (𝑡) , (140)
2𝑚 + 1 (131) 0
𝐿 𝑚+1 (𝑡) = 𝑡𝐿 (𝑡)
𝑚+1 𝑚
𝑚 where 𝑃 is the (2𝑘−1 𝑀 × 2𝑘−1 𝑀) operational matrix for
− 𝐿 (𝑡) , 𝑚 = 1, 2, 3, . . . . integration and is given in [23] as
𝑚 + 1 𝑚−1
The set of Legendre wavelets are an orthonormal set. 𝐿 𝐻 ⋅⋅⋅ 𝐻 𝐻
[0 𝐿 ⋅⋅⋅ 𝐻 𝐻]
[ ]
5.3.1. Function Approximation. A function 𝑓(𝑥) ∈ 𝐿2 [0, 1] [ .. ] .
𝑃 = [ ... ..
. d
..
. .] (141)
can be expanded as [ ]
[0 0 ⋅⋅⋅ 𝐿 𝐻]
∞ ∞ [0 0 ⋅⋅⋅ 0 𝐿]
𝑓 (𝑥) = ∑ ∑ 𝑐𝑛,𝑚 𝜓𝑛,𝑚 (𝑥) , (132)
𝑛=1 𝑚=0 In (141), 𝐻 and 𝐿 are (𝑀 × 𝑀) matrices given in [23] as
2 0 ⋅⋅⋅ 0
[
1 [0 0 ⋅⋅⋅ 0] ]
𝐻 = 𝑘 [ .. .. .. ] ,
2 [. . d .]
[0 0 ⋅⋅⋅ 0]
1
1 0 0 ⋅⋅⋅ 0 0
[ √3 ]
[ ]
[ √3 √3 ]
[− 0 0 ⋅⋅⋅ 0 0 ]
[ 3 3√5 ]
[ ]
[ √ √5 ]
[ 0 − 5
(142)
0 ⋅⋅⋅ 0 0 ]
[ 5√3 5√7 ]
[ ]
1 [
[ √7 ]
].
𝐿= 𝑘[ 0 0 − 0 ⋅⋅⋅ 0 0 ]
2 [ 7√5 ]
[ . .. .. .. .. ]
[ .. ]
[ . . . d . ]
[ √2 𝑀 − 3 ]
[ ]
[ 0 0 0 0 ⋅⋅⋅ 0 ]
[ (2 𝑀 − 3) √2 𝑀 − 1 ]
[ ]
[ −√2 𝑀 − 1 ]
0 0 0 0 ⋅⋅⋅ 0
[ (2 𝑀 − 1) √2 𝑀 − 3 ]
The integration of the product of two Legendre wavelets Substituting (146) into (145), we have
vector functions is obtained as
1
𝑌𝑇 Ψ (𝑥) = 𝐹𝑇 Ψ (𝑥)
∫ Ψ (𝑡) Ψ𝑇 (𝑡) 𝑑𝑡 = 𝐼, (143) 1
𝑇
0
+ ∫ 𝑌∗ Ψ (𝑡) Ψ𝑇 (𝑡) 𝐾Ψ (𝑥) 𝑑𝑡
0
where 𝐼 is an identity matrix.
The product of two Legendre wavelet vector functions is = 𝐹𝑇 Ψ (𝑥)
defined as
1
𝑇
+ 𝑌∗ (∫ Ψ (𝑡) Ψ𝑇 (𝑡) 𝑑𝑡) 𝐾Ψ (𝑥) (147)
𝑇
Ψ (𝑡) Ψ (𝑡) 𝐶 = 𝐶 Ψ (𝑡) , ̃𝑇 (144) 0
𝑇
̃ is (2𝑘−1 𝑀 × 2𝑘−1 𝑀) = 𝐹𝑇 Ψ (𝑥) + 𝑌∗ 𝐾Ψ (𝑥)
where 𝐶 is a vector given in (135) and 𝐶
matrix, which is called the product operation of Legendre 𝑇
wavelet vector functions [23, 24]. = (𝐹𝑇 + 𝑌∗ 𝐾) Ψ (𝑥)
𝑇
󳨐⇒ 𝑌𝑇 − 𝑌∗ 𝐾 − 𝐹𝑇 = 0.
5.3.3. Solution of Fredholm Integral Equation of Second Kind.
Consider the nonlinear Fredholm-Hammerstein integral Equation (147) is a system of algebraic equations. Solving
equation of second kind of the form
(147), we can obtain the solution 𝑦(𝑥) ≈ 𝑌𝑇 Ψ(𝑥).
1
𝑝
𝑦 (𝑥) = 𝑓 (𝑥) + ∫ 𝑘 (𝑥, 𝑡) [𝑦 (𝑡)] 𝑑𝑡, (145) 5.4. Homotopy Perturbation Method. Consider the following
0
nonlinear Fredholm integral equation of second kind of the
where 𝑓 ∈ 𝐿2 [0, 1], 𝑘 ∈ 𝐿2 ([0, 1] × [0, 1]), 𝑦 is an unknown form
function, and 𝑝 is a positive integer. 1
We can approximate the following functions as 𝑢 (𝑥) = 𝑓 (𝑥) + ∫ 𝐾 (𝑥, 𝑡) 𝐹 (𝑢 (𝑡)) 𝑑𝑡,
0 (148)
𝑇
𝑓 (𝑥) ≈ 𝐹 Ψ (𝑥) , 0 ≤ 𝑥 ≤ 1.
𝑦 (𝑥) ≈ 𝑌𝑇 Ψ (𝑥) , For solving (148) by Homotopy perturbation method (HPM)

(146) [14–16], we consider (148) as
𝑘 (𝑥, 𝑡) ≈ Ψ𝑇 (𝑡) 𝐾Ψ (𝑥) ,
1
𝑝
[𝑦 (𝑥)] ≈ 𝑌 Ψ (𝑥) . ∗𝑇 𝐿 (𝑢) = 𝑢 (𝑥) − 𝑓 (𝑥) − ∫ 𝐾 (𝑥, 𝑡) 𝐹 (𝑢 (𝑡)) 𝑑𝑡 = 0. (149)
0
As a possible remedy, we can define 𝐻(𝑢, 𝑝) by Equating the terms with identical power of 𝑝 in (156), we have
𝐻 (𝑢, 0) = 𝑁 (𝑢) ,
(150) 𝑝0 : 𝑢0 (𝑥) − 𝑓 (𝑥) = 0 󳨐⇒ 𝑢0 (𝑥) = 𝑓 (𝑥)
𝐻 (𝑢, 1) = 𝐿 (𝑢) ,
1
where 𝑁(𝑢) is an integral operator with known solution 𝑢0 . 𝑝1 : 𝑢1 (𝑥) − ∫ 𝐾 (𝑥, 𝑡) 𝐻0 𝑑𝑡 = 0 󳨐⇒ 𝑢1 (𝑥)
0
We may choose a convex homotopy by
1
𝐻 (𝑢, 𝑝) = (1 − 𝑝) 𝑁 (𝑢) + 𝑝𝐿 (𝑢) = 0 (151) = ∫ 𝐾 (𝑥, 𝑡) 𝐻0 𝑑𝑡
0
and continuously trace an implicitly defined curve from a 1 (157)
starting point 𝐻(𝑢0 , 0) to a solution function 𝐻(𝑈, 1). The 𝑝2 : 𝑢2 (𝑥) − ∫ 𝐾 (𝑥, 𝑡) 𝐻1 𝑑𝑡 = 0 󳨐⇒ 𝑢2 (𝑥)
embedding parameter 𝑝 monotonically increases from zero 0
to unit as the trivial problem 𝐿(𝑢) = 0. The embedding 1
parameter 𝑝 ∈ (0, 1] can be considered as an expanding = ∫ 𝐾 (𝑥, 𝑡) 𝐻1 𝑑𝑡
parameter. The HPM uses the homotopy parameter 𝑝 as an 0
expanding parameter; that is, ..
2
.
𝑢 = 𝑢0 + 𝑝𝑢1 + 𝑝 𝑢2 + ⋅ ⋅ ⋅ . (152a)
When 𝑝 → 1, (152a) corresponding to (151) become the and in general form we have
approximate solution of (149) as follows:
𝑈 = lim 𝑢 = 𝑢0 + 𝑢1 + 𝑢2 + ⋅ ⋅ ⋅ . 𝑢0 (𝑥) = 𝑓 (𝑥) ,
𝑝→1 (152b)
1 (158)
The series in (152b) converges in most cases, and the rate of 𝑢𝑛+1 (𝑥) = ∫ 𝐾 (𝑥, 𝑡) 𝐻𝑛 𝑑𝑡, 𝑛 = 0, 1, 2, . . . .
0
convergence depends on 𝐿(𝑢) [14].
Consider
Hence, we can obtain the approximate solution of aforesaid
𝑁 (𝑢) = 𝑢 (𝑥) − 𝑓 (𝑥) . (153) equation (148) from (152b).
The nonlinear term 𝐹(𝑢) can be expressed in He polynomials
[25] as 5.5. Adomian Decomposition Method. Adomian decomposi-
∞
tion method (ADM) [16–18] has been applied to a wide class
𝐹 (𝑢) = ∑ 𝑝𝑚 𝐻𝑚 (𝑢0 , 𝑢1 , . . . , 𝑢𝑚 ) of functional equations. This method gives the solution as
𝑚=0 an infinite series usually converging to an accurate solution.
(154) Let us consider the nonlinear Fredholm integral equation of
= 𝐻0 (𝑢0 ) + 𝑝𝐻1 (𝑢0 , 𝑢1 ) second kind as follows:
+ ⋅ ⋅ ⋅ + 𝑝𝑚 𝐻𝑚 (𝑢0 , 𝑢1 , . . . 𝑢𝑚 ) + ⋅ ⋅ ⋅ , 𝑏
𝑢 (𝑥) = 𝑓 (𝑥) + ∫ 𝐾 (𝑥, 𝑡) (𝐿𝑢 (𝑡) + 𝑁𝑢 (𝑡)) 𝑑𝑡,
where 𝑎 (159)
𝐻𝑚 (𝑢0 , 𝑢1 , . . . , 𝑢𝑚 ) 𝑎 ≤ 𝑥 ≤ 𝑏,
𝑚 󵄨󵄨
1 𝜕𝑚 󵄨󵄨 (155)
= (𝐹 ( ∑ 𝑝 𝑢𝑘 ))󵄨󵄨󵄨 ,
𝑘
𝑚 ≥ 0. where 𝐿(𝑢(𝑡)) and 𝑁(𝑢(𝑡)) are the linear and nonlinear terms,
𝑚! 𝜕𝑝𝑚 󵄨󵄨
𝑘=0 󵄨𝑝=0 respectively.
The Adomian decomposition method (ADM) consists of
Substituting (152a), (153), and (154) into (151), we have representing 𝑢(𝑥) as a series
(1 − 𝑝) ((𝑢0 + 𝑝𝑢1 + ⋅ ⋅ ⋅ ) − 𝑓 (𝑥))
∞
𝑢 (𝑥) = ∑ 𝑢𝑚 (𝑥) . (160)
+ 𝑝 ( (𝑢0 + 𝑝𝑢1 + ⋅ ⋅ ⋅ ) − 𝑓 (𝑥) 𝑚=0
1 ∞ In the view of ADM, the nonlinear term 𝑁𝑢 can be repre-

− ∫ 𝐾 (𝑥, 𝑡) ∑ 𝑝𝑚 𝐻𝑚 (𝑢0 , 𝑢1 , . . . , 𝑢𝑚 ) 𝑑𝑡) = 0 sented as
0 𝑚=0
∞
󳨐⇒ (𝑢0 + 𝑝𝑢1 + ⋅ ⋅ ⋅ ) − 𝑓 (𝑥) 𝑁𝑢 = ∑ 𝐴 𝑛 , (161)
𝑛=0
1 ∞
− 𝑝 ∫ 𝐾 (𝑥, 𝑡) ∑ 𝑝𝑚 𝐻𝑚 (𝑢0 , 𝑢1 , . . . , 𝑢𝑚 ) 𝑑𝑡 = 0. 𝜕𝑛 ∞ 󵄨󵄨
1 󵄨󵄨
0 𝑚=0 where 𝐴 𝑛 = ( 𝑁 ( ∑ 𝜆 𝑢𝑘 ))󵄨󵄨󵄨
𝑘
. (162)
(156) 𝑛! 𝜕𝜆𝑛 󵄨󵄨
𝑘=0 󵄨 𝜆=0
Now substituting (160) and (161) into (159), we have and Adomian decomposition method (ADM) can be also
applied to approximate the solution of nonlinear Fredholm
∞
integral equation of second kind. The solutions obtained by
∑ 𝑢𝑚 (𝑥) = 𝑓 (𝑥)
𝑚=0
HPM and ADM are applicable for not only weakly nonlinear
equations, but also strong ones. The approximate solutions by
𝑏 ∞ ∞
these aforesaid methods highly agree with exact solutions.
+ ∫ 𝐾 (𝑥, 𝑡) (𝐿 ( ∑ 𝑢𝑚 (𝑡)) + ∑ 𝐴 𝑚 ) 𝑑𝑡,
𝑎 𝑚=0 𝑚=0
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http://dx.doi.org/10.1155/2013/742643
Research Article
Classification of Exact Solutions for Generalized Form
of 𝐾(𝑚, 𝑛) Equation
Hasan Bulut
Department of Mathematics, Faculty of Science, Firat University, 23119 Elazig, Turkey
Correspondence should be addressed to Hasan Bulut; hbulut@firat.edu.tr
Received 24 May 2013; Revised 1 August 2013; Accepted 18 August 2013
Academic Editor: Santanu Saha Ray
Copyright © 2013 Hasan Bulut. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The classification of exact solutions, including solitons and elliptic solutions, to the generalized 𝐾(𝑚, 𝑛) equation by the complete
discrimination system for polynomial method has been obtained. From here, we find some interesting results for nonlinear partial
differential equations with generalized evolution.
1. Introduction of the KdV equation, where, in particular, the case 𝑙 = 𝑚 =

𝑛 = 1 leads to the KdV equation. The Korteweg de Vries
In science and engineering applications, it is often very equation is one of the most important equations in applied
difficult to obtain analytical solutions of partial differential mathematics and physics. There have been several kinds of
equations. Recently, many exact solutions of partial differ- solutions, such as compactons, that are studied in the context
ential equations have been examined by the use of trial of 𝐾(𝑚, 𝑛) equation, for various situations. We now offer a
equation method. Also there are a lot of important methods more general trial equation method for discussion as follows.
that have been defined such as Hirota method, tanh-coth
method, sine-cosine method, the trial equation method, and
the extended trial equation method [1–15] to find exact 2. The Extended Trial Equation Method
solutions to nonlinear partial differential equations. There Step 1. For a given nonlinear partial differential equation
are a lot of nonlinear evolution equations that are solved by
the use of various mathematical methods. Soliton solutions, 𝑃 (𝑢, 𝑢𝑡 , 𝑢𝑥 , 𝑢𝑥𝑥 , . . .) = 0, (2)
singular solitons, and other solutions have been found by
using these approaches. These obtained solutions have been take the general wave transformation
encountered in various areas of applied mathematics and are
very important. 𝑁
In Section 2, we introduce an extended trial equation 𝑢 (𝑥1 , 𝑥2 , . . . , 𝑥𝑁, 𝑡) = 𝑢 (𝜂) , 𝜂 = 𝜆 ( ∑𝑥𝑗 − 𝑐𝑡) , (3)
method for nonlinear evolution equations with higher order 𝑗=1
nonlinearity. In Section 3, as applications, we procure some

where 𝜆 ≠0 and 𝑐 ≠0. Substituting (3) into (2) yields a
exact solutions to nonlinear partial differential equations
nonlinear ordinary differential equation:
such as the generalized form of 𝐾(𝑚, 𝑛) equation [16–18]:
(𝑞𝑙 )𝑡 + 𝑎𝑞𝑚 𝑞𝑥 + 𝑏(𝑞𝑛 )𝑥𝑥𝑥 = 0, (1) 𝑁 (𝑢, 𝑢󸀠 , 𝑢󸀠󸀠 , . . .) = 0. (4)
where 𝑎, 𝑏 ∈ 𝑅 are constants since 𝑙, 𝑚, and 𝑛 ∈ 𝑍+ . Here, Step 2. Take the finite series and trial equation as follows:
the first term is the generalized evolution term, while the 𝛿
second term represents the nonlinear term and the third term 𝑢 = ∑𝜏𝑖 Γ𝑖 , (5)
is the dispersion term. This equation is the generalized form 𝑖=0
where Let 𝑙 = 𝑛, applying balance and using the following transfor-

mation:
2 Φ (Γ) 𝜉𝜃 Γ𝜃 + ⋅ ⋅ ⋅ + 𝜉1 Γ + 𝜉0
(Γ󸀠 ) = Λ (Γ) = = . (6)
Ψ (Γ) 𝜁𝜖 Γ𝜖 + ⋅ ⋅ ⋅ + 𝜁1 Γ + 𝜁0
𝑢 = V1/(𝑚−𝑛+1) . (13)
Using (5) and (6), we can write
2
󸀠 2 Φ (Γ) 𝛿 Equation (12) turns into the following equation:
(𝑢 ) = (∑𝑖𝜏 Γ𝑖−1 ) ,
Ψ (Γ) 𝑖=0 𝑖
Φ󸀠 (Γ) Ψ (Γ) − Φ (Γ) Ψ󸀠 (Γ) 𝛿 − 𝑐 (𝑚 + 1) (𝑚 + 1 − 𝑛)2 V2 + 𝑎(𝑚 + 1 − 𝑛)2 V3

𝑢󸀠󸀠 = (∑𝑖𝜏𝑖 Γ𝑖−1 ) (7)
2Ψ2 (Γ) 2
𝑖=0
+ 𝑏𝑛 (𝑚 + 1) (2𝑛 − 𝑚 − 1) (V󸀠 ) (14)
𝛿
Φ (Γ)
+ (∑𝑖 (𝑖 − 1) 𝜏𝑖 Γ𝑖−2 ) , + 𝑏𝑛 (𝑚 + 1) (𝑚 + 1 − 𝑛) VV󸀠󸀠 = 0.
Ψ (Γ) 𝑖=0
where Φ(Γ) and Ψ(Γ) are polynomials. Substituting these Substituting (7) into (14) and using balance principle yield
relations into (4) yields an equation of polynomial Ω(Γ) of
Γ:
𝜃 = 𝜖 + 𝛿 + 2. (15)
Ω (Γ) = 󰜚𝑠 Γ𝑠 + ⋅ ⋅ ⋅ + 󰜚1 Γ + 󰜚0 = 0. (8)
According to the balance principle, we can find a relation of

𝜃, 𝜖, and 𝛿. We can calculate some values of 𝜃, 𝜖, and 𝛿. After this solution procedure, we obtain the results as follows.
Step 3. Letting the coefficients of Ω(Γ) all be zero will yield Case 1. If we take 𝜖 = 0, 𝛿 = 1, and 𝜃 = 3, then
an algebraic equations system:
2
󰜚𝑖 = 0, 𝑖 = 0, . . . , 𝑠. (9) 󸀠 2
(𝜏1 ) (𝜉3 Γ3 + 𝜉2 Γ2 + 𝜉1 Γ + 𝜉0 )
(V ) = ,
𝜁0
Solving this system, we will determine the values of (16)
𝜉0 , . . . , 𝜉𝜃 ; 𝜁0 , . . . , 𝜁𝜖 ; and 𝜏0 , . . . , 𝜏𝛿 . 𝜏1 (3𝜉3 Γ2 + 2𝜉2 Γ + 𝜉1 )
V󸀠󸀠 = ,
Step 4. Reduce (6) to the elementary integral form 2𝜁0
𝑑Γ Ψ (Γ)
± (𝜂 − 𝜂0 ) = ∫ = ∫√ 𝑑Γ. (10) where 𝜉3 ≠0 and 𝜁0 ≠0. Respectively, solving the algebraic
√Λ (Γ) Φ (Γ) equation system (9) yields
Using a complete discrimination system for polynomial to
classify the roots of Φ(Γ), we solve (10) and obtain the exact
solutions to (4). Furthermore, we can write the exact traveling 𝜉12 (3 + 3𝑚 − 5𝑛) (1 + 𝑚 + 𝑛)
𝜉0 = − ,
wave solutions to (2), respectively. 16𝜉2 (1 + 𝑚 − 2𝑛)2
𝜉1 = 𝜉1 , 𝜉2 = 𝜉2 ,
3. Application to the Generalized Form of
𝐾(𝑚,𝑛) Equation 8𝜉22 (1 + 𝑚 − 2𝑛) (1 + 𝑚 − 𝑛)
𝜉3 = − ,
𝜉1 (1 + 𝑚 + 𝑛)2
In order to look for travelling wave solutions of (1), we make
(17)
the transformation 𝑞(𝑥, 𝑡) = 𝑢(𝜂), 𝜂 = 𝑥 − 𝑐𝑡, where 𝑐 is the 4 (1 + 𝑚 − 2𝑛) 𝜉2 𝜏0
wave speed. Therefore it can be converted to the ODE 𝜏0 = 𝜏0 , 𝜏1 = − ,
(1 + 𝑚 + 𝑛) 𝜉1
󸀠 𝑎 󸀠 󸀠󸀠󸀠 𝑏𝑛𝜉2 (1 + 𝑚)
−𝑐(𝑢𝑙 (𝜂)) + (𝑢𝑚+1 (𝜂)) + 𝑏(𝑢𝑛 (𝜂)) = 0, (11) 𝜁0 = − ,
𝑚+1 𝑎 (1 + 𝑚 − 𝑛) 𝜏0
where prime denotes the derivative with respect to 𝜂. Then, 𝑎𝑛 (5 + 5𝑚 − 7𝑛) 𝜏0
integrating this equation with respect to 𝜂 one time and 𝑐= .
(1 + 𝑚) (1 + 𝑚 − 𝑛) (1 + 𝑚 + 𝑛)
setting the integration constant to zero, we obtain
𝑎 󸀠󸀠
−𝑐𝑢𝑙 (𝜂) + 𝑢𝑚+1 (𝜂) + 𝑏(𝑢𝑛 (𝜂)) = 0. (12) Substituting these results into (6) and (10), we have
𝑚+1
𝐴 𝑑Γ
± (𝜂 − 𝜂0 ) = ∫ ,
2 𝜉1 (1 + 𝑚 + 𝑛) 2
+ 𝑚 + 𝑛)2
𝜉12 (1 𝜉13 (3 + 3𝑚 − 5𝑛) (1 + 𝑚 + 𝑛)3
√ Γ3 − 2
Γ − 2 Γ+
8𝜉2 (1 + 𝑚 − 2𝑛) (1 + 𝑚 − 𝑛) 8𝜉2 (1 + 𝑚 − 2𝑛) (1 + 𝑚 − 𝑛) 128𝜉22 (1 + 𝑚 − 2𝑛)3 (1 + 𝑚 − 𝑛)
(18)
where 𝑢 (𝑥, 𝑡) = [𝜏0 + 𝜏1 𝛼1 + 𝜏1 (𝛼2 − 𝛼1 )

𝑏𝑛𝜉1 (1 + 𝑚) (1 + 𝑚 + 𝑛)2
𝐴=√ . (19) √𝛼2 − 𝛼1
2𝑎𝜉2 𝜏0 (1 + 𝑚 − 𝑛)2 (1 + 𝑚 − 2𝑛) × sech2 ( (𝑥 − (𝑎𝑛 (5 + 5𝑚 − 7𝑛) 𝜏0 )
𝐴
Integrating (18), we obtain the solutions to (1) as follows: × ((1 + 𝑚) (1 + 𝑚 − 𝑛)
𝐴 × (1 + 𝑚 + 𝑛))−1
± (𝜂 − 𝜂0 ) = − ,
√Γ − 𝛼1
1/(𝑚−𝑛+1)
× 𝑡 − 𝜂0 )) ] ,
𝐴 Γ − 𝛼2
± (𝜂 − 𝜂0 ) = arctan √ , 𝛼2 > 𝛼1 ,
√𝛼2 − 𝛼1 𝛼2 − 𝛼1
𝑢 (𝑥, 𝑡)
󵄨󵄨 √Γ − 𝛼 − 𝛼 − 𝛼 󵄨󵄨
𝐴 󵄨 √ 1 2 󵄨󵄨
± (𝜂 − 𝜂0 ) = ln 󵄨󵄨󵄨󵄨 2
󵄨󵄨 , 𝛼1 > 𝛼2 , = [𝜏0 + 𝜏1 𝛼1 + 𝜏1 (𝛼1 − 𝛼2 )
√𝛼1 − 𝛼2 󵄨󵄨 √Γ − 𝛼2 + √𝛼1 − 𝛼2 󵄨󵄨󵄨
𝐴 √𝛼1 − 𝛼2
± (𝜂 − 𝜂0 ) = − 𝐹 (𝜑, 𝑙) , 𝛼1 > 𝛼2 > 𝛼3 , × cosech2 ( (𝑥 − (𝑎𝑛 (5 + 5𝑚 − 7𝑛) 𝜏0 )
√ 1 − 𝛼3
𝛼 2𝐴
(20)
× ((1 + 𝑚) (1 + 𝑚 − 𝑛)
where
× (1 + 𝑚 + 𝑛))−1
𝜑 𝑑𝜓 Γ − 𝛼3
𝐹 (𝜑, 𝑙) = ∫ , 𝜑 = arcsin √ , 1/(𝑚−𝑛+1)
0 √1 − 𝑙2 sin2 𝜓 𝛼2 − 𝛼3 × 𝑡 − 𝜂0 ))] ,
𝑢 (𝑥, 𝑡)
2 𝛼 − 𝛼3
𝑙 = 2 .
𝛼1 − 𝛼3 = [𝜏0 + 𝜏1 𝛼1 + (𝜏1 (𝛼2 − 𝛼1 ))
(21)
Also 𝛼1 , 𝛼2 , and 𝛼3 are the roots of the polynomial equation √𝛼2 − 𝛼1

× (𝑠𝑛2 (±
𝐴
𝜉2 2 𝜉1 𝜉
Γ3 + Γ + Γ + 0 = 0. (22) × (𝑥 −
𝑎𝑛 (5 + 5𝑚 − 7𝑛) 𝜏0
𝜉3 𝜉3 𝜉3
(1 + 𝑚) (1 + 𝑚 − 𝑛) (1 + 𝑚 + 𝑛)
Substituting solutions (20) into (5) and (13), we have
× 𝑡 − 𝜂0 ) ,
𝑢 (𝑥, 𝑡)
1/(𝑚−𝑛+1)
𝛼1 − 𝛼3 −1
= [𝜏0 + 𝜏1 𝛼1 )) ] .
𝛼1 − 𝛼2
(23)
+ (𝐴2 𝜏1 ( (𝑥 − (𝑎𝑛 (5 + 5𝑚 − 7𝑛) 𝜏0 )
If we take 𝜏0 = −𝜏1 𝛼1 and 𝜂0 = 0, then solutions (23) can
× ((1 + 𝑚) (1 + 𝑚 − 𝑛) (1 + 𝑚 + 𝑛))−1 reduce to rational function solution
1/(𝑚−𝑛+1) 2/(𝑚−𝑛+1)
2 −1 ̃
𝐴
× 𝑡 − 𝜂0 ) ) )] , 𝑢 (𝑥, 𝑡) = ( ) , (24)
𝑥 − 𝑐𝑡
1-soliton wave solution where 𝐴 1 = √−6𝑏𝑛𝜉0 (1+𝑚)(1+𝑚+𝑛)/𝑎𝜉1 𝜏1 (1+𝑚−𝑛)2. Inte-

𝐵̃ grating (30), we obtain the solutions to (1) as follows:
𝑢 (𝑥, 𝑡) = 2/(𝑚−𝑛+1)
, (25)
cosh (𝐵 (𝑥 − 𝑐𝑡))
singular soliton solution 2𝐴 1
± (𝜂 − 𝜂0 ) = − ,
Γ − 𝛼1
̃
𝐶
𝑢 (𝑥, 𝑡) = , (26)
sinh2/(𝑚−𝑛+1)
(𝐶 (𝑥 − 𝑐𝑡)) 4𝐴 1 Γ − 𝛼2
± (𝜂 − 𝜂0 ) = √ , 𝛼1 > 𝛼2 ,
𝛼1 − 𝛼2 Γ − 𝛼1
and elliptic soliton solution
2𝐴 1 󵄨󵄨 Γ − 𝛼 󵄨󵄨
𝐵̃ 󵄨 1 󵄨󵄨
± (𝜂 − 𝜂0 ) = ln 󵄨󵄨󵄨 󵄨,
𝑢 (𝑥, 𝑡) = , (27) 𝛼1 − 𝛼2 󵄨󵄨 Γ − 𝛼2 󵄨󵄨󵄨
𝑠𝑛2/(𝑚−𝑛+1) (𝜑, 𝑙)
± (𝜂 − 𝜂0 )
where 𝐴 ̃ = 𝐴√𝜏1 , 𝐵̃ = (𝜏1 (𝛼2 − 𝛼1 ))1/(𝑚−𝑛+1) , 𝐵 =
√𝛼2 − 𝛼1 /𝐴, 𝐶̃ = (𝜏1 (𝛼1 − 𝛼2 ))1/(𝑚−𝑛+1) , 𝐶 = √𝛼1 − 𝛼2 /2𝐴, 4𝐴 1
=
𝜑 = ±(√𝛼2 − 𝛼1 /𝐴)(𝑥 − 𝑐𝑡), 𝑙2 = (𝛼1 − 𝛼3 )/(𝛼1 − 𝛼2 ), and √(𝛼1 − 𝛼2 ) (𝛼1 − 𝛼3 )
𝑐 = 𝑎𝑛(5 + 5𝑚 − 7𝑛)𝜏1 𝛼1 /(1 + 𝑚)(1 + 𝑚 − 𝑛)(1 + 𝑚 + 𝑛). Here,
𝐵̃ and 𝐶
̃ are the amplitudes of the solitons, while 𝐵 and 𝐶 are 󵄨󵄨 󵄨
󵄨󵄨 √(Γ − 𝛼2 ) (𝛼1 − 𝛼3 ) − √(Γ − 𝛼3 ) (𝛼1 − 𝛼2 ) 󵄨󵄨󵄨
the inverse widths of the solitons and 𝑐 is the velocity. Thus, 󵄨󵄨 󵄨󵄨
× ln 󵄨󵄨󵄨 󵄨󵄨 ,
we can say that the solitons exist for 𝜏1 > 0. 󵄨󵄨 √(Γ − 𝛼 ) (𝛼 − 𝛼 ) + √(Γ − 𝛼 ) (𝛼 − 𝛼 ) 󵄨󵄨󵄨
󵄨󵄨 2 1 3 3 1 2 󵄨󵄨
Remark 1. If we choose the corresponding values for some 𝛼1 > 𝛼2 > 𝛼3 ,

parameters, solution (25) is in full agreement with solution
(21) mentioned in [17]. 4𝐴 1
± (𝜂 − 𝜂0 ) = 𝐹 (𝜑, 𝑙) ,
Case 2. If we take 𝜖 = 0, 𝛿 = 2, and 𝜃 = 4, then √(𝛼1 − 𝛼4 ) (𝛼2 − 𝛼3 )
2
2 (𝜏1 + 2𝜏2 Γ) (𝜉4 Γ4 + 𝜉3 Γ3 + 𝜉2 Γ2 + 𝜉1 Γ + 𝜉0 ) 𝛼1 > 𝛼2 > 𝛼3 > 𝛼4 ,
(V󸀠 ) = , (28) (31)
𝜁0
where 𝜉4 ≠0 and 𝜁0 ≠0. Respectively, solving the algebraic
equation system (9) yields where
𝜉12 𝜉13
𝜉0 = 𝜉0 , 𝜉1 = 𝜉1 , 𝜉2 = , 𝜉3 = , (Γ − 𝛼2 ) (𝛼1 − 𝛼4 )
3𝜉0 24𝜉02 𝜑1 = arcsin √ ,
(Γ − 𝛼1 ) (𝛼2 − 𝛼4 )
𝜉14 𝑏𝑛 (𝑚 + 1) (𝑚 + 𝑛 + 1) 𝜉13 (32)
𝜉4 = , 𝜁0 = − , (𝛼 − 𝛼3 ) (𝛼2 − 𝛼4 )
576𝜉03 24𝑎(𝑚 − 𝑛 + 1)2 𝜉02 𝜏1 𝑙12 = 1 .
(29) (𝛼2 − 𝛼3 ) (𝛼1 − 𝛼4 )
2𝜉 𝜏 𝜉𝜏
𝜏0 = 0 1 , 𝜏1 = 𝜏1 , 𝜏2 = 1 1 ,
𝜉1 12𝜉0
Also 𝛼1 , 𝛼2 , 𝛼3 , and 𝛼4 are the roots of the polynomial
2𝑎𝑛𝜉0 𝜏1 equation
𝑐=− .
(𝑚 + 1) (𝑚 + 𝑛 + 1) 𝜉1
Substituting these resultss into (6) and (10), we get 𝜉3 3 𝜉2 2 𝜉1 𝜉
Γ4 + Γ + Γ + Γ + 0 = 0. (33)
𝜉4 𝜉4 𝜉4 𝜉4
± (𝜂 − 𝜂0 )
Substituting solutions (31) into (5) and (13), we have

= 2𝐴 1 ∫ ( (𝑑Γ)
𝑢 (𝑥, 𝑡)
424𝜉0 3 192𝜉2
× (Γ + ( ) Γ + ( 2 0 ) Γ2
𝜉1 𝜉1
= [𝜏0 + 𝜏1 𝛼1 ± (2𝜏1 𝐴 1 )
−1/2
576𝜉3 576𝜉4
+ ( 3 0 ) Γ + ( 4 0 )) ),
𝜉1 𝜉1 2𝑎𝑛𝜉0 𝜏1 −1
× (𝑥 + 𝑡−𝜂0 )
(30) (𝑚+1) (𝑚+𝑛+1) 𝜉1
𝑢 (𝑥, 𝑡)
+ 𝜏2 (𝛼1 ± (2𝐴 1 )
2𝑎𝑛𝜉0 𝜏1 = [𝜏0 + 𝜏1 𝛼1 + ((𝛼1 − 𝛼2 ) 𝜏1 )

× (𝑥 + [
(𝑚 + 1) (𝑚 + 𝑛 + 1) 𝜉1
𝛼1 − 𝛼2
−1 2 1/(𝑚−𝑛+1) × ( exp [
2𝐴 1
× 𝑡 − 𝜂0 ) ) ] ,
× (𝑥+(2𝑎𝑛𝜉0 𝜏1 )
𝑢 (𝑥, 𝑡) −1
× ((𝑚+1) (𝑚+𝑛+1)𝜉1)
−1
= [𝜏0 + 𝜏1 𝛼1 × 𝑡 − 𝜂0 ) ] − 1)
[
+ (16𝐴21 (𝛼2 − 𝛼1 ) 𝜏1 ) + 𝜏2 (𝛼1 + (𝛼1 − 𝛼2 )
× (16𝐴21 − [ (𝛼1 − 𝛼2 ) 𝛼1 − 𝛼2
× ( exp [
2𝐴 1
2 −1
2𝑎𝑛𝜉0 𝜏1
× (𝑥+ 𝑡−𝜂0)] ) × (𝑥 + (2𝑎𝑛𝜉0 𝜏1 )
(𝑚+1)(𝑚+𝑛+1) 𝜉1
−1
× ((𝑚+1) (𝑚+𝑛+1) 𝜉1)
+ 𝜏2 (𝛼1 + (16𝐴21 (𝛼2 −𝛼1 )) 2 1/(𝑚−𝑛+1)
−1
× 𝑡 − 𝜂0 ) ] − 1) ) ] ,
× (16𝐴21 − [ (𝛼1 − 𝛼2 )
𝑢 (𝑥, 𝑡)
× (𝑥 +
(𝑚 + 1) (𝑚 + 𝑛 + 1) 𝜉1 [
= [𝜏0 + 𝜏1 𝛼1 − (2 (𝛼1 − 𝛼2 ) (𝛼1 − 𝛼3 ) 𝜏1 )
2 1/(𝑚−𝑛+1)
2 −1 [
× 𝑡−𝜂0)] ) ) ] ,
]
× (2𝛼1 − 𝛼2 − 𝛼3 + (𝛼3 − 𝛼2 )
𝑢 (𝑥, 𝑡)
= [𝜏0 + 𝜏1 𝛼2 + ((𝛼2 − 𝛼1 ) 𝜏1 ) [ √(𝛼1 − 𝛼2 ) (𝛼1 − 𝛼3 )

× cosh [
[ 2𝐴 1
𝛼 − 𝛼2 [
× (exp [ 1
2𝐴 1 2𝑎𝑛𝜉0 𝜏1
× (𝑥 +
2𝑎𝑛𝜉0 𝜏1 −1 (𝑚 + 1) (𝑚 + 𝑛 + 1) 𝜉1
× (𝑥+ 𝑡 − 𝜂0)]− 1)
(𝑚 + 1)(𝑚 + 𝑛 + 1) 𝜉1 −1
]
× 𝑡 − 𝜂0 ) ])
+ 𝜏2 (𝛼2 + (𝛼2 − 𝛼1 )
]
𝛼1 − 𝛼2
× (exp [
2𝐴 1
+ 𝜏2 (𝛼1 − (2 (𝛼1 − 𝛼2 ) (𝛼1 − 𝛼3 ))
× (𝑥 +
(𝑚+1) (𝑚+𝑛+1) 𝜉1
2 1/(𝑚−𝑛+1)
−1
× (2𝛼1 − 𝛼2 − 𝛼3 + (𝛼3 − 𝛼2 )
× 𝑡−𝜂0 ) ]−1) ) ] ,
]
[ √(𝛼1 − 𝛼2 ) (𝛼1 − 𝛼3 )
× cosh [ × 𝑡 − 𝜂0 ) ,
2𝐴 1
[
2𝑎𝑛𝜉0 𝜏1 (𝛼2 − 𝛼3 ) (𝛼1 − 𝛼4 )

× (𝑥 + )
(𝑚 + 1) (𝑚 + 𝑛 + 1) 𝜉1 (𝛼1 − 𝛼3 ) (𝛼2 − 𝛼4 )
2 1/(𝑚−𝑛+1)
1/(𝑚−𝑛+1) −1
−1 2
]
] ] +𝛼4 − 𝛼2 ) ) ]
] .
]
× 𝑡 − 𝜂0 )]
]) ) ] ,
] ]
] ] (34)
𝑢 (𝑥, 𝑡) For simplicity, if we take 𝜂0 = 0, then we can write
solutions (34) as follows:
[ 1/(𝑚−𝑛+1)
=[
[𝜏0 + 𝜏1 𝛼2 + ((𝛼1 − 𝛼2 ) (𝛼4 − 𝛼2 ) 𝜏1 )
2
2𝐴 1 𝑖
𝑢 (𝑥, 𝑡) = [∑𝜏𝑖 (𝛼1 ± )] ,
𝑖=0 𝑥 − 𝑐𝑡
[
𝑢 (𝑥, 𝑡)
× ( (𝛼1 − 𝛼4 )
2
= [∑𝜏𝑖 (𝛼1 + (16𝐴21 (𝛼1 − 𝛼2 ))
𝑖=0
√(𝛼1 − 𝛼3 ) (𝛼2 − 𝛼4 )
2 1/(𝑚−𝑛+1)
× 𝑠𝑛 ( 𝑖
4𝐴 1 2 −1
×(16𝐴21 −[(𝛼1 − 𝛼2 ) (𝑥 − 𝑐𝑡)] ) ) ] ,
× (𝑥 + 2 𝑖 1/(𝑚−𝑛+1)
(𝑚 + 1) (𝑚 + 𝑛 + 1) 𝜉1 𝛼2 − 𝛼1
𝑢 (𝑥, 𝑡) = [∑𝜏𝑖 (𝛼2 + )] ,
𝑖=0 exp [𝐵1 (𝑥 − 𝑐𝑡)] − 1
× 𝑡 − 𝜂0 ) ,
2 𝑖 1/(𝑚−𝑛+1)
𝛼1 − 𝛼2
𝑢 (𝑥, 𝑡) = [∑𝜏𝑖 (𝛼1 + )] ,
(𝛼2 − 𝛼3 ) (𝛼1 − 𝛼4 ) 𝑖=0 exp [𝐵1 (𝑥 − 𝑐𝑡)] − 1
) 2
(𝛼1 − 𝛼3 ) (𝛼2 − 𝛼4 )
𝑢 (𝑥, 𝑡) = [∑𝜏𝑖 (𝛼1 − (2 (𝛼1 − 𝛼2 ) (𝛼1 − 𝛼3 ))
𝑖=0
−1
× (2𝛼1 − 𝛼2 − 𝛼3 + (𝛼3 − 𝛼2 )
+ 𝛼4 − 𝛼2 )
1/(𝑚−𝑛+1)
−1 𝑖
×cosh [𝐶1 (𝑥−𝑐𝑡)]) ) ] ,
+ 𝜏2 (𝛼2 + ((𝛼1 − 𝛼2 ) (𝛼4 − 𝛼2 ) 𝜏1 ) 2

𝑢 (𝑥, 𝑡) = [∑𝜏𝑖 (𝛼2 + ((𝛼1 − 𝛼2 ) (𝛼4 − 𝛼2 ))
𝑖=0
× ((𝛼1 − 𝛼4 ) 𝑠𝑛2 (𝜑, 𝑙)

× ( (𝛼1 − 𝛼4 )
1/(𝑚−𝑛+1)
−1 𝑖
+ 𝛼4 − 𝛼2 ) ) ] ,
√(𝛼1 − 𝛼3 ) (𝛼2 − 𝛼4 )
× 𝑠𝑛2 ( (35)
4𝐴 1
where 𝐵1 = (𝛼1 − 𝛼2 )/2𝐴 1 , 𝐶1 = √(𝛼1 − 𝛼2 )(𝛼1 − 𝛼3 )/2𝐴 1 ,
𝜑1 = (√(𝛼1 − 𝛼3 )(𝛼2 − 𝛼4 )/4𝐴 1 )(𝑥 − 𝑐𝑡), 𝑙12 = (𝛼2 − 𝛼3 )(𝛼1 −
2𝑎𝑛𝜉0 𝜏1 𝛼4 )/(𝛼1 − 𝛼3 )(𝛼2 − 𝛼4 ), and 𝑐 = −2𝑎𝑛𝜉0 𝜏1 /(𝑚 + 1)(𝑚 + 𝑛 + 1)𝜉1 .
× (𝑥 +
(𝑚 + 1) (𝑚 + 𝑛 + 1) 𝜉1 Here, 𝐴 1 is the amplitude of the soliton, while 𝑐 is the velocity
and 𝐵1 and 𝐶1 are the inverse widths of the solitons. Thus, we where 𝐴 2 = √−𝑏𝑛(1+𝑚)(1+𝑚+𝑛)/2𝑎𝜏3 (1+𝑚−𝑛)2 . Integrat-
can say that the solitons exist for 𝜏1 > 0. ing (38), we obtain the solutions to (1) as follows:
Case 3. If we take 𝜖 = 0, 𝛿 = 3, and 𝜃 = 5, then 2𝐴 2
± (𝜂 − 𝜂0 ) = − ,
√(Γ − 𝛼1 )3
2 2
(V󸀠 ) = (𝜏1 + 2𝜏2 Γ + 3𝜏3 Γ2 )
3𝐴 2 arctanh [√(Γ − 𝛼2 ) / (𝛼1 − 𝛼2 )]
× (𝜉5 Γ5 + 𝜉4 Γ4 + 𝜉3 Γ3 + 𝜉2 Γ2 + 𝜉1 Γ + 𝜉0 ) (36) ± (𝜂 − 𝜂0 ) = 3/2
(𝛼1 − 𝛼2 )
−1
× (𝜁0 ) , 3𝐴 2 √Γ − 𝛼2
− , 𝛼1 > 𝛼2 ,
(𝛼1 − 𝛼2 ) (Γ − 𝛼1 )
equation system (9) yields 6𝐴 2 arctan [√(Γ − 𝛼1 ) / (𝛼1 − 𝛼2 )]
± (𝜂 − 𝜂0 ) = − 3/2
(𝛼1 − 𝛼2 )
6𝐴 2
3 − ,
𝜉5 (𝜏22 − 4𝜏1 𝜏3 ) (2𝜏23 − 9𝜏1 𝜏2 𝜏3 + 2√(𝜏22 − 3𝜏1 𝜏3 ) ) √Γ − 𝛼1 (𝛼1 − 𝛼2 )
𝜉0 = ,
81𝜏35
3 6𝐴 2 arctanh [√(Γ − 𝛼3 ) / (𝛼2 − 𝛼3 )]
𝜉5 (4𝜏24 + 9𝜏1 𝜏22 𝜏3 −108𝜏12 𝜏32 +4𝜏2 √(𝜏22 − 3𝜏1 𝜏3 ) ) ± (𝜂 − 𝜂0 ) =
𝜉1 = − , 𝛼1 − 𝛼2
81𝜏34
3 1 1
𝜉5 (−11𝜏23 + 63𝜏1 𝜏2 𝜏3 − 2√(𝜏22 − 3𝜏1 𝜏3 ) ) ×( − ),
√𝛼2 − 𝛼3 √𝛼1 − 𝛼3
𝜉2 = ,
27𝜏33
𝛼1 > 𝛼2 > 𝛼3 ,
𝜉5 (𝜏22 + 7𝜏1 𝜏3 ) 5𝜉5 𝜏2 ± (𝜂 − 𝜂0 )
𝜉3 = , 𝜉4 = ,
3𝜏32 3𝜏3
−6𝐴 2
=
9𝑏𝑛 (𝑚 + 1) (𝑚 + 𝑛 + 1) 𝜉5 √Γ − 𝛼1 (𝛼1 − 𝛼2 ) (𝛼1 − 𝛼3 )
𝜁0 = − ,
2𝑎𝜏3 (𝑚 − 𝑛 + 1)2
× [√(Γ − 𝛼2 ) (Γ − 𝛼3 ) + 𝑖 (𝐸 (𝜑, 𝑙) − 𝐹 (𝜑, 𝑙))] ,
3
2𝜏23 − 9𝜏1 𝜏2 𝜏3 + 2√(𝜏22 − 3𝜏1 𝜏3 )
𝜏0 = − , (39)
27𝜏32
where
𝜑
𝜏1 = 𝜏1 , 𝜏2 = 𝜏2 ,
𝐸 (𝜑, 𝑙) = ∫ √1 − 𝑙2 sin2 𝜓 𝑑𝜓,
0
𝜏3 = 𝜏3 , 𝜉5 = 𝜉5 ,
Γ − 𝛼1
8𝑎𝑛√(𝜏22 − 3𝜏1 𝜏3 )
3 𝜑2 = − arcsin √ ,
𝛼2 − 𝛼1
𝑐=− .
27 (𝑚 + 1) (𝑚 + 𝑛 + 1) 𝜏32
𝛼1 − 𝛼2
(37) 𝑙22 = , (40)
𝛼1 − 𝛼3
Substituting these results into (6) and (10), we get ± (𝜂 − 𝜂0 )

−6𝑖𝐴 2
± (𝜂 − 𝜂0 ) = (𝐹 (𝜑, 𝑙) − 𝜋 (𝜑, 𝑛, 𝑙)) ,
√𝛼2 − 𝛼3 (𝛼1 − 𝛼2 )
= 3𝐴 2 ∫ ( (𝑑Γ) 𝛼1 > 𝛼2 > 𝛼3 > 𝛼4 ,

where
𝜉4 4 𝜉3 3 (38)
× (Γ5 + Γ + Γ 𝛼3 − 𝛼2 𝛼2 − 𝛼4
𝜉5 𝜉5 𝜑3 = − arcsin √ , 𝑙32 = ,
Γ − 𝛼2 𝛼2 − 𝛼3
(41)
𝜉 𝜉 𝜉 −1/2 𝛼 − 𝛼1
+ 2 Γ2 + 1 Γ + 0 ) ) , 𝑛= 2 .
𝜉5 𝜉5 𝜉5 𝛼2 − 𝛼3
Also 𝛼1 , 𝛼2 , 𝛼3 , 𝛼4 , and 𝛼5 are the roots of the polynomial where 𝐴 3 = √𝑏𝑛(1 + 𝑚)(1 + 𝑚 + 𝑛)/2𝑎𝜏1 (1 + 𝑚 − 𝑛)2 . Inte-
equation grating (45), we obtain the solution to (1) as follows:
𝜉4 4 𝜉3 3 𝜉2 2 𝜉1 𝜉
Γ5 + Γ + Γ + Γ + Γ + 0 = 0. (42) 𝜁1
𝜉5 𝜉5 𝜉5 𝜉5 𝜉5 ± (𝜂 − 𝜂0 ) = −𝐴 3 √
𝜁0 + 𝜁1 𝛼1
Case 4. If we take 𝜖 = 1, 𝛿 = 1, and 𝜃 = 4, then
𝜁0 + 𝜁1 Γ
× arctanh [√ ]
2 𝜏12 4 3 2
(𝜉4 Γ + 𝜉3 Γ + 𝜉2 Γ + 𝜉1 Γ + 𝜉0 ) 𝜁0 + 𝜁1 𝛼1
(V󸀠 ) = , (43)
𝜁0 + 𝜁1 Γ
𝐴3 𝜁 +𝜁 Γ
− √ 0 1 ,
where 𝜉4 ≠0 and 𝜁1 ≠0. Respectively, solving the algebraic Γ − 𝛼1 𝜁1
equation system (9) yields
2𝐴 3
± (𝜂 − 𝜂0 ) =
𝜁0 𝜏02 2
(𝑀 + 2𝑎(1 + 𝑚 − 𝑛) (2𝜁1 𝜏0 + 𝜁0 𝜏1 )) 𝛼1 − 𝛼2
𝜉0 = ,
𝑏𝑛 (1 + 𝑚) (1 + 𝑚 + 𝑛) 𝜁1 𝜏12 𝜁0 + 𝜁1 𝛼1
× (−√
𝜉3 = 𝜉3 , 𝜁1
2𝑎(1 + 𝑚 − 𝑛)2 𝜁1 𝜏1 𝜁0 + 𝜁1 Γ (46)

𝜉4 = − , × arctanh [√ ]
𝑏𝑛 (1 + 𝑚) (1 + 𝑚 + 𝑛) 𝜁0 + 𝜁1 𝛼1
𝜉1 = (𝜏0 (4𝑎(1 + 𝑚 − 𝑛)2 𝜁12 𝜏02
+ √𝜁0 + 𝜁1 𝛼2
+ 2𝜁0 𝜏1 (𝑀 + 2𝑎(1 + 𝑚 − 𝑛)2 𝜁0 𝜏1 )
𝜁0 + 𝜁1 Γ
2 × arctanh [√ ]) ,
+𝜁1 𝜏0 (𝑀 + 8𝑎(1 + 𝑚 − 𝑛) 𝜁0 𝜏1 ))) 𝜁0 + 𝜁1 𝛼2
−1 ± (𝜂 − 𝜂0 ) = 2𝐴 3
× (𝑏𝑛 (1 + 𝑚) (1 + 𝑚 + 𝑛) 𝜁1 𝜏12 ) , (44)
𝜉2 = (6𝑎(1 + 𝑚 − 𝑛)2 𝜁12 𝜏02 (Γ − 𝛼1 ) (𝜁0 + 𝜁1 Γ)
× (√ 2
𝜁1 (Γ − 𝛼2 )
+ 2𝜁1 𝜏0 (𝑀 + 2𝑎(1 + 𝑚 − 𝑛)2 𝜁0 𝜏1 )
+𝜁0 𝜏1 (𝑀 + 2𝑎(1 + 𝑚 − 𝑛)2 𝜁0 𝜏1 )) +𝑖√𝛼1 −𝛼2 (𝐸 (𝜑, 𝑙)−𝐹 (𝜑, 𝑙)) ) ,

−1
× (𝑏𝑛 (1 + 𝑚) (1 + 𝑚 + 𝑛) 𝜁1 𝜏12 ) ,
where
𝜁0 = 𝜁0 , 𝜁1 = 𝜁1 ,
𝜁1 (𝛼1 − Γ)
𝜏0 = 𝜏0 , 𝜏1 = 𝜏1 , 𝜑4 = − arcsin √ ,
𝜁0 + 𝜁1 𝛼1
𝑛 (𝑀 + 2𝑎(1 + 𝑚 − 𝑛)2 (3𝜁1 𝜏0 + 𝜁0 𝜏1 )) 𝜁0 + 𝜁1 𝛼1
𝑐= 2
, 𝑙42 = , (47)
(1 + 𝑚) (1 + 𝑚 + 𝑛) (1 + 𝑚 − 𝑛) 𝜁1 𝜁1 (𝛼1 − 𝛼2 )
where 𝑀 = 𝑏𝑛(1 + 𝑚)(1 + 𝑚 + 𝑛)𝜉3 . Substituting these results −2𝐴 3

± (𝜂 − 𝜂0 ) = 𝐸 (𝜑, 𝑙) ,
into (6) and (10), we get √𝛼2 − 𝛼1
± (𝜂 − 𝜂0 ) where
𝜁0 𝛼2 − 𝛼1
= 𝐴 3 ∫ ((Γ + ) 𝜑5 = arcsin [√ ],
𝜁1 Γ − 𝛼1
𝜉3 3 𝜉 (45) 𝜁0 + 𝜁1 𝛼1
× (Γ4 + ( ) Γ + ( 2 ) Γ2 𝑙52 = ,
𝜉4 𝜉4 𝜁1 (𝛼1 − 𝛼2 )
−1 1/2 −2𝑖𝐴 3
𝜉1 𝜉 ± (𝜂 − 𝜂0 ) =
+ ( ) Γ + ( 0 )) ) 𝑑Γ, (𝛼1 − 𝛼2 ) √𝜁1 (𝜁0 + 𝜁1 𝛼2 )
𝜉4 𝜉4
× (𝜁0 (𝐹 (𝜑, 𝑙) − 𝜋 (𝜑, 𝑛, 𝑙)) where 𝐴 4 = √−2𝑏𝑛(1 + 𝑚)(1 + 𝑚 + 𝑛)/𝑎𝜏22 (1 + 𝑚 − 𝑛)2 .

+𝜁1 (𝛼2 𝐹 (𝜑, 𝑙) − 𝛼2 𝜋 (𝜑, 𝑛, 𝑙))) , Integrating (52), we obtain the solution to (1) as follows:
(48)
−2𝐴 4 𝜁 + 𝜁 Γ 3/2
± (𝜂 − 𝜂0 ) = ( 0 1 ) ,
where 3√𝜁1 (𝜁0 + 𝜁1 𝛼1 ) Γ − 𝛼1
𝜁0 + 𝜁1 𝛼2 ± (𝜂 − 𝜂0 )
𝜑6 = − arcsin √ ,
𝜁1 (𝛼2 − Γ) −𝐴 4 (𝜁0 + 𝜁1 𝛼2 )
(49) =
3/2
𝜁1 (𝛼2 − 𝛼3 ) 𝜁1 (𝛼2 − 𝛼1 ) 2(𝛼1 − 𝛼2 ) √𝜁1 (𝜁0 + 𝜁1 𝛼1 )
𝑙62 = , 𝑛1 = .
𝜁0 + 𝜁1 𝛼2 𝜁0 + 𝜁1 𝛼2 󵄨
× ln 󵄨󵄨󵄨󵄨 (Γ − 𝛼1 )
Case 5. If we take 𝜖 = 1, 𝛿 = 2, and 𝜃 = 5, then
2 × (𝜁0 (Γ + 𝛼1 − 2𝛼2 )
󸀠 2
(𝜏1 + 2𝜏2 Γ) (𝜉5 Γ5 + 𝜉4 Γ4 + 𝜉3 Γ3 + 𝜉2 Γ2 + 𝜉1 Γ + 𝜉0 )
(V ) = ,
𝜁0 + 𝜁1 Γ +2√(𝜁0 +𝜁1 Γ) (𝜁0 +𝜁1 𝛼1 ) (Γ−𝛼2 ) (𝛼1 −𝛼2 )
(50) −1 󵄨
+𝜁1 (2Γ𝛼1 − 𝛼2 (Γ + 𝛼1 ))) 󵄨󵄨󵄨󵄨
equation system (9) yields 𝐴4 (𝜁 +𝜁 Γ) (Γ−𝛼2 )
− √ 0 1 , 𝛼1 > 𝛼2 ,
𝜏2 (−2𝜉5 𝜏1 + 𝜉4 𝜏2 ) (𝛼1 −𝛼2 ) (Γ−𝛼1 ) 𝜁1
𝜉0 = 0 ,
𝜏23 ± (𝜂 − 𝜂0 )
𝜏0 (2𝜉4 𝜏1 𝜏2 + 𝜉5 (−4𝜏12 + 𝜏0 𝜏2 )) −2𝐴 4 𝜁 +𝜁 Γ
𝜉1 = , = √ 0 1
𝜏23 (𝛼1 − 𝛼2 ) 𝜁1 (Γ − 𝛼1 )
𝜉4 𝜏2 (𝜏12 + 2𝜏0 𝜏2 ) − 2𝜉5 (𝜏13 + 𝜏0 𝜏1 𝜏2 ) 2𝐴 4 𝜁0 + 𝜁1 𝛼2
𝜉2 = , − √
𝜏23 3/2 𝜁1
(𝛼1 − 𝛼2 )
−3𝜉5 𝜏12 + 2𝜏2 (𝜉5 𝜏0 + 𝜉4 𝜏1 )
𝜉3 = , (Γ − 𝛼1 ) (𝜁0 + 𝜁1 𝛼2 )
𝜏22 × arctan [√ ],
(51) (𝛼1 − 𝛼2 ) (𝜁0 + 𝜁1 Γ)
−2𝑏𝑛 (1 + 𝑚) (1 + 𝑚 + 𝑛) (𝜉4 𝜏2 − 2𝜉5 𝜏1 )
𝜁0 = , ± (𝜂 − 𝜂0 )
𝑎(1 + 𝑚 − 𝑛)2 𝜏22
−2𝑏𝑛 (1 + 𝑚) (1 + 𝑚 + 𝑛) 𝜉5 −𝐴 4 𝜁 +𝜁 𝛼
𝜁1 = , = √ 0 1 2
𝑎(1 + 𝑚 − 𝑛)2 𝜏22 𝛼1 − 𝛼3 𝜁1 (𝛼2 − 𝛼3 )
󵄨
𝜉4 = 𝜉4 , 𝜉5 = 𝜉5 , × ln 󵄨󵄨󵄨󵄨 (𝛼2 − Γ)
𝜏0 = 𝜏0 , 𝜏1 = 𝜏1 , 𝜏2 = 𝜏2 , × (𝜁0 (Γ + 𝛼2 − 2𝛼3 )
𝑎𝑛 (𝜏12 − 4𝜏0 𝜏2 )
𝑐=− .
2 (1 + 𝑚) (1 + 𝑚 + 𝑛) 𝜏2 +2√(𝜁0 +𝜁1 Γ) (𝜁0 +𝜁1 𝛼2 ) (Γ−𝛼3 ) (𝛼2 −𝛼3 )
−1 󵄨
+𝜁1 (2Γ𝛼2 − 𝛼3 (Γ + 𝛼2 ))) 󵄨󵄨󵄨󵄨
Substituting these results into (6) and (10), we get
± (𝜂 − 𝜂0 ) = 𝐴 4
𝐴4
−
𝜁0 𝛼1 − 𝛼3
× ∫ ( (Γ + )
𝜁1
𝜁0 + 𝜁1 𝛼1
×√
𝜉 𝜉 (52) 𝜁1 (𝛼1 − 𝛼3 )
× (Γ5 + 4 Γ4 + 3 Γ3
𝜉5 𝜉5 󵄨
× ln 󵄨󵄨󵄨󵄨 (𝜁0 (Γ + 𝛼1 − 2𝛼3 )
1/2
𝜉2 2 𝜉1 𝜉 −1
+ Γ + Γ + 0) ) 𝑑Γ,
𝜉5 𝜉5 𝜉5 + 2√(𝜁0 + 𝜁1 Γ) (𝜁0 + 𝜁1 𝛼1 ) (Γ − 𝛼3 ) (𝛼1 − 𝛼3 )
+ 𝜁1 (2Γ𝛼1 − 𝛼3 (Γ + 𝛼1 ))) Here, 𝜏𝑖 (𝑖 = 0, . . . , 𝛿), 𝜔𝑗 (𝑗 = 0, . . . , 𝜇), 𝜉𝜍 (𝜍 = 0, . . . , 𝜃), and

−1 󵄨󵄨 𝜁𝜎 (𝜎 = 0, . . . , 𝜖) are the constants to be specified.
× (Γ − 𝛼2 ) 󵄨󵄨󵄨 , 𝛼1 > 𝛼2 > 𝛼3 ,
Step 2. Taking trial equations (56) and (57), we derive the
−2𝐴 4
± (𝜂 − 𝜂0 ) = following equations:
𝛼1 − 𝛼3
󸀠 󸀠 2
𝜁 +𝜁 𝛼 2 Φ (Γ) (𝐴 (Γ) 𝐵 (Γ) − 𝐴 (Γ) 𝐵 (Γ))
× √ 0 1 3 𝐸 (𝜑, 𝑙) , (𝑢󸀠 ) = , (58)
𝜁1 (𝛼1 − 𝛼2 ) Ψ (Γ) 𝐵4 (Γ)
𝑢󸀠󸀠 = (𝐴󸀠 (Γ) 𝐵 (Γ) − 𝐴 (Γ) 𝐵󸀠 (Γ))

𝛼1 > 𝛼2 > 𝛼3 ,
(53) × {(Φ󸀠 (Γ) Ψ (Γ) − Φ (Γ) Ψ󸀠 (Γ)) 𝐵 (Γ)
where
−4Φ (Γ) Ψ (Γ) 𝐵󸀠 (Γ)}
(Γ − 𝛼3 ) (𝛼2 − 𝛼1 )
𝜑7 = arcsin √ , + 2Φ (Γ) Ψ (Γ) 𝐵 (Γ) (𝐴󸀠󸀠 (Γ) 𝐵 (Γ) − 𝐴 (Γ) 𝐵󸀠󸀠 (Γ))
(Γ − 𝛼1 ) (𝛼2 − 𝛼3 )
−1
(𝛼3 − 𝛼2 ) (𝜁0 + 𝜁1 𝛼1 ) × (2𝐵3 (Γ) Ψ2 (Γ))
𝑙72 = ,
(𝛼1 − 𝛼2 ) (𝜁0 + 𝜁1 𝛼3 ) (59)
± (𝜂 − 𝜂0 ) and other derivation terms such as 𝑢󸀠󸀠󸀠 .
2𝐴 4 (𝛼2 − 𝛼4 ) (54)
Step 3. Substituting 𝑢󸀠 , 𝑢󸀠󸀠 , and other derivation terms into
=
(𝛼1 − 𝛼2 ) (𝛼3 − 𝛼4 ) √𝜁1 (𝛼2 − 𝛼3 ) (𝜁0 + 𝜁1 𝛼4 ) (5) yields the following equation:
(𝜁 + 𝜁 Γ) (𝛼3 − 𝛼4 ) Ω (Γ) = 󰜚𝑠 Γ𝑠 + ⋅ ⋅ ⋅ + 󰜚1 Γ + 󰜚0 = 0. (60)

×( 0 1 𝜋 (𝜑, 𝑛, 𝑙)
𝛼1 − 𝛼2 According to the balance principle we can determine a
relation of 𝜃, 𝜖, 𝛿, and 𝜇.
(𝜁0 + 𝜁1 𝛼2 ) (𝛼4 − 𝛼3 )
+ 𝐹 (𝜑, 𝑙)) ,
𝛼2 − 𝛼4 Step 4. Letting the coefficients of Ω(Γ) all be zero will yield
where an algebraic equations system 󰜚𝑖 = 0 (𝑖 = 0, . . . , 𝑠). Solving
this equations system, we will determine the values 𝜏0 , . . . 𝜏𝛿 ;
(Γ − 𝛼3 ) (𝛼2 − 𝛼1 ) 𝜔0 , . . . , 𝜔𝜇 ; 𝜉0 , . . . , 𝜉𝜃 ; and 𝜁0 , . . . , 𝜁𝜖 .
𝜑8 = arcsin √ ,
(Γ − 𝛼1 ) (𝛼2 − 𝛼3 )
Step 5. Substituting the results obtained in Step 4 into (57)
(𝛼 − 𝛼2 ) (𝜁0 + 𝜁1 𝛼1 ) and integrating (57), we can find the exact solutions of (3).
𝑙82 = 3 ,
(𝛼1 − 𝛼2 ) (𝜁0 + 𝜁1 𝛼3 ) (55)
5. Conclusions and Remarks
(𝛼 − 𝛼2 ) (𝛼3 − 𝛼4 )
𝑛2 = − 1 , In this study, we proposed an extended trial equation method
(𝛼2 − 𝛼3 ) (𝛼1 − 𝛼4 )
and used it to obtain some soliton and elliptic function
𝛼1 > 𝛼2 > 𝛼3 > 𝛼4 . solutions to the generalized 𝐾(𝑚, 𝑛) equation. Otherwise,
we discussed a more general trial equation method. The
4. Discussion proposed method can also be applied to other nonlinear
differential equations with nonlinear evolution.
Thus we introduce a more general extended trial equation
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http://dx.doi.org/10.1155/2013/562140
Research Article
Numerical Solution of the Fractional Partial
Differential Equations by the Two-Dimensional
Fractional-Order Legendre Functions
Fukang Yin, Junqiang Song, Yongwen Wu, and Lilun Zhang

College of Computer, National University of Defense Technology, Changsha 410073, China
Correspondence should be addressed to Fukang Yin; yinfukang@nudt.edu.cn
Received 13 May 2013; Revised 8 September 2013; Accepted 8 September 2013
Copyright © 2013 Fukang Yin et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs). The
basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-
order Legendre functions (2D-FLFs). The operational matrices of integration and derivative for 2D-FLFs are first derived. Then,
by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs
coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.
1. Introduction properties of fractional calculus. Recently, Rida and Yousef

[27] presented a fractional extension of the classical Legendre
Fractional partial differential equations play a significant role polynomials by replacing the integer order derivative in Rod-
in modeling physical and engineering processes. Therefore, rigues formula with fractional order derivatives. The defect is
there is an urgent need to develop efficient and fast con- that the complexity of these functions made them unsuitable
vergent methods for FPDEs. Recently, several different tech- for solving FDEs. Subsequently, Kazem et al. [28] presented
niques, including Adomian’s decomposition method (ADM) the orthogonal fractional order Legendre functions based on
[1, 2], homotopy perturbation method (HPM) [3–5], varia- shifted Legendre polynomials to find the numerical solution
tional iteration method (VIM) [6–8], spectral methods [9– of FDEs and drew a conclusion that their method is accurate,
13], orthogonal polynomials method [14–17], and wavelets effective, and easy to implement.
method [18–21] have been presented and applied to solve Benefiting from their “exponential-convergence” prop-
FPDEs. erty when smooth solutions are involved, spectral methods
The method based on the orthogonal functions is a won- have been widely and effectively used for the numerical solu-
derful and powerful tool for solving the FDEs and has enjoyed tion of partial differential equations. The basic idea of spectral
many successes in this realm. The operational matrix of frac- methods is to expand a function into sets of smooth global
tional integration has been determined for some types of functions, called the trial functions. Because of their special
orthogonal polynomials, such as Chebyshev polynomials properties, the orthogonal polynomials are usually chosen to
[16], Legendre polynomials [22], Laguerre polynomials [23– be trial functions. Spectral methods can obtain very accurate
25], and Jacobi polynomials [26]. Moreover, the operational approximations for a smooth solution while only need a few
matrix of fractional derivative for Chebyshev polynomials degrees of freedom. Recently, Chebyshev spectral method [9],
[9] and Legendre polynomials [9, 14] also has been derived. Legendre spectral method [10], and adaptive pseudospectral
However, since these polynomials using integer power series method [11] were proposed for solving fractional boundary
to approximate fractional ones, it cannot accurately represent value problems. Moreover, generalized Laguerre spectral
algorithms and Legendre spectral Galerkin method were Lemma 4. Let 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ 𝑁, 𝑡 > 0, ℎ ∈ 𝐶𝜇𝑛 , 𝜇 ≥ −1.
developed by Baleanu et al. [12] and Bhrawy and Alghamdi Then
[13] for fractional initial value problems, respectively.
𝑛−1
Motivated and inspired by the ongoing research in or- 𝑡𝑘
thogonal polynomials methods and spectral methods, we (𝐽𝛼 𝐷𝛼 ) ℎ (𝑡) = ℎ (𝑡) − ∑ ℎ(𝑘) (0+ ) . (3)
𝑘=0
𝑘!
construct two-dimensional fractional-order Legendre func-
tions and derive the operational matrices of integration and
derivative for the solution of FPDEs. To the best of the 2.2. Fractional-Order Legendre Functions. In this section,
authors’ knowledge, such approach has not been employed we introduce the fractional-order Legendre functions which
for solving FPDEs. were first proposed by Kazem et al. [28]. The normalized
The rest of the paper is organized as follows. In Section 2, eigenfunctions problem for FLFs is
we introduce some mathematical preliminaries of the frac- 󸀠
tional calculus theory and fractional-order Legendre func- ((𝑥 − 𝑥1+𝛼 ) 𝐿󸀠𝛼 2
𝑖 (𝑥)) + 𝛼 𝑖 (𝑖 + 1) 𝑥
𝛼−1 𝛼
𝐿 𝑖 (𝑥) = 0,
tions. In Section 3, a basis of 2D-FLFs is defined and some (4)
properties are given. Section 4 is devoted to the operational 𝑥 ∈ (0, 1) ,
matrices of fractional derivative and integration for 2D-FLFs.
Some numerical examples are presented in Section 5. Finally, which is a singular Sturm-Liouville problem. The fractional-
we conclude the paper with some remarks. order Legendre polynomials, denoted by FL𝛼𝑖 (𝑥), are defined
on the interval [0, 1] and can be determined with the aid of
following recurrence formulae:
2. Preliminaries and Notations
FL𝛼0 (𝑥) = 1, FL𝛼1 (𝑥) = 2𝑥𝛼 − 1,
2.1. Fractional Calculus Theory. Some necessary definitions
and Lemma of the fractional calculus theory [29, 30] are listed (2𝑖 + 1) (2𝑥𝛼 − 1) 𝛼
here for our subsequent development. FL𝛼𝑖+1 (𝑥) = FL𝑖 (𝑥) (5)
𝑖+1
Definition 1. A real function ℎ(𝑡), 𝑡 > 0, is said to be in the 𝑖
space 𝐶𝜇 , 𝜇 ∈ 𝑅, if there exists a real number 𝑝 > 𝜇, such that − FL𝛼 (𝑥) , 𝑖 = 1, 2, . . . ,
𝑖 + 1 𝑖−1
ℎ(𝑡) = 𝑡𝑝 ℎ1 (𝑡), where ℎ1 (𝑡) ∈ 𝐶(0, ∞), and it is said to be in
the space 𝐶𝜇𝑛 if and only if ℎ(𝑛) ∈ 𝐶𝜇 , 𝑛 ∈ 𝑁. and the analytic form of FL𝛼𝑖 (𝑥) of degree 𝑖 is given by
𝑖
Definition 2. Riemann-Liouville fractional integral operator (−1)𝑖+𝑠 (𝑖 + 𝑠)!
FL𝛼𝑖 (𝑥) = ∑𝑏𝑠,𝑖 𝑥𝑠𝛼 , 𝑏𝑠,𝑖 = , (6)
(𝐽𝛼 ) of order 𝛼 ≥ 0, of a function 𝑓 ∈ 𝐶𝜇 , 𝜇 ≥ −1 is defined as 𝑠=0 (𝑖 − 𝑠)!(𝑠!)2
1 𝑡 where FL𝛼𝑖 (0) = (−1)𝑖 and FL𝛼𝑖 (1) = 1. The orthogonality

𝐽𝛼 𝑓 (𝑡) = ∫ (𝑡 − 𝜏)𝛼−1 𝑓 (𝜏) 𝑑𝜏, 𝑡 > 0, condition is
Γ (𝛼) 0 (1)
1
0
𝐽 𝑓 (𝑡) = 𝑓 (𝑡) , 1
∫ FL𝛼𝑛 (𝑥) FL𝛼𝑚 (𝑥) 𝜔 (𝑥) 𝑑𝑥 = 𝛿 , (7)
0 (2𝑛 1) 𝛼 𝑛𝑚
+
where Γ(𝛼) is the well-known Gamma function. Some prop-
where 𝜔(𝑥) = 𝑥𝛼−1 is the weight function and 𝛿 is the
erties of the operator 𝐽𝛼 can be found, for example, in [29, 30].
Kronecker delta. For more details, please see [28].
Definition 3. The fractional derivative of 𝑓(𝑥) in the Caputo
sense is defined as
3. 2D-FLFs
(𝐷𝛼 𝑓) (𝑥) In this section, the definitions and theorems of 2D-FLFs are
given by Liu’s method described in [31].
1
{
{
{
{ Γ (𝑚 − 𝛼)
{
{ 3.1. Definitions and Properties of the 2D-FLFs
{
{
{ 𝑥 𝑓(𝑚) (𝜉)
= { ×∫ 𝛼−𝑚+1
𝑑𝜉, (𝛼 > 0, 𝑚 − 1 < 𝛼 < 𝑚) , ∞
Definition 5. Let {FL𝛼𝑛 (𝑥)}𝑛=0 be the fractional Legendre poly-
{
{ 0 (𝑥 − 𝜉)
{
{ 𝛽 ∞
{
{
{ 𝑚 nomials on [0, 1]; we call {FL𝛼𝑖 (𝑥)FL𝑗 (𝑦)} the two-dimen-
{ 𝑑 𝑓 (𝑥) 𝑖,𝑗=0
, 𝛼 = 𝑚, sional fractional Legendre polynomials on [0, 1] × [0, 1].
{ 𝑑𝑥𝑚
(2) 𝛽 ∞
Theorem 6. The basis {𝐹𝐿𝛼𝑖 (𝑥)𝐹𝐿𝑗 (𝑦)} is orthogonal on
𝑖,𝑗=0
where 𝑓 : 𝑅 → 𝑅, 𝑥 → 𝑓(𝑥) denotes a continuous (but not [0, 1] × [0, 1] with the weight function 𝜔(𝑥, 𝑦) = 𝜔(𝑥)𝜔(𝑦) =
necessarily differentiable) function. 𝑥𝛼−1 𝑦𝛽−1 .
Proof. Let 𝑖 ≠𝑚 or 𝑗 ≠𝑛 Proof. By multiplying 𝜔(𝑥, 𝑦)FL𝛼𝑛 (𝑥)FL𝛽𝑚 (𝑦) on both sides of
(10), where 𝑛 and 𝑚 are fixed and integrating termwise with
regard to 𝑥 and 𝑦 on [0, 1] × [0, 1], then
1 1
𝛽
∫ ∫ 𝜔 (𝑥, 𝑦) FL𝛼𝑖 (𝑥) FL𝑗 (𝑦) FL𝛼𝑚 (𝑥) FL𝛽𝑛 (𝑦) 𝑑𝑥 𝑑𝑦 1 1
0 0 ∫ ∫ 𝑓 (𝑥, 𝑦) 𝜔 (𝑥, 𝑦) FL𝛼𝑛 (𝑥) FL𝛽𝑚 (𝑦) 𝑑𝑥 𝑑𝑦
0 0
1
=∫ 𝜔 (𝑥) FL𝛼𝑖 (𝑥) FL𝛼𝑚 (𝑥) 𝑑𝑥 (8) ∞ ∞ 1 1
𝛽
0 = ∑ ∑𝑎𝑖𝑗 ∫ ∫ 𝜔 (𝑥, 𝑦) FL𝛼𝑖 (𝑥) FL𝑗 (𝑦)
𝑖=0 𝑗=0 0 0
1
𝛽
× ∫ 𝜔 (𝑦) FL𝑗 (𝑦) FL𝛽𝑛 (𝑦) 𝑑𝑦 = 0.
0 × FL𝛼𝑛 (𝑥) FL𝛽𝑚 (𝑦) 𝑑𝑥 𝑑𝑦
∞ ∞ 1
= ∑ ∑𝑎𝑖𝑗 ∫ 𝜔 (𝑥) FL𝛼𝑖 (𝑥) FL𝛼𝑛 (𝑥) 𝑑𝑥
Theorem 7. Consider 𝑖=0 𝑗=0 0
1
𝛽
1 1
𝛽 2 × ∫ 𝜔 (𝑦) FL𝑗 (𝑦) FL𝛽𝑚 (𝑦) 𝑑𝑦
∫ ∫ 𝜔 (𝑥, 𝑦) [𝐹𝐿𝛼𝑖 (𝑥) 𝐹𝐿 𝑗 (𝑦)] 𝑑𝑥 𝑑𝑦 0
0 0
1 1 2
2
1 1 = 𝑎𝑛𝑚 ∫ 𝜔 (𝑥) [FL𝛼𝑛 (𝑥)] 𝑑𝑥 ∫ 𝜔 (𝑦) [FL𝛽𝑚 (𝑦)] 𝑑𝑦
= , 0 0
(2𝑖 + 1) 𝛼 (2𝑗 + 1) 𝛽
1 1
1 1 2 = 𝑎𝑛𝑚 .
∫ ∫ 𝜔 (𝑥, 𝑦) [𝐹𝐿𝛼𝑖
𝛽
(𝑥) 𝐹𝐿 𝑗 (𝑦)] 𝑑𝑥 𝑑𝑦 (9) (2𝑛 + 1) 𝛼 (2𝑚 + 1) 𝛽
0 0 (13)
1 1 2
2 𝛽
= ∫ 𝜔 (𝑥) [𝐹𝐿𝛼𝑖 (𝑥)] 𝑑𝑥 ∫ 𝜔 (𝑦) [𝐹𝐿 𝑗 (𝑦)] 𝑑𝑦 Finally one can get (11).
0 0
If the infinite series in (10) is truncated, then it can be
1 1 written as
= .
(2𝑖 + 1) 𝛼 (2𝑗 + 1) 𝛽
𝑚 𝑚󸀠
𝛽
𝑓 (𝑥, 𝑦) ≈ ∑ ∑𝑎𝑖𝑗 FL𝛼𝑖 (𝑥) FL𝑗 (𝑦) = 𝐶𝑇 Ψ (𝑥𝛼 , 𝑦𝛼 ) , (14)
3.2. 2D-FLFs Expansion 𝑖=0 𝑗=0
Definition 8. A function of two independent variables 𝑓(𝑥, 𝑦) where 𝐶 and Ψ(𝑥𝛼 , 𝑦𝛽 ) are given by
which is integrable in square [0, 1] × [0, 1] can be expanded
as 𝐶 = [𝑐0,0 , 𝑐0,1 , . . . , 𝑐0,𝑚󸀠 −1 , 𝑐1,0 , 𝑐1,1 , . . . ,
𝑇
𝑐1,𝑚󸀠 −1 , . . . , 𝑐𝑚−1,0 , 𝑐𝑚−1,1 , . . . , 𝑐𝑚−1,𝑚󸀠 −1 ] ,
∞ ∞
𝛽 (15)
𝑓 (𝑥, 𝑦) = ∑ ∑𝑎𝑖𝑗 FL𝛼𝑖 (𝑥) FL𝑗 (𝑦) , (10)
𝑖=0 𝑗=0
Ψ (𝑥𝛼 , 𝑦𝛽 ) = [𝜓0,0 , 𝜓0,1 , . . . , 𝜓0,𝑚󸀠 −1 , 𝜓1,0 , 𝜓1,1 , . . . ,
𝑇
where 𝜓1,𝑚󸀠 −1 , . . . , 𝜓𝑚−1,0 , 𝜓𝑚−1,1 , . . . , 𝜓𝑚−1,𝑚󸀠 −1 ] ,
(16)
𝛽
𝑎𝑖𝑗 = (2𝑖 + 1) (2𝑗 + 1) 𝛼𝛽 where 𝜓𝑖𝑗 = FL𝛼𝑖 (𝑥)FL𝑗 (𝑦), 𝑖 = 0, 1, . . . , 𝑚, and 𝑗 =
1 1 (11) 0, 1, . . . , 𝑚󸀠 .
𝛽
× ∫ ∫ 𝑓 (𝑥, 𝑦) 𝜔 (𝑥, 𝑦) FL𝛼𝑖 (𝑥) FL𝑗 (𝑦) 𝑑𝑥 𝑑𝑦. According to the definition of FLFs, one can find that
0 0 fractional Legendre polynomials are identical to Legendre
polynomials shifted to [0, 1] when using the transform 𝑥𝛼 →
𝛽 𝑥, 𝑦𝛽 → 𝑦. Therefore, in a similar method described in [31],
Theorem 9. If the series ∑∞ ∞ 𝛼
𝑖=0 ∑𝑗=0 𝑎𝑖𝑗 𝐹𝐿 𝑖 (𝑥)𝐹𝐿 𝑗 (𝑦) converges we can easily get the convergence and stability theorems of
uniformly to 𝑓(𝑥, 𝑦) on the square [0, 1] × [0, 1], then we have proposed method.
Lemma 10. If the function 𝑓(𝑥, 𝑦) is a continuous function on

𝑎𝑖𝑗 = (2𝑖 + 1) (2𝑗 + 1) 𝛼𝛽 [0, 1] × [0, 1] and the series ∑∞ ∞ 𝛼 𝛽
𝑖=0 ∑𝑗=0 𝑎𝑖𝑗 𝐹𝐿 𝑖 (𝑥)𝐹𝐿 𝑗 (𝑦) con-
𝛽
1 1
𝛽
(12) verges uniformly to 𝑓(𝑥, 𝑦), then ∑∞ ∞ 𝛼
𝑖=0 ∑𝑗=0 𝑎𝑖𝑗 𝐹𝐿 𝑖 (𝑥)𝐹𝐿 𝑗 (𝑦)
×∫ ∫ 𝑓 (𝑥, 𝑦) 𝜔 (𝑥, 𝑦) 𝐹𝐿𝛼𝑖 (𝑥) 𝐹𝐿 𝑗 (𝑦) 𝑑𝑥 𝑑𝑦. is the 2D-FLFs expansion of 𝑓(𝑥, 𝑦).
0 0
Proof (by contradiction). Let Proof. Consider

󵄨󵄨 ∞ ∞ 󵄨󵄨 ∞ ∞
󵄨󵄨 󵄨
󵄨󵄨∑ ∑𝑎 FL𝛼 (𝑥) FL𝛽 (𝑦)󵄨󵄨󵄨 ≤ ∑ ∑ 󵄨󵄨󵄨𝑎 󵄨󵄨󵄨 󵄨󵄨󵄨FL𝛼 (𝑥)󵄨󵄨󵄨 󵄨󵄨󵄨󵄨FL𝛽 (𝑦)󵄨󵄨󵄨󵄨
∞ ∞
𝛽 󵄨󵄨󵄨 𝑖𝑗 𝑖 𝑗 󵄨󵄨 󵄨 󵄨
󵄨󵄨 𝑖=0𝑗=0 󵄨 𝑖𝑗 󵄨 󵄨 𝑖 󵄨 󵄨󵄨 𝑗 󵄨󵄨
𝑓 (𝑥, 𝑦) = ∑∑ 𝑏𝑖𝑗 FL𝛼𝑖 (𝑥) FL𝑗 (𝑦) , 󵄨󵄨𝑖=0𝑗=0 󵄨
𝑖=0 𝑗=0
∞ ∞
󵄨 󵄨
≤ ∑ ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 .
(17)
∞ ∞
𝛽
𝑓 (𝑥, 𝑦) ∼ ∑ ∑𝑎𝑖𝑗 FL𝛼𝑖 (𝑥) FL𝑗 (𝑦) . 𝑖=0 𝑗=0
𝑖=0 𝑗=0 (21)
𝛽
Then ∑∞ ∞ 𝛼
𝑖=0 ∑𝑗=0 𝑎𝑖𝑗 FL𝑖 (𝑥)FL𝑗 (𝑦) converges uniformly to the
Then there is at least one coefficient such that 𝑎𝑛𝑚 ≠𝑏𝑛𝑚 . function 𝑓(𝑥, 𝑦).
However,
Theorem 14. If a continuous function 𝑓(𝑥, 𝑦), defined on [0,
𝑏𝑛𝑚 = (2𝑛 + 1) (2𝑚 + 1) 𝛼𝛽 1] × [0, 1], has bounded mixed partial derivative 𝐷𝑥2𝛼 𝐷𝑦2𝛽 𝑓(𝑥,
𝑦), then the 2D-FLFs expansion of the function converges
1 1
uniformly to the function.
× ∫ ∫ 𝑓 (𝑥, 𝑦) 𝜔 (𝑥, 𝑦) FL𝛼𝑛 (𝑥) FL𝛽𝑚 (𝑦) 𝑑𝑥 𝑑𝑦 (18)
0 0
Proof. Let 𝑓(𝑥, 𝑦) be a function defined on [0, 1] × [0, 1] such
= 𝑎𝑛𝑚 . that
󵄨󵄨 2𝛼 2𝛽 󵄨
󵄨󵄨𝐷𝑥 𝐷𝑦 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ≤ 𝑀, (22)
󵄨 󵄨
Lemma 11. If two continuous functions defined on [0, 1] × where 𝑀 is a positive constant and
[0, 1] have the identical 2D-FLFs expansions, then these two 𝑎𝑖𝑗 = (2𝑖 + 1) (2𝑗 + 1) 𝛼𝛽
functions are identical.
1 1 (23)
𝛽
Proof. Suppose that 𝑓(𝑥, 𝑦) and 𝑔(𝑥, 𝑦) can be expanded by × ∫ ∫ 𝑓 (𝑥, 𝑦) 𝜔 (𝑥, 𝑦) FL𝛼𝑖 (𝑥) FL𝑗 (𝑦) 𝑑𝑥 𝑑𝑦.
0 0
2D-FLFs as follows:
By employing the transform 𝑋 = 2𝑥𝛼 − 1 and 𝑌 = 2𝑦𝛽 − 1,
∞ ∞ one can obtain
𝛽
𝑓 (𝑥, 𝑦) ∼ ∑ ∑𝑎𝑖𝑗 FL𝛼𝑖 (𝑥) FL𝑗 (𝑦) , 2𝑖 + 1 2𝑗 + 1 1 1
𝑖=0 𝑗=0 𝑎𝑖𝑗 = ∫ ∫ 𝑓 (𝑋, 𝑌) 𝑝𝑖 (𝑋) 𝑝𝑗 (𝑌) 𝑑𝑋 𝑑𝑌.
2 2 −1 −1
(19) (24)
∞ ∞
𝛽
𝑔 (𝑥, 𝑦) ∼ ∑∑ 𝑎𝑖𝑗 FL𝛼𝑖 (𝑥) FL𝑗 (𝑦) . Consequently, in a similar method described in [31],
𝑖=0 𝑗=0
Theorem 14 can be proved.
By subtracting the above two equations with each other, one 4. Operational Matrices of 2D-FLFs
has
4.1. Integration Operational Matrices of 2D-FLFs
∞ ∞
𝑓 (𝑥, 𝑦) − 𝑔 (𝑥, 𝑦) ∼ ∑∑ (𝑎𝑖𝑗 − 𝑎𝑖𝑗 ) FL𝛼𝑖 (𝑥) FL𝑗 (𝑦)
𝛽 Lemma 15. The Riemann-Liouville fractional integration of
𝑖=0 𝑗=0 order 𝛾 > 0 of the 2D-FLFs 𝜓𝑖𝑗 can be obtained in the form
(20) of
∞ ∞
𝛽
= 0 = ∑∑0 × FL𝛼𝑖 (𝑥) FL𝑗 (𝑦) . 𝛽
𝑖
Γ (1 + 𝑠𝛼) 𝑠𝛼+𝛾
𝑖=0 𝑗=0 𝐽𝑥𝛾 {𝜓𝑖𝑗 (𝑥𝛼 , 𝑦𝛽 )} = 𝐹𝐿 𝑗 (𝑦) ∑𝑏𝑠𝑖 𝑥 . (25)
𝑠=0 Γ (1 + 𝑠𝛼 + 𝛾)
Proof. Consider
Then Lemma 11 can be proved.
𝛽
𝐽𝑥𝛾 {𝜓𝑖𝑗 (𝑥𝛼 , 𝑦𝛽 )} = 𝐽𝑥𝛾 {FL𝛼𝑖 (𝑥) FL𝑗 (𝑦)}
Theorem 12. If the 2D-FLFs expansion of a continuous func-
tion 𝑓(𝑥, 𝑦) converges uniformly, then the 2D-FLFs expansion 𝛽
= 𝐽𝑥𝛾 {FL𝛼𝑖 (𝑥)} FL𝑗 (𝑦)
converges to the function 𝑓(𝑥, 𝑦).
𝑖
𝛽 (26)
Proof. Theorem 12 can be proved by Theorems 7 and 9. = 𝐽𝑥𝛾 {∑𝑏𝑠𝑖 𝑥𝑠𝛼 } FL𝑗 (𝑦)
𝑠=0
Theorem 13. If the sum of the absolute values of the 2D-FLFs 𝑖
coefficients of a continuous function 𝑓(𝑥, 𝑦) forms a convergent 𝛽 Γ (1 + 𝑠𝛼) 𝑠𝛼+𝛾
= FL𝑗 (𝑦) ∑𝑏𝑠𝑖 𝑥 .
series, then the 2D-FLFs expansion is absolutely uniformly 𝑠=0 Γ (1 + 𝑠𝛼 + 𝛾)
convergent, and converges to the function 𝑓(𝑥, 𝑦).
Lemma 16. Let 𝛾 > 0; then one has where P𝛾𝑥 is the 𝑚𝑚󸀠 × 𝑚𝑚󸀠 operational matrix of Riemann-
1 1 Liouville fractional integration of order 𝛾 > 0, and has the form
∫ ∫ 𝐽𝑥𝛾 {𝜓𝑖𝑗 } 𝜓𝑖󸀠 𝑗󸀠 𝜔 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦 as follows:
0 0
𝑖 𝑖 󸀠 𝐸0,0 𝐸0,1 ⋅⋅⋅ 𝐸0,𝑚−1

{
{ 𝑏𝑠𝑖 𝑏𝑠󸀠 𝑖󸀠 [ 𝐸1,0 𝐸1,1 ⋅⋅⋅ 𝐸1,𝑚−1 ]
{
{ ∑ ∑ [ ]
{
{ (𝑠 + 𝑠 󸀠 + 1) 𝛼 + 𝛾 (27) P𝛾𝑥 = [ .. .. .. ], (30)
{𝑠=0𝑠󸀠 =0
{ [ . . d . ]
={ Γ (1 + 𝑠𝛼) 1 𝐸 𝐸 ⋅ ⋅ ⋅ 𝐸𝑚−1,𝑚−1 ]
{
{ × , 𝑗 = 𝑗󸀠 [ 𝑚−1,0 𝑚−1,1
{
{ Γ (1 + 𝑠𝛼 + 𝛾) (2𝑗 + 1) 𝛽
{
{
{ in which 𝐸𝑖,𝑖󸀠 is 𝑚󸀠 × 𝑚󸀠 matrix and the elements are defined as
{0, 𝑗 ≠𝑗󸀠 . follows:
Proof. Using previous Lemma 15 and (6), one can have
1 1
𝑖 𝑖󸀠 𝑏𝑠𝑖 𝑏𝑠󸀠 𝑖󸀠 (2𝑖󸀠 + 1) 𝛼 Γ (1 + 𝑠𝛼)
∫ ∫ 𝐽𝑥𝛾 𝛼 𝛽
{𝜓𝑖𝑗 (𝑥 , 𝑦 )} 𝜓𝑖󸀠 𝑗󸀠 (𝑥 , 𝑦 ) 𝜔 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦 𝛼 𝛽 𝐸𝑖,𝑖󸀠 = 𝐼∑ ∑ ,
0 0 𝑠=0𝑠󸀠 =0 (𝑠 + 𝑠󸀠 + 1) 𝛼 + 𝛾 Γ (1 + 𝑠𝛼 + 𝛾) (31)
1 1 󸀠
𝛽 𝛽 𝑖, 𝑖 = 0, 1, . . . , 𝑚 − 1,
= ∫ ∫ 𝜔 (𝑥, 𝑦) FL𝛼𝑖󸀠 (𝑥𝛼 ) FL𝑗󸀠 (𝑦𝛽 ) FL𝑗 (𝑦𝛽 )
0 0
𝑖
and 𝐼 is 𝑚󸀠 × 𝑚󸀠 identity matrix.
Γ (1 + 𝑠𝛼) 𝑠𝛼+𝛾
× ∑𝑏𝑠𝑖 𝑥 𝑑𝑥 𝑑𝑦
𝑠=0 Γ (1 + 𝑠𝛼 + 𝛾) Proof. Using (29) and orthogonality property of FLFs, one
can get
1 1
𝛽 𝛽
= ∫ ∫ 𝜔 (𝑦) FL𝑗 (𝑦𝛽 ) FL𝑗󸀠 (𝑦𝛽 )
0 0 P𝛾𝑥 = ⟨𝐽𝑥𝛾 Ψ (𝑥𝛼 , 𝑦𝛽 ) , Ψ𝑇 (𝑥𝛼 , 𝑦𝛽 )⟩ 𝐻−1 , (32)
𝑖 𝑖󸀠
Γ (1 + 𝑠𝛼) where ⟨𝐽𝑥𝛾 Ψ(𝑥𝛼 , 𝑦𝛽 ), Ψ𝑇(𝑥𝛼 , 𝑦𝛽 )⟩ and 𝐻−1 are two 𝑚󸀠 × 𝑚󸀠
× ∑ ∑ 𝑏𝑠𝑖 𝑏𝑠󸀠 𝑖󸀠
𝑠=0𝑠󸀠 =0 Γ (1 + 𝑠𝛼 + 𝛾) matrices defined as
󸀠 ⟨𝐽𝑥𝛾 Ψ (𝑥𝛼 , 𝑦𝛽 ) , Ψ𝑇 (𝑥𝛼 , 𝑦𝛽 )⟩
× 𝑥(𝑠+𝑠 +1)𝛼+𝛾−1 𝑑𝑥 𝑑𝑦
1 1
1
=∫
𝛽
𝜔 (𝑦) FL𝑗 (𝑦 𝛽 𝛽
) FL𝑗󸀠 (𝑦 )𝛽 = {∫ ∫ 𝐽𝑥𝛾 {Ψ𝑘 (𝑥𝛼 , 𝑦𝛽 )}
0 0
0
(28) 𝑚𝑚󸀠
1 𝑖 𝑖󸀠
Γ (1 + 𝑠𝛼) 𝛼 𝛽
× Ψ𝑘󸀠 (𝑥 , 𝑦 ) 𝜔 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦}
× (∫ ∑ ∑ 𝑏𝑠𝑖 𝑏𝑠󸀠 𝑖󸀠
0 𝑠=0𝑠󸀠 =0 Γ (1 + 𝑠𝛼 + 𝛾) 𝑘,𝑘󸀠
󸀠
{ 𝑖 𝑖 Γ (1 + 𝑠𝛼)
󸀠 = {∑ ∑ 𝑏𝑠𝑖 𝑏𝑠󸀠 𝑖󸀠
× 𝑥(𝑠+𝑠 +1)𝛼+𝛾−1 𝑑𝑥) 𝑑𝑦 Γ (1 + 𝑠𝛼 + 𝛾)
{𝑠=0𝑠 =0
󸀠
𝑚;𝑚󸀠
𝑖 𝑖󸀠 }
𝑏𝑠𝑖 𝑏𝑠󸀠 𝑖󸀠 Γ (1 + 𝑠𝛼) 1 1
= ∑∑ × ,
𝑠=0𝑠󸀠 =0 (𝑠 + 𝑠 󸀠 + 1) 𝛼 + 𝛾 Γ (1 + 𝑠𝛼 + 𝛾) (𝑠 + 𝑠 + 1) 𝛼 + 𝛾 (2𝑗 + 1) 𝛽 }
󸀠
}𝑖,𝑖󸀠 ;𝑗=𝑗󸀠
1
𝛽 𝛽 𝑚;𝑚󸀠
× ∫ 𝜔 (𝑦) FL𝑗 (𝑦𝛽 ) FL𝑗󸀠 (𝑦𝛽 ) 𝑑𝑦 𝐻−1 = {(2𝑖󸀠 + 1) (2𝑗 + 1) 𝛼𝛽}𝑖,𝑖󸀠 ;𝑗=𝑗󸀠 .
0
󸀠
(33)
𝑖 𝑖
{
{ 𝑏𝑠𝑖 𝑏𝑠󸀠 𝑖󸀠
{
{ ∑∑
{
{𝑠=0𝑠󸀠 =0 (𝑠 + 𝑠 󸀠 + 1) 𝛼 + 𝛾 Now by substituting above equations in (32), Theorem 12
{ can be proved.
={ Γ (1 + 𝑠𝛼) 1
{
{ × , 𝑗 = 𝑗󸀠
{
{ Γ (1 + 𝑠𝛼 + 𝛾) (2𝑗 + 1) 𝛽
{
{ In a similar way as previous, one can obtain the oper-
{0, 𝑗 ≠𝑗󸀠 . ational matrix of Riemann-Liouville fractional integration
with respect to variable 𝑦.
Theorem 18. Let Ψ(𝑥𝛼 , 𝑦𝛽 ) be the 2D-FLFs vector defined in

(16); one has
(16); then one has
𝐽𝑥𝛾 Ψ (𝑥𝛼 , 𝑦𝛽 ) ≃ P𝛾𝑥 Ψ (𝑥𝛼 , 𝑦𝛽 ) , (29) 𝐽𝑦𝛾 Ψ (𝑥𝛼 , 𝑦𝛽 ) ≃ P𝛾𝑦 Ψ (𝑥𝛼 , 𝑦𝛽 ) , (34)
where P𝛾𝑦 is the 𝑚𝑚󸀠 × 𝑚𝑚󸀠 operational matrix of Riemann- Proof. Using previous Lemma 19 and (6), one can have
Liouville fractional integration of order 𝛾 > 0, and has the form
as follows: 1 1
∫ ∫ 𝐷𝑥𝛾 {𝜓𝑖𝑗 (𝑥𝛼 , 𝑦𝛽 )} 𝜓𝑖󸀠 𝑗󸀠 (𝑥𝛼 , 𝑦𝛽 ) 𝜔 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦
0 0
𝐸 𝑂 ⋅⋅⋅ 𝑂
[𝑂 𝐸 ⋅⋅⋅ 𝑂] 1 1
[ ] = ∫ ∫ 𝜔 (𝑥, 𝑦) FL𝛼𝑖󸀠 (𝑥𝛼 ) FL𝑗󸀠 (𝑦𝛽 ) FL𝑗 (𝑦𝛽 )
𝛽 𝛽
P𝛾𝑦 = [ .. .. .. ] , (35)
[. . d .] 0 0
[𝑂 𝑂 ⋅⋅⋅ 𝐸] 𝑖
Γ (1 + 𝑠𝛼) 𝑠𝛼−𝛾
× ∑𝑏𝑠𝑖󸀠 𝑥 𝑑𝑥 𝑑𝑦
𝑠=0 Γ (1 + 𝑠𝛼 − 𝛾)
in which 𝐸 is 𝑚󸀠 × 𝑚󸀠 matrix and the elements are defined as
follows: 1 1
𝛽 𝛽
= ∫ ∫ 𝜔 (𝑦) FL𝑗 (𝑦𝛽 ) FL𝑗󸀠 (𝑦𝛽 )
0 0
𝑗 𝑗󸀠 𝑏𝑟𝑗 𝑏𝑟󸀠 𝑗󸀠 (2𝑗󸀠 + 1) 𝛽 Γ (1 + 𝑟𝛽) 𝑖 𝑖󸀠
𝐸𝑗,𝑗󸀠 = ∑ ∑ , Γ (1 + 𝑠𝛼)
𝑟=0𝑟󸀠 =0 (𝑟 + 𝑟󸀠 + 1) 𝛽 + 𝛾 Γ (1 + 𝑟𝛽 + 𝛾) (36) × ∑ ∑ 𝑏𝑠𝑖󸀠 𝑏𝑠󸀠 𝑖󸀠
𝑠=0𝑠󸀠 =0 Γ (1 + 𝑠𝛼 − 𝛾)
󸀠 󸀠
𝑗, 𝑗 = 0, 1, . . . , 𝑚 − 1. 󸀠
× 𝑥(𝑠+𝑠 +1)𝛼−𝛾−1 𝑑𝑥 𝑑𝑦
4.2. Derivative Operational Matrices of 2D-FLFs 1
𝛽 𝛽
= ∫ 𝜔 (𝑦) FL𝑗 (𝑦𝛽 ) FL𝑗󸀠 (𝑦𝛽 )
Lemma 19. The FLFs Caputo fractional derivative of 𝛾 > 0 0
(40)
can be obtained in the form of 1 𝑖 𝑖󸀠
Γ (1 + 𝑠𝛼)
× (∫ ∑ ∑ 𝑏𝑠𝑖󸀠 𝑏𝑠󸀠 𝑖󸀠
𝑖 0 𝑠=0𝑠󸀠 =0 Γ (1 + 𝑠𝛼 − 𝛾)
𝛽 Γ (1 + 𝑠𝛼) 𝑠𝛼−𝛾
𝐷𝑥𝛾 {𝜓𝑖𝑗 (𝑥𝛼 , 𝑦𝛽 )} = 𝐹𝐿 𝑗 (𝑦𝛽 ) ∑𝑏𝑠𝑖󸀠 𝑥 ,
𝑠=0 Γ (1 + 𝑠𝛼 − 𝛾)
󸀠
(37) × 𝑥(𝑠+𝑠 +1)𝛼−𝛾−1 𝑑𝑥) 𝑑𝑦
󸀠 󸀠
where 𝑏𝑠,𝑖 = 0 when 𝑠𝛼 ∈ 𝑁0 and 𝑠𝛼 < 𝛾 in other case 𝑏𝑠,𝑖 = 𝑏𝑠,𝑖 . 𝑖󸀠
𝑖
𝑏𝑠𝑖󸀠 𝑏𝑠󸀠 𝑖󸀠 Γ (1 + 𝑠𝛼)
= ∑∑ 󸀠 + 1) 𝛼 − 𝛾 Γ (1 + 𝑠𝛼 − 𝛾)
Proof. Consider 𝑠=0𝑠 =0
󸀠 (𝑠 + 𝑠
𝛽 1
𝐷𝑥𝛾 {𝜓𝑖𝑗 (𝑥𝛼 , 𝑦𝛽 )} = 𝐷𝑥𝛾 {FL𝛼𝑖 (𝑥𝛼 ) FL𝑗 (𝑦𝛽 )} 𝛽
× ∫ 𝜔 (𝑦) FL𝑗 (𝑦𝛽 ) FL𝑗󸀠 (𝑦𝛽 ) 𝑑𝑦
𝛽
0
𝛽
= FL𝑗 (𝑦𝛽 ) 𝐷𝑥𝛾 {FL𝛼𝑖 (𝑥𝛼 )} 󸀠
{
{
𝑖 𝑖
𝑏𝑠𝑖󸀠 𝑏𝑠󸀠 𝑖󸀠
{
{ ∑ ∑
{
{ 󸀠
𝑖
𝛽 {𝑠=0𝑠󸀠 =0 (𝑠 + 𝑠 + 1) 𝛼 − 𝛾
= 𝐷𝑥𝛾 {∑𝑏𝑠𝑖 𝑥𝑠𝛼 } FL𝑗 (𝑦𝛽 ) ={ Γ (1 + 𝑠𝛼) 1
𝑠=0 {
{
{ × , 𝑗 = 𝑗󸀠
{
{ Γ (1 + 𝑠𝛼 − 𝛾) (2𝑗 + 1) 𝛽
{
𝛽
𝑖
Γ (1 + 𝑠𝛼) 𝑠𝛼−𝛾 {0, 𝑗 ≠𝑗󸀠 .
= FL𝑗 (𝑦𝛽 ) ∑𝑏𝑠𝑖󸀠 𝑥 .
𝑠=0 Γ (1 + 𝑠𝛼 − 𝛾)
(38)
(16); one has
Lemma 20. Let 𝛾 > 0, 𝛼 ∉ 𝑁; then one has
𝐷𝑥𝛾 Ψ (𝑥𝛼 , 𝑦𝛽 ) ≃ D𝛾𝑥 Ψ (𝑥𝛼 , 𝑦𝛽 ) , (41)
1 1
∫ ∫ 𝐷𝑥𝛾 {𝜓𝑖𝑗 } 𝜓𝑖󸀠 𝑗󸀠 𝜔 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦
0 0
where D𝛾𝑥 is the 𝑚𝑚󸀠 × 𝑚𝑚󸀠 operational matrix of Caputo
󸀠
{
𝑖 𝑖
𝑏𝑠𝑖 𝑏𝑠󸀠 𝑖󸀠 fractional derivative of order 𝛾 > 0, and has the form as follows:
{
{ ∑∑
{
{ (𝑠 + 𝑠 󸀠 + 1) 𝛼 − 𝛾
{𝑠=0𝑠󸀠 =0
{ (39)
{
{ 𝑂 𝑂 ⋅⋅⋅ 𝑂
={ Γ (1 + 𝑠𝛼) 1 [ 𝐹1,0
{
{ × , 𝑗 = 𝑗󸀠 [ 𝑂 ⋅ ⋅ ⋅ 𝑂] ]
{
{ Γ (1 + 𝑠𝛼 − 𝛾) (2𝑗 + 1) 𝛽 D𝛾𝑥 = [ .. .. ]
{
{ [ .
.. (42)
{
{ . d .]
{0, 𝑗 ≠𝑗󸀠 . [𝐹𝑚−1,0 𝐹𝑚−1,1 ⋅ ⋅ ⋅ 𝑂]
in which 𝐹𝑖,𝑖󸀠 is 𝑚󸀠 × 𝑚󸀠 matrix and the elements are defined as in which 𝐹 is 𝑚󸀠 × 𝑚󸀠 matrix and the elements are defined as
follows: follows:
𝑗 𝑗󸀠 𝑏𝑟𝑗󸀠 𝑏𝑟󸀠 𝑗󸀠 (2𝑗󸀠 + 1) 𝛽

𝑖 𝑖󸀠 𝑏𝑠𝑖󸀠 𝑏𝑠󸀠 𝑖󸀠 (2𝑖󸀠 + 1) 𝛼
Γ (1 + 𝑠𝛼) Γ (1 + 𝑟𝛽)
𝐹𝑖,𝑖󸀠 = 𝐼∑ ∑ , 𝐹𝑗,𝑗󸀠 = ∑ ∑ ,
(𝑠 + 𝑠 󸀠 + 1) 𝛼 − 𝛾 Γ (1 + 𝑠𝛼 − 𝛾)
𝑟=0𝑟󸀠 =0 (𝑟 + 𝑟󸀠 + 1) 𝛽 + 𝛾 Γ (1 + 𝑟𝛽 + 𝛾) (48)
𝑠=0𝑠󸀠 =0 (43)
𝑖, 𝑖󸀠 = 0, 1, . . . , 𝑚 − 1, 𝑗, 𝑗󸀠 = 0, 1, . . . , 𝑚󸀠 − 1.
and 𝐼 is a 𝑚󸀠 × 𝑚󸀠 identity matrix. 5. Applications and Results

Proof. Using (41) and the orthogonality property of FLFs, one Consider the following FPDEs:
can have 𝛽
𝐷𝑥𝛼 𝑢 (𝑥, 𝑡) + 𝐷𝑡 𝑢 (𝑥, 𝑡)
D𝛾𝑥 = ⟨𝐷𝑥𝛾 Ψ (𝑥𝛼 , 𝑦𝛽 ) , Ψ𝑇 (𝑥𝛼 , 𝑦𝛽 )⟩ 𝐻−1 , (44) + 𝑁 [𝑢 (𝑥, 𝑡)] + 𝐿 [𝑢 (𝑥, 𝑡)] = 𝑔 (𝑥, 𝑡) , 𝛼, 𝛽 ∈ (0, 1] ,
(49)
where ⟨𝐷𝑥𝛾 Ψ(𝑥𝛼 , 𝑦𝛽 ), Ψ𝑇 (𝑥𝛼 , 𝑦𝛽 )⟩ and 𝐻−1 are two 𝑚𝑚󸀠 ×
𝑚𝑚󸀠 matrices defined as where 𝐿 and 𝑁 are linear operator and nonlinear operator;
respectively. 𝐷𝛼 and 𝐷𝛽 are the Caputo fractional derivatives
⟨𝐷𝑥𝛾 Ψ (𝑥𝛼 , 𝑦𝛽 ) , Ψ𝑇 (𝑥𝛼 , 𝑦𝛽 )⟩ of order 𝛼 and 𝛽, respectively; 𝑔 is a known analytic function.
𝛽
By employing operator 𝐽𝑡 on both sides of (49) and then
1 1
using the Lemma 4, one can have
= {∫ ∫ 𝐷𝑥𝛾 {Ψ𝑘 (𝑥𝛼 , 𝑦𝛽 )}
0 0
𝛽
𝑢 (𝑥, 𝑡) + 𝐽𝑡 {𝐷𝑥𝛼 𝑢 (𝑥, 𝑡) + 𝑁𝑢 (𝑥, 𝑡) + 𝐿𝑢 (𝑥, 𝑡)}
𝑚𝑚󸀠
× Ψ𝑘󸀠 (𝑥𝛼 , 𝑦𝛽 ) 𝜔 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑦} 𝑚−1
𝑥𝑘 (50)
𝛽
𝑘,𝑘󸀠 − ∑ 𝑢(𝑘) (𝑥, 0) − 𝐽𝑡 𝑔 (𝑥, 𝑡) = 0.
𝑘=0
𝑘!
󸀠
{ 𝑖 𝑖 𝑏𝑠𝑖󸀠 𝑏𝑠󸀠 𝑖󸀠 (45)
= {∑ ∑ We first express unknown function 𝑢(𝑥, 𝑡) and derivative
(𝑠 + 𝑠󸀠 + 1) 𝛼 − 𝛾
{𝑠=0𝑠 =0
󸀠
term 𝐷𝑥𝛼 𝑢(𝑥, 𝑡) as
𝑚;𝑚󸀠
Γ (1 + 𝑠𝛼) 1 } 𝑢 (𝑥, 𝑡) = 𝐶𝑇 Ψ (𝑥𝛼 , 𝑡𝛽 ) , 𝐷𝑥𝛼 𝑢 (𝑥, 𝑡) = 𝐶𝑇 D𝛼𝑥 Ψ (𝑥𝛼 , 𝑡𝛽 ) .
×
Γ (1 + 𝑠𝛼 − 𝛾) (2𝑗 + 1) 𝛽 } (51)
}𝑖,𝑖󸀠 ;𝑗=𝑗󸀠
Now for the nonlinear part, by employing the nonlinear
𝑚;𝑚󸀠
𝐻−1 = {(2𝑖󸀠 + 1) (2𝑗 + 1) 𝛼𝛽}𝑖,𝑖󸀠 ;𝑗=𝑗󸀠 . term approximation method described in [32] and then by
using transform 𝑥 → 𝑥𝛼 , 𝑡 → 𝑡𝛽 , one can get the 2D-FLFs
expansion of nonlinear term as
Now by substituting above equations in (44), Theorem 21
can be proved.
𝑁𝑢 (𝑥, 𝑡) = 𝑁𝑇 Ψ (𝑥𝛼 , 𝑡𝛽 ) , (52)
In a similar way as above, one can get Caputo fractional where 𝑁𝑇 is coefficient matrix of nonlinear term which must
derivative of order 𝛾 > 0 with respect to variable 𝑦. be computed and its order is 𝑚𝑚󸀠 × 𝑚𝑚󸀠 .
For the linear part, we have
(16); one can have 𝐿𝑢 (𝑥, 𝑡) = 𝐿𝑇 Ψ (𝑥𝛼 , 𝑡𝛽 ) , (53)
𝐷𝑦𝛾 Ψ (𝑥𝛼 , 𝑦𝛽 ) ≃ D𝛾𝑦 Ψ (𝑥𝛼 , 𝑦𝛽 ) , (46) where 𝐿 is a matrix of order 𝑚𝑚󸀠 × 𝑚𝑚󸀠 .
After substituting (51)–(53) into (50), one can obtain
where D𝛾𝑦 is the 𝑚𝑚󸀠 × 𝑚𝑚󸀠 operational matrix of Caputo
𝐶𝑇 + (𝐶𝑇 D𝛼𝑥 + 𝑁𝑇 + 𝐿𝑇 ) P𝛽𝑦 − 𝐶guess
𝑇
= 0. (54)
fractional derivative of order 𝛾 > 0, and has the form as follows:
According to the Wu’s [33] technology for determining the
𝐹 𝑂 ⋅⋅⋅ 𝑂 initial iteration value, the initial iteration value is chosen as
[𝑂 𝐹 ⋅⋅⋅ 𝑂] 𝑢guess = ∑𝑚−1 (𝑘) 𝑘 𝛽 𝑇 𝛼 𝛽
[ ] 𝑘=0 𝑢 (𝑥, 0)(𝑥 /𝑘!) + 𝐽𝑡 {𝑔(𝑥, 𝑡)} = 𝐶guess Ψ(𝑥 , 𝑡 ).
D𝛾𝑦 = [ .. .. .. ] , (47)
[. . d .] The coefficient matrix 𝐶𝑇 can be computed by using the
[𝑂 𝑂 ⋅ ⋅ ⋅ 𝐹] MATLAB function fsolve( ) or the method described in [34].
𝛽 = 0.25 𝛽 = 0.25
−13
×10
2 5
Absolute error
u 1
0 0
1 1
1 1
0.5 0.5
0.5 0.5
t t
0 0 x 0 0 x
(a) (b)
𝛽 = 0.50 𝛽 = 0.50
×10−15
2 4
u 1 Absolute error 2
0 0
1 1
1 1
0.5 0.5
0.5 t 0.5
t
0 0 x 0 0 x
(c) (d)
Figure 1: Numerical results for Example 23 when 𝛽 = 0.25, 0.50.
Now, the present method is applied to solve the linear and By employing 2D-FLFs method, one can get
nonlinear FPDEs, and their results are compared with the
2 𝛽
solution of other methods. The accuracy of our approach is 𝐶𝑇 [𝐼 + (D𝛼𝑥 − (D𝛼𝑥 ) ) P𝑡 ] = 𝐶guess
𝑇
, (57)
estimated by the following error functions:
where 𝛼 = 1. Then we can get 𝐶𝑇 = 𝐶guess
𝑇
inv(𝐼 + (D𝛼𝑥 −
𝑒𝑗 = (𝑢exact )𝑗 − (𝑢approx )𝑗 , 𝑒 = 𝑢exact − 𝑢approx , 2 𝛽
(D𝛼𝑥 ) )P𝑡 ).
𝑁 󵄨
Figures 1(a) and 1(b) show the numerical results for 𝛽 =
󵄨 󵄨 󵄨 2 󵄨󵄨 0.25 with 𝑚 = 3, 𝑚󸀠 = 9 and 𝛽 = 0.5 with 𝑚 = 3, 𝑚󸀠 = 5,
‖𝑒‖𝐿 ∞ = max 󵄨󵄨󵄨󵄨𝑒𝑗 󵄨󵄨󵄨󵄨 , ‖𝑒‖𝐿 2 = √ ∑ 󵄨󵄨󵄨(𝑒𝑗 ) 󵄨󵄨󵄨,
1≤𝑗≤𝑁
𝑗=1
󵄨 󵄨 (55) respectively. It should be found that the accuracy of 2D-FLFs
method is very high while only a small number of 2D-FLFs
are needed.
1 𝑁 󵄨󵄨󵄨 2 󵄨󵄨
‖𝑒‖RMS = √ ∑ 󵄨󵄨(𝑒𝑗 ) 󵄨󵄨󵄨.
𝑁 𝑗=1 󵄨 󵄨 Example 24. Consider nonlinear fractional Klein-Gordon
equation [36, 37]:
Example 23. Consider the one-dimensional linear inhomo-
𝛽
geneous fractional Burger’s equation [35]: 𝐷𝑡 𝑢 (𝑥, 𝑡) − 𝐷𝑥𝛼 𝑢 (𝑥) + 𝑢3 (𝑥) = 𝑔 (𝑥, 𝑡) ,
(58)
𝛽
𝜕 𝑢 (𝑥, 𝑡) 𝜕𝑢 (𝑥, 𝑡) 𝜕 𝑢 (𝑥, 𝑡) 2 𝑥 ≥ 0, 𝑡 > 0, 𝛼, 𝛽 ∈ (1, 2] ,
+ −
𝜕𝑡𝛽 𝜕𝑥 𝜕𝑥2
(56) subject to the initial conditions
2𝑡2−𝛽
= + 2𝑥 − 2, 0 < 𝛽 ≤ 1, 𝑢 (𝑥, 0) = 0, 𝑢𝑡 (𝑥, 0) = 0, (59)
Γ (3 − 𝛽)
with the initial condition 𝑢(𝑥, 0) = 𝑥2 and the exact solution and 𝑔(𝑥, 𝑡) = Γ(𝛽 + 1)𝑥𝛼 − Γ(𝛼 + 1)𝑡𝛽 + 𝑥3𝛼 𝑡3𝛽 . The exact
being 𝑢(𝑥, 𝑡) = 𝑥2 + 𝑡2 . solution of (58) is 𝑢(𝑥, 𝑡) = 𝑥𝛼 𝑡𝛽 .
𝛼 = 1.25, 𝛽 = 1.25 𝛼 = 1.50, 𝛽 = 1.50
1 1
u 0.5 u 0.5
0 0
1 1
1 1
0.5 0.5 0.5 0.5
t x t x
0 0 0 0
(a) (b)
𝛼 = 1.75, 𝛽 = 1.75 𝛼 = 2.00, 𝛽 = 2.00
1 1
u 0.5 u 0.5
0 0
1 1
1 1
0.5 0.5 0.5 0.5
t x t x
0 0 0 0
(c) (d)
Figure 2: Numerical results of Example 24 for different values of 𝛼 and 𝛽.
By employing 2D-FLFs method with 𝑚 = 3 and 𝑚󸀠 = 3, 𝑚󸀠 = 9. Moreover, Table 2 shows the approximate solutions
one can have for (61) obtained for different values of 𝛽 using the frac-
𝛽
tional variational iteration method (FVIM) [39] and 2D-FLFs
𝐶𝑇 + (−𝐶𝑇 D𝛼𝑥 + 𝑁𝑇 ) P𝑡 − 𝐶guess
𝑇
= 0. (60) method. The values of 𝛽 = 1 are the only case for which we
know the exact solution 𝑢(𝑥, 𝑡) = 𝑥𝑡. It should be noted that
The numerical results of Example 24 for different values only the fourth-order term of the FVIM was used in eval-
of 𝛼 and 𝛽 are shown in Figure 2. In addition, 𝐿 2 and uating the approximate solutions for Table 2. From Table 2,
𝐿 ∞ errors are presented in Table 1. From Table 1, one can it clearly appears that 2D-FLFs method is more accurate than
conclude that the solutions of 2D-FLFs method are in good FVIM and the obtained results are in good agreement with
agreement with the exact results. Compared with homotopy exact solution.
analysis method (HAM) [36] and homotopy perturbation
method (HPM) [37], 2D-FLFs method can get high accuracy Example 26. We finally consider the linear time-fractional
solution while only need a few terms of 2D-FLFs. wave equation:
Example 25. Consider the nonlinear time-fractional advec- 𝜕2𝛽 𝑢 1 2 𝜕2 𝑢

tion partial differential equation [37–39] = 𝑥 , 𝑡 > 0, 𝑥 ∈ 𝑅, 0.5 < 𝛽 ≤ 1, (63)
𝜕𝑡2𝛽 2 𝜕𝑥2
𝛽
𝐷𝑡 𝑢 (𝑥, 𝑡) + 𝑢 (𝑥, 𝑡) 𝑢𝑥 (𝑥, 𝑡) = 𝑥 + 𝑥𝑡2 , subject to the initial conditions
(61)
𝑡 > 0, 𝑥 ∈ 𝑅, 0 < 𝛽 ≤ 1, 𝜕𝑢 (𝑥, 0)
𝑢 (𝑥, 0) = 𝑥, = 𝑥2 . (64)
𝜕𝑡
subject to the initial condition
Table 3 gives a comparison of the approximate solutions
𝑢 (𝑥, 0) = 0. (62) at different values of 𝛽 using the FVIM [39] and 2D-FLFs
method. Figure 4 shows the numerical solutions of 2D-FLFs
Figure 3 gives the approximation solutions of (61) for method for (63) at different values of 𝛽 with 𝑚 = 3, 𝑚󸀠 = 9.
𝛽 = 0.50 with 𝑚 = 4, 𝑚󸀠 = 5 and 𝛽 = 0.75 with 𝑚 = 4, The values of 𝛽 = 1 are the only case for which we know
𝛽 = 0.50 𝛽 = 0.75
1.2 1.2
1 1
0.8 0.8
u 0.6 u 0.6
0.4 0.4
0.2 0.2
0 0
1 1
1 1
0.8 0.8
0.5 0.6 0.5 0.6
0.4 0.4
t 0.2 t 0.2
0 0 x 0 0 x
(a) (b)
Figure 3: Numerical results of Example 25 for different value of 𝛽.
𝛽 = 0.750 𝛽 = 0.875
4 4
u 2 u 2
0 0
1 1
1 1
0.5 0.5
0.5 0.5
t 0 0 x t 0 0 x
(a) (b)
𝛽 = 1.000 𝛽 = 1.000
×10−10
4 1
Error
u 2 0.5
0 0
1 1
1 1
0.5 0.5
0.5 0.5
t 0 0 x t 0 0 x
(c) (d)
Figure 4: Numerical results of Example 26 for different value of 𝛽.
Table 1: Errors of Example 24 for different values of 𝛼 and 𝛽 with 𝑀 = 𝑀󸀠 = 4.
Error 𝛼 = 𝛽 = 1.25 𝛼 = 𝛽 = 1.50 𝛼 = 𝛽 = 1.75 𝛼 = 𝛽 = 2.00

𝐿2 5.6437𝑒 − 015 1.2075𝑒 − 015 3.4584𝑒 − 015 8.9917𝑒 − 016
𝐿∞ 4.4409𝑒 − 016 1.1102𝑒 − 016 3.3307𝑒 − 016 1.1102𝑒 − 016
Table 2: Numerical values when 𝛽 = 0.50, 0.75, and 1.0 for (61).
𝛽 = 0.50 𝛽 = 0.75 𝛽 = 1.00

𝑡 𝑥
FVIM 2D-FLFs FVIM 2D-FLFs FVIM 2D-FLFs Exact
0.25 0.12422501 0.12225461 0.09230374 0.09224583 0.06250058 0.062500 0.062500
0.50 0.24845002 0.24450922 0.18460748 0.18449165 0.12500117 0.125000 0.125000
0.25
0.75 0.37267504 0.36676383 0.27691122 0.27673748 0.18750175 0.187500 0.187500
1.00 0.49690005 0.48901844 0.36921496 0.36898331 0.25000234 0.250000 0.250000
0.25 0.18377520 0.16584130 0.15148283 0.14985508 0.12507592 0.125000 0.125000
0.50 0.36755040 0.33168259 0.30296566 0.29971016 0.25015184 0.250000 0.250000
0.50
0.75 0.55132559 0.49752389 0.45444848 0.44956524 0.37522776 0.375000 0.375000
1.00 0.73510079 0.66336518 0.60593131 0.59942032 0.50030368 0.500000 0.500000
0.25 0.27227270 0.20678964 0.21407798 0.20119503 0.18881843 0.187500 0.187500
0.50 0.54454540 0.41357929 0.42815596 0.40239005 0.37763687 0.375000 0.375000
0.75
0.75 0.81681810 0.62036893 0.64223394 0.60358508 0.56645530 0.562500 0.562500
1.00 1.08909080 0.82715857 0.85631192 0.80478011 0.75527373 0.750000 0.750000
Table 3: Numerical values when 𝛽 = 0.750, 0.875, and 1.000 for (63).
𝛽 = 0.750 𝛽 = 0.875 𝛽 = 1.000

𝑡 𝑥
FVIM 2D-FLFs FVIM 2D-FLFs FVIM Exact
0.25 0.26622298 0.26622021 0.26593959 0.26594005 0.26578827 0.26578827
0.50 0.56489190 0.56488083 0.56375836 0.56376020 0.56315308 0.56315308
0.25
0.75 0.89600678 0.89598187 0.89345630 0.89346046 0.89209443 0.89209443
1.00 1.25956762 1.25952332 1.25503343 1.25504082 1.25261232 1.25261232
0.25 0.28474208 0.28474415 0.28340402 0.28340659 0.28256846 0.28256846
0.50 0.63896831 0.63897662 0.63361610 0.63362636 0.63027383 0.63027383
0.50
0.75 1.06267869 1.06269739 1.05063622 1.05065931 1.04311611 1.04311611
1.00 1.55587323 1.55590647 1.53446439 1.53450544 1.52109530 1.52109531
0.25 0.30690489 0.30690747 0.30361709 0.30361656 0.30139478 0.30139480
0.50 0.72761955 0.72762986 0.71446834 0.71446625 0.70557913 0.70557918
0.75
0.75 1.26214400 1.26216719 1.23255378 1.23254905 1.21255304 1.21255316
1.00 1.91047821 1.91051944 1.85787338 1.85786498 1.82231652 1.82231673
the exact solution 𝑢(𝑥, 𝑡) = 𝑥 + 𝑥2 sinh(𝑡). As previous, only Acknowledgments

the fourth-order term of the FVIM was used in evaluating the
numerical solutions for Table 3. In the case of 𝛽 = 1, it can This work is supported by the National Natural Science
be found that absolute error of 2D-FLFs is not bigger than Foundation of China (Grant no. 11272352). The authors are
1.0𝑒 − 10 which is very small compared with that obtained by grateful to the anonymous referees for their comments which
FVIM. substantially improved the quality of this paper.
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http://dx.doi.org/10.1155/2013/390132
Research Article
Persistence Property and Estimate on Momentum Support for
the Integrable Degasperis-Procesi Equation
Zhengguang Guo1 and Liangbing Jin2

1
College of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China
2
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
Correspondence should be addressed to Zhengguang Guo; gzgmath@gmail.com
Received 24 April 2013; Accepted 4 October 2013
Academic Editor: T. Raja Sekhar
Copyright © 2013 Z. Guo and L. Jin. This is an open access article distributed under the Creative Commons Attribution License,
It is shown that a strong solution of the Degasperis-Procesi equation possesses persistence property in the sense that the solution
with algebraically decaying initial data and its spatial derivative must retain this property. Moreover, we give estimates of measure
for the momentum support.
1. Introduction the addition of linear dispersion terms) both the Camassa-

Holm and DP equations are physically relevant; otherwise the
Recently, Degasperis and Procesi [1] consider the following DP equation would be of purely theoretical interest.
family of third order dispersive conservation laws: When 𝑐1 = −3𝑐3 /2𝛼2 and 𝑐2 = 𝑐3 /2 in (1), we recover the
Camassa-Holm equation derived physically by Camassa and
𝑢𝑡 + 𝑐0 𝑢𝑥 + 𝛾𝑢𝑥𝑥𝑥 − 𝛼2 𝑢𝑥𝑥𝑡 = (𝑐1 𝑢2 + 𝑐2 𝑢𝑥2 + 𝑐3 𝑢𝑢𝑥𝑥 )𝑥 , (1) Holm in [3] by approximating directly the Hamiltonian for
Euler’s equations in the shallow water regime, where 𝑢(𝑥, 𝑡)
where 𝛼, 𝛾, 𝑐0 , 𝑐1 , 𝑐2 , and 𝑐3 are real constants. Within this represents the free surface above a flat bottom. There is also
family, only three equations that satisfy asymptotic integra- a geometric approach which is used to prove the least action
bility condition up to third order are singled out, namely, the principle holding for the Camassa-Holm equation, compared
KdV equation with [4]. It is worth pointing out that a fundamental aspect of
the Camassa-Holm equation, the fact that it is a completely
𝑢𝑡 + 𝑢𝑥 + 𝑢𝑢𝑥 + 𝑢𝑥𝑥𝑥 = 0, (2) integrable system, was shown in [5, 6]. Some satisfactory
results have been obtained for this shallow water equation
the Camassa-Holm equation recently, we refer the readers to see [7–19].
Although, the DP equation (4) has a similar form to the
𝑢𝑡 − 𝑢𝑥𝑥𝑡 + 3𝑢𝑢𝑥 = 2𝑢𝑥 𝑢𝑥𝑥 + 𝑢𝑢𝑥𝑥𝑥 , (3)
Camassa-Holm equation and admits exact peakon solutions
analogous to the Camassa-Holm peakons [20], these two
and a new equation (the Degasperis-Procesi equation, the DP
equations are pretty different. The isospectral problem for
equation, for simplicity) which can be written as (after rescal-
equation (4) is
ing) the dispersionless form [1]
𝑢𝑡 − 𝑢𝑥𝑥𝑡 + 4𝑢𝑢𝑥 = 3𝑢𝑥 𝑢𝑥𝑥 + 𝑢𝑢𝑥𝑥𝑥 . (4) Ψ𝑥 − Ψ𝑥𝑥𝑥 − 𝜆𝑦Ψ = 0, (5)
It is worth noting that in [2] both the Camassa-Holm and while for Camassa-Holm equation it is
DP equations are derived as members of a one-parameter
family of asymptotic shallow water approximations to the 1
Ψ𝑥𝑥 − Ψ − 𝜆𝑦Ψ = 0, (6)
Euler equations: this is important because it shows that (after 4
where 𝑦 = 𝑢 − 𝑢𝑥𝑥 for both cases. This implies that the inside Theorem 1 (see [23]). Given 𝑢(𝑥, 𝑡 = 0) = 𝑢0 ∈ 𝐻𝑠 (R), 𝑠 >
structures of the DP equation (4) and the Camassa-Holm 3/2, then there exist a 𝑇 and a unique solution 𝑢 to (4) (also
equation are truly different. However, we not only have some (10)) such that
similar results [21–23], but also have considerable differences
in the scattering/inverse scattering approach, compared with 𝑢 (𝑥, 𝑡) ∈ 𝐶 ([0, 𝑇) ; 𝐻𝑠 (R)) ∩ 𝐶1 ([0, 𝑇) ; 𝐻𝑠−1 (R)) . (14)
the discussion in [5, 6] and in the paper [24].
Analogous to the Camassa-Holm equation, (4) can be It should be mentioned that due to the form of (10) (no
written in Hamiltonian form and has infinitely many con- derivative appears in the convolution term), Coclite and
servation laws. Here we list some of the simplest conserved Karlsen [25] established global existence and uniqueness
quantities [20]: result for entropy weak solutions belonging to the class
𝐿1 (R) ∩ 𝐵𝑉(R).
𝐻−1 = ∫ 𝑢3 𝑑𝑥, 𝐻0 = ∫ 𝑦 𝑑𝑥, 𝐻1 = ∫ 𝑦V 𝑑𝑥,
R R R
2. Unique Continuation
1/3
𝐻5 = ∫ 𝑦 𝑑𝑥, 𝐻7 = ∫ (𝑦𝑥2 𝑦−7/3 + 9𝑦 −1/3
) 𝑑𝑥,
R R The purpose of this section is to show that the solution to (10)
(7) and its first-order spatial derivative retain algebraic decay at
infinity as their initial values do. Precisely, we prove.
where V = (4−𝜕𝑥2 )−1 𝑢. So they are different from the invariants
of the Camassa-Holm equation Theorem 2. Assume that for some 𝑇 > 0 and 𝑠 > 3/2, 𝑢 ∈
𝐶([0, 𝑇]; 𝐻𝑠 (R)) is a strong solution of the initial value problem
𝐸 (𝑢) = ∫ (𝑢2 + 𝑢𝑥2 ) 𝑑𝑥, 𝐹 (𝑢) = ∫ (𝑢3 + 𝑢𝑢𝑥2 ) 𝑑𝑥. (8) associated with (10), and that 𝑢0 (𝑥) = 𝑢(𝑥, 0) satisfies that for
R R some 𝜃 > 1
Set 𝑄 = (1 − 𝜕𝑥2 ); then the operator 𝑄−1 in R can be 󵄨󵄨 󵄨 󵄨󵄨 󵄨
󵄨󵄨𝑢0 (𝑥)󵄨󵄨󵄨 , 󵄨󵄨𝜕𝑥 𝑢0 (𝑥)󵄨󵄨󵄨 = 𝑂 (𝑥 )
−𝜃
expressed by as 𝑥 ↑ ∞. (15)
1 Then
𝑄−1 𝑓 = 𝐺 ∗ 𝑓 = ∫ 𝑒−|𝑥−𝑦| 𝑓 (𝑦) 𝑑𝑦. (9)
2 R 󵄨󵄨 󵄨
󵄨󵄨𝜕𝑥 𝑢 (𝑥, 𝑡)󵄨󵄨󵄨 = 𝑂 (𝑥 )
−𝜃
|𝑢 (𝑥, 𝑡)| , as 𝑥 ↑ ∞, (16)
Equation (4) can be written as
3 uniformly in the time interval [0, 𝑇].
𝑢𝑡 + 𝑢𝑢𝑥 + 𝜕𝑥 𝐺 ∗ ( 𝑢2 ) = 0, (10)
2
Notation. We will say that
while the Camassa-Holm equation can be written as
󵄨󵄨 󵄨
󵄨󵄨 󵄨 󵄨𝑓 (𝑥)󵄨󵄨
1
𝑢𝑡 + 𝑢𝑢𝑥 + 𝜕𝑥 𝐺 ∗ (𝑢 + 𝑢𝑥2 ) = 0.
2
(11) 󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨 = 𝑂 (𝑥 )
−𝜃
as 𝑥 ↑ ∞ if lim 󵄨 −𝜃 󵄨 = 𝐿, (17)
𝑥→∞ 𝑥
2
On the other hand, the DP equation can also be expressed in where 𝐿 is a nonnegative constant.
the following momentum form:
Proof. We introduce the following notations:
𝑦𝑡 + 𝑦𝑥 𝑢 = −3𝑦𝑢𝑥
(12) 3
𝑦 = (1 − 𝜕𝑥2 ) 𝑢. 𝐹 (𝑢) = 𝑢2 , (18)
2
This formulation is important to motivate us to consider 𝑀 = sup ‖𝑢 (𝑡)‖𝐻𝑠 . (19)
the measure of momentum support which is the second 𝑡∈[0,𝑇]
object of this paper, since we found that (12) is similar to the

vorticity equation of the three-dimensional Euler equation Multiplying (10) by 𝑢2𝑝−1 with 𝑝 ∈ 𝑍+ and integrating the
for incompressible perfect fluids (𝑈 is the speed, and 𝜔 is its result in the 𝑥-variable, one gets
vorticity) ∞
∫ 𝑢2𝑝−1 (𝑢𝑡 + 𝑢𝑢𝑥 + 𝜕𝑥 𝐺 ∗ 𝐹 (𝑢)) 𝑑𝑥 = 0. (20)
𝜔𝑡 + (𝑈 ⋅ ∇) 𝜔 = (𝜔 ⋅ ∇) 𝑈, −∞
div 𝑈 = 0, (13) The first term in (20) is

curl 𝑈 = 𝜔. ∞ ∞
1 𝑑𝑢2𝑝
∫ 𝑢2𝑝−1 𝑢𝑡 𝑑𝑥 = ∫ 𝑑𝑥
The stretching term (𝜔 ⋅ ∇)𝑈 in (13) is similar to the term −∞ −∞ 2𝑝 𝑑𝑡
−3𝑦𝑢𝑥 in (12).
1 𝑑 ∞ 2𝑝 2𝑝−1 𝑑
One can follow the argument for the Camassa-Holm = ∫ 𝑢 𝑑𝑥 = ‖𝑢 (𝑡)‖2𝑝 ‖𝑢 (𝑡)‖2𝑝 ,
equation [8] to establish the following well posedness theo- 2𝑝 𝑑𝑡 −∞ 𝑑𝑡
rem for the Degasperis-Procesi equation. (21)
and for the rest, we have Thus, for any appropriate function 𝑓 one finds that
󵄨󵄨 ∞ 󵄨󵄨󵄨 󵄨󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨
󵄨󵄨
󵄨󵄨∫ 𝑢
2𝑝−1
𝑢𝑢𝑥 𝑑𝑥󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ 𝑢2𝑝 𝑢𝑥 𝑑𝑥󵄨󵄨󵄨
󵄨 󵄨󵄨𝜑𝑁𝜕𝑥 𝐺 ∗ 𝑓2 (𝑥)󵄨󵄨󵄨
󵄨 󵄨
󵄨󵄨 −∞ 󵄨󵄨 󵄨󵄨 −∞ 󵄨󵄨
󵄨󵄨 1 ∞ 󵄨󵄨
󵄩 󵄩 󵄨 󵄨
2𝑝
≤ 󵄩󵄩󵄩𝑢𝑥 (𝑡)󵄩󵄩󵄩∞ ‖𝑢 (𝑡)‖2𝑝 , = 󵄨󵄨󵄨 𝜑𝑁 (𝑥) ∫ sgn (𝑥 − 𝑦) 𝑒−|𝑥−𝑦| 𝑓2 (𝑦) 𝑑𝑦󵄨󵄨󵄨
󵄨󵄨 2 −∞ 󵄨󵄨
󵄨󵄨 ∞ 󵄨󵄨
󵄨󵄨 󵄨 2𝑝−1 󵄩 󵄩 𝜑𝑁 (𝑥) ∞ −|𝑥−𝑦| 1
󵄨󵄨∫ 𝑢
2𝑝−1
𝜕𝑥 𝐺 ∗ 𝐹 (𝑢) 𝑑𝑥󵄨󵄨󵄨 ≤ ‖𝑢 (𝑡)‖2𝑝 󵄩󵄩󵄩𝜕𝑥 𝐺 ∗ 𝐹 (𝑢) (𝑡)󵄩󵄩󵄩2𝑝 . ≤ ∫ 𝑒 𝜑 (𝑦) 𝑓 (𝑦) 𝑓 (𝑦) 𝑑𝑦
󵄨󵄨 −∞ 󵄨󵄨 2 −∞ 𝜑𝑁 (𝑦) 𝑁
(22)
𝜑𝑁 (𝑥) ∞ 𝑒−|𝑥−𝑦| 󵄩 󵄩 󵄩 󵄩
From the above inequalities, we get ≤( ∫ 𝑑𝑦) 󵄩󵄩󵄩𝜑𝑁𝑓󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩∞
2 −∞ 𝜑𝑁 (𝑦)
𝑑 󵄩 󵄩 󵄩 󵄩
󵄩 󵄩 󵄩 󵄩
‖𝑢 (𝑡)‖2𝑝 ≤ 󵄩󵄩󵄩𝑢𝑥 (𝑡)󵄩󵄩󵄩∞ ‖𝑢 (𝑡)‖2𝑝 + 󵄩󵄩󵄩𝜕𝑥 𝐺 ∗ 𝐹 (𝑢)󵄩󵄩󵄩2𝑝 , (23) ≤ 𝐶0 󵄩󵄩󵄩𝜑𝑁𝑓󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩∞ .
𝑑𝑡 (32)
and therefore, by Sobolev embedding theorem and Gronwall’s
Combining with (30), we get
inequality, there exists a constant 𝑀 such that
𝑡
𝑡 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩
‖𝑢 (𝑡)‖2𝑝
󵄩 󵄩
≤ (‖𝑢 (0)‖2𝑝 + ∫ 󵄩󵄩󵄩𝜕𝑥 𝐺 ∗ 𝐹 (𝑢)󵄩󵄩󵄩2𝑝 𝑑𝜏) 𝑒𝑀𝑡 . (24) 󵄩󵄩𝑢 (𝑡) 𝜑𝑁󵄩󵄩󵄩∞ ≤ 𝐶1 (󵄩󵄩󵄩𝑢0 𝜑𝑁󵄩󵄩󵄩∞ + ∫ 󵄩󵄩󵄩𝜑𝑁𝑢󵄩󵄩󵄩∞ 𝑑𝜏) , (33)
0
0
where 𝐶1 = 𝐶1 (𝑀; 𝑇, ) > 0. By Gronwall’s inequality, there

Since 𝑓 ∈ 𝐿1 (R) ∩ 𝐿∞ (R) implies ̃ for any 𝑡 ∈ [0, 𝑇] such that
exists a constant 𝐶
󵄩 󵄩 󵄩 󵄩
lim 󵄩󵄩𝑓󵄩󵄩󵄩𝑞 = 󵄩󵄩󵄩𝑓󵄩󵄩󵄩∞ ,
𝑞 → ∞󵄩 (25) 󵄩󵄩 󵄩 ̃󵄩󵄩󵄩󵄩𝑢0 ⋅ max (1, 𝑥𝜃 )󵄩󵄩󵄩󵄩 .
̃󵄩󵄩󵄩𝑢0 𝜑𝑁󵄩󵄩󵄩 ≤ 𝐶
󵄩󵄩𝜑𝑁𝑢󵄩󵄩󵄩∞ ≤ 𝐶 (34)
󵄩 󵄩∞ 󵄩 󵄩∞
taking the limits in (24) (note that 𝜕𝑥 𝐺 ∈ 𝐿1 and 𝐹(𝑢) ∈ 𝐿1 ∩
Finally, taking the limit as 𝑁 goes to infinity in (34) we find
𝐿∞ ) from (25) we get
that for any 𝑡 ∈ [0, 𝑇]
𝑡
󵄩 󵄩 󵄨󵄨 󵄨 ̃󵄩󵄩 󵄩
‖𝑢 (𝑡)‖∞ ≤ (‖𝑢 (0)‖∞ + ∫ 󵄩󵄩󵄩𝜕𝑥 𝐺 ∗ 𝐹 (𝑢)󵄩󵄩󵄩∞ 𝑑𝜏) 𝑒𝑀𝑡 . (26) 󵄨󵄨𝑢 (𝑥, 𝑡) 𝑥𝜃 󵄨󵄨󵄨 ≤ 𝐶 󵄩󵄩𝑢0 ⋅ max (1, 𝑥𝜃 )󵄩󵄩󵄩 . (35)
0 󵄨 󵄨 󵄩 󵄩∞
We will now repeat the above arguments using the barrier From (15), we get |𝑢(𝑥, 𝑡)| = 𝑂(𝑥−𝜃 ) as 𝑥 ↑ ∞.
function Next, differentiating (10) in the 𝑥-variable produces the
equation
{ 1, 𝑥 ≤ 1,
{ 𝜃
𝜑𝑁 (𝑥) = {𝑥 , 𝑥 ∈ (1, 𝑁) , 3
{ 𝜃
(27) 𝑢𝑥𝑡 + 𝑢𝑢𝑥𝑥 + 𝑢𝑥2 + 𝜕𝑥2 𝐺 ∗ ( 𝑢2 ) = 0. (36)
2
{𝑁 , 𝑥 ≥ 𝑁,
Again, multiplying (36) by 𝑢𝑥2𝑝−1 , (𝑝 ∈ Z+ ), integrating the
where 𝑁 ∈ Z+ . Observe that for all 𝑁 we have
result in the 𝑥-variable, and using integration by parts
󸀠
0 ≤ 𝜑𝑁 (𝑥) ≤ 𝜃𝜑𝑁 (𝑥) a.e. 𝑥 ∈ R. (28) 2𝑝
∞ ∞(𝑢 )
2𝑝−1
∫ 𝑢𝑢𝑥𝑥 (𝑢𝑥 ) 𝑑𝑥 = ∫ 𝑢 𝑥 𝑑𝑥
Using notation in (18), from (10) we obtain −∞ −∞ 2𝑝
(37)
(𝑢𝜑𝑁)𝑡 + (𝑢𝜑𝑁) 𝑢𝑥 + 𝜑𝑁𝜕𝑥 𝐺 ∗ 𝐹 (𝑢) = 0. (29) 1 ∞ 2𝑝
= − ∫ 𝑢𝑥 (𝑢𝑥 ) 𝑑𝑥,
2𝑝 −∞
Hence, as in the weightless case (26), we get
one gets the inequality
󵄩󵄩 󵄩 𝑀𝑡 󵄩 󵄩
󵄩󵄩𝑢 (𝑡) 𝜑𝑁󵄩󵄩󵄩∞ ≤ 𝑒 󵄩󵄩󵄩𝑢 (0) 𝜑𝑁󵄩󵄩󵄩∞
𝑑 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 2 󵄩
𝑡 (30) 󵄩󵄩𝑢𝑥 (𝑡)󵄩󵄩󵄩2𝑝 ≤ 2󵄩󵄩󵄩𝑢𝑥 (𝑡)󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝑢𝑥 (𝑡)󵄩󵄩󵄩2𝑝 + 󵄩󵄩󵄩󵄩𝜕𝑥 𝐺 ∗ 𝐹 (𝑢)󵄩󵄩󵄩󵄩2𝑝 ,
󵄩 󵄩 𝑑𝑡
+𝑒 𝑀𝑡
∫ 󵄩󵄩󵄩𝜑𝑁𝜕𝑥 𝐺 ∗ 𝐹 (𝑢)󵄩󵄩󵄩∞ 𝑑𝜏.
0 (38)
A simple calculation shows that there exists 𝐶0 > 0 depend- and therefore as before
ing only on 𝜃 such that, for any 𝑁 ∈ Z+ ,
𝑡
󵄩󵄩 󵄩 󵄩 󵄩 󵄩 2 󵄩
󵄩󵄩𝑢𝑥 (𝑡)󵄩󵄩󵄩2𝑝 ≤ (󵄩󵄩󵄩𝑢𝑥 (0)󵄩󵄩󵄩2p + ∫ 󵄩󵄩󵄩󵄩𝜕𝑥 𝐺 ∗ 𝐹 (𝑢)󵄩󵄩󵄩󵄩2𝑝 𝑑𝜏) 𝑒 .
2𝑀𝑡
∞
1 1
𝜑𝑁 (𝑥) ∫ 𝑒−|𝑥−𝑦| 𝑑𝑦 ≤ 𝐶0 . (31) 0
2 −∞ 𝜑𝑁 (𝑦) (39)
Since 𝜕𝑥2 𝐺 = 𝐺 − 𝛿, we can use (25) and pass to the limit in Moreover, the exponential behavior of 𝑢 in 𝑥 outside this
(39) to obtain support is obvious. The comparison of the DP equation and
𝑡
the incompressible Euler equation above implies that the
󵄩󵄩 󵄩 󵄩 󵄩 󵄩 2 󵄩 momentum 𝑦(𝑥, 𝑡) in (12) plays a similar role as the vorticity
󵄩󵄩𝑢𝑥 (𝑡)󵄩󵄩󵄩∞ ≤ (󵄩󵄩󵄩𝑢𝑥 (0)󵄩󵄩󵄩∞ + ∫ 󵄩󵄩󵄩󵄩𝜕𝑥 𝐺 ∗ 𝐹 (𝑢)󵄩󵄩󵄩󵄩∞ 𝑑𝜏) 𝑒 ;
2𝑀𝑡
0 does in (13). This motivates us to estimate the size of supp
(40) 𝑦(𝑡, ⋅) for strong solutions. The approach is inspired by the
work of Kim [28] and the recent work [29].
from (36) we get We first introduce the particle trajectory method. Let 𝑢 ∈
𝐶([0, 𝑇], 𝐻3 (R)) ∩ 𝐶1 ([0, 𝑇], 𝐻2 (R)) be a strong solution of
𝜕𝑡 (𝑢𝑥 𝜑𝑁) + 𝑢𝑢𝑥𝑥 𝜑𝑁 + (𝑢𝑥 𝜑𝑁) 𝑢𝑥 + 𝜑𝑁𝜕𝑥2 𝐺 ∗ 𝐹 (𝑢) = 0. (4) guaranteed by the well posedness Theorem 1. Let 𝑠 ∈
(41) [0, 𝑇], 𝑞(𝑡; 𝛼, 𝑠) be the solution of the following initial value
problem:
We need to eliminate the second derivatives in the second
term in (41). Thus, combining integration by parts and (28), 𝑑𝑞 (𝑡; 𝛼, 𝑠)
we find = 𝑢 (𝑠 + 𝑡, 𝑞 (𝑡; 𝛼, 𝑠)) , 𝑠, 𝑠 + 𝑡 ∈ [0, 𝑇] , 𝛼 ∈ R,
𝑑𝑡
󵄨󵄨 ∞ 󵄨󵄨
󵄨󵄨 2𝑝−1 󵄨 𝑞 (0; 𝛼, 𝑠) = 𝛼, 𝛼 ∈ R.
󵄨󵄨∫ 𝑢𝑢𝑥𝑥 𝜑𝑁(𝑢𝑥 𝜑𝑁) 𝑑𝑥󵄨󵄨󵄨
󵄨󵄨 −∞ 󵄨󵄨 (47)
󵄨󵄨 ∞ 󵄨󵄨
󵄨 2𝑝−1 󵄨
= 󵄨󵄨󵄨∫ 𝑢(𝑢𝑥 𝜑𝑁) 󸀠
(𝜕𝑥 (𝑢𝑥 𝜑𝑁) − 𝑢𝑥 𝜑𝑁 ) 𝑑𝑥󵄨󵄨󵄨 Then, 𝑞(𝑡; ⋅, 𝑠) : R → R is an increasing diffeomorphism. It
󵄨󵄨 −∞ 󵄨󵄨 is shown [21, 23] that
󵄨󵄨 ∞ 2𝑝 󵄨󵄨󵄨
󵄨󵄨 (𝑢 𝜑 ) 󵄨
= 󵄨󵄨󵄨∫ 𝑢 (𝜕𝑥 ( 𝑥 𝑁 ) − 𝑢𝑥 𝜑𝑁 󸀠
(𝑢𝑥 𝜑𝑁)
2𝑝−1
) 𝑑𝑥󵄨󵄨󵄨 𝑦 (𝑞 (𝑡; 𝑥, 0) , 𝑡) 𝑞𝑥3 (𝑡; 𝑥, 0) = 𝑦 (𝑥, 0) ; (48)
󵄨󵄨 −∞ 2𝑝 󵄨󵄨
󵄨 󵄨
this implies that the support of 𝑦 propagates along the flow.
󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩2𝑝
≤ 𝜅 ⋅ (‖𝑢 (𝑡)‖∞ + 󵄩󵄩𝜕𝑥 𝑢 (𝑡)󵄩󵄩∞ ) 󵄩󵄩𝜕𝑥 𝑢𝜑𝑁󵄩󵄩2𝑝 . Set 𝐷(𝑡) to be the support of 𝑦(⋅, 𝑡). Let 𝜓 ∈ 𝐿2 (𝐷(𝑠)), and let
(42) 𝜓𝑡 ∈ 𝐿2 (𝐷(𝑠 + 𝑡)) be given by the following:
Since 𝜕𝑥2 𝐺 = 𝐺 − 𝛿, the argument in (32) also shows that 𝜓𝑡 (𝑞 (𝑡; 𝛼, 𝑠)) = 𝜓 (𝛼) . (49)
󵄨󵄨 󵄨
󵄨󵄨𝜑𝑁𝜕𝑥2 𝐺 ∗ 𝑓2 (𝑥)󵄨󵄨󵄨 ≤ 𝐶0 󵄩󵄩󵄩󵄩𝜑𝑁𝑓󵄩󵄩󵄩󵄩∞ 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩∞ . (43)
Moreover, we also want to mention the standard argument
󵄨 󵄨 on the first Dirichlet eigenvalue problem. Let Ω be an open
Similarly, we get interval in R, and, 𝜆 1 (Ω) be the first Dirichlet eigenvalue of
the Laplacian on Ω. Then we have
󵄩󵄩 󵄩
󵄩󵄩𝑢𝑥 (𝑡) 𝜑𝑁󵄩󵄩󵄩∞ 󵄩 󵄩2 󵄩 󵄩
𝜆 1 (Ω) = inf {󵄩󵄩󵄩󵄩𝜙󸀠 󵄩󵄩󵄩󵄩𝐿2 (Ω) | 𝜙 ∈ 𝐻01 (Ω) with 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐿2 (Ω) = 1} .
𝑡 (44)
󵄩 󵄩 󵄩 󵄩
≤ 𝐶2 (󵄩󵄩󵄩𝑢𝑥 (0) 𝜑𝑁󵄩󵄩󵄩∞ + ∫ 󵄩󵄩󵄩𝑢 (𝜏) 𝜑𝑁󵄩󵄩󵄩∞ 𝑑𝜏) , (50)
0
It is just (𝜋/|Ω|)2 and the normalized eigenfunctions are the
where 𝐶2 = 𝐶2 (𝑀; 𝑇).
suitable translations of
Then, taking the limit as 𝑁 goes to infinity, we find that
for any 𝑡 ∈ [0, 𝑇] 2 1/2 𝜋𝑥
±( ) sin ( ). (51)
󵄨󵄨 󵄨 󵄩 󵄩 𝑡
󵄩 󵄩 |Ω| |Ω|
󵄨󵄨𝑢𝑥 (𝑡) 𝑥𝜃 󵄨󵄨󵄨 ≤ 𝐶2 (󵄩󵄩󵄩𝑢𝑥 (0) 𝑥𝜃 󵄩󵄩󵄩 + ∫ 󵄩󵄩󵄩𝑢 (𝜏) 𝑥𝜃 󵄩󵄩󵄩 𝑑𝜏) . (45)
󵄨 󵄨 󵄩 󵄩∞ 󵄩 󵄩∞
0 Theorem 3. Let 𝑦 ∈ 𝐶([0, 𝑇]; 𝐻1 (R)) ∩ 𝐶1 ([0, 𝑇]; 𝐿2 (R)) be
a strong solution of (12). Let 𝐷(𝑡) be the support of 𝑦(⋅, 𝑡) for
Since |𝑢(𝑥, 𝑡)| = 𝑂(𝑥−𝜃 ) as 𝑥 ↑ ∞ and (15), we get 𝑡 ∈ [0, 𝑇] with its initial 𝐷(0) being connected.
󵄨󵄨 󵄨
󵄨󵄨𝜕𝑥 𝑢 (𝑥, 𝑡)󵄨󵄨󵄨 = 𝑂 (𝑥 ) ,
−𝜃
as 𝑥 ↑ ∞. (46) (I) Suppose there exists a positive constant 𝐾 such that
𝑢𝑥 (𝑥, 𝑘) > −𝐾 for (𝑥, 𝑡) ∈ R × [0, 𝑇]. Then
This completes the proof.
|𝐷 (0)| 𝑒−(exp(5𝐾𝑇/2)‖𝑦0 ‖𝐿2 (R) )𝑡
3. Measure of Momentum Support (52)
≤ |𝐷 (𝑡)| ≤ |𝐷 (0)| 𝑒(exp(5𝐾𝑇/2)‖𝑦0 ‖𝐿2 (R) )𝑡 .
It is known that, for the Degasperis-Procesi equation, the
momentum density 𝑦(𝑥, 𝑡) with compactly supported initial (II) 𝑦0 does not change sign or
data 𝑦0 (𝑥) will retain this property; that is, 𝑦(𝑥, 𝑡) is also
compactly supported [21]. However, the same argument for 𝑦0 (𝑥) ≤ 0, 𝑥 ∈ (−∞, 𝑥0 ) ,
𝑢(𝑥, 𝑡) is false [21]. Note that a detailed description of solution (53)
𝑢(𝑥, 𝑡) outside of the support of 𝑦(𝑥, 𝑡) is given in [26, 27]. 𝑦0 (𝑥) ≥ 0, 𝑥 ∈ (𝑥0 , ∞) ,
and 𝑦0 ∈ 𝐻1 (R) ∩ 𝐿1 (R); then, for all 𝑡 ≥ 0 where we denote 𝑞(𝑡; 𝑥, 𝑠) with 𝑠 = 0 by 𝑞(𝑥, 𝑡). By direct com-
putation, we have
|𝐷 (0)| 𝑒−‖𝑦0 ‖𝐿1 (R)𝑡 ≤ |𝐷 (𝑡)| ∞ 𝑞(𝑥 ,𝑡)
󵄨 󵄨 0
(54) ∫ 󵄨󵄨󵄨𝑦 (𝑥, 𝑡)󵄨󵄨󵄨 𝑑𝑥 = ∫ 𝑦 (𝑥, 𝑡) 𝑑𝑥 − ∫ 𝑦 (𝑥, 𝑡) 𝑑𝑥.
≤ |𝐷 (0)| 𝑒‖𝑦0 ‖𝐿1 (R)𝑡 . R 𝑞(𝑥 ,𝑡) 0 −∞
(64)
Proof. (I) The relation of momenta 𝑦 and 𝑢 gives Next, we prove that ‖𝑦(𝑥, 𝑡)‖𝐿1 (R) is decreasing with respect to
time. To this end, one gets, by differentiating (64) with respect
1 to 𝑡 and integrating by parts,
𝑢 (𝑥, 𝑡) = ∫ 𝑒−|𝑥−𝜉| 𝑦 (𝜉, 𝑡) 𝑑𝜉, (55)
2 R 𝑑 ∞
󵄨 󵄨
∫ 󵄨󵄨󵄨𝑦 (𝑥, 𝑡)󵄨󵄨󵄨 𝑑𝑥 = ∫ 𝑦𝑡 (𝑥, 𝑡) 𝑑𝑥
1 𝑑𝑡 R
𝑢𝑥 (𝑥, 𝑡) = ∫ sgn (𝜉 − 𝑥) 𝑒−|𝑥−𝜉| 𝑦 (𝜉, 𝑡) 𝑑𝜉. (56) 𝑞(𝑥0 ,𝑡)
2 R
𝑞(𝑥0 ,𝑡)
Then, we have by (12) and the lower bound of 𝑢𝑥 −∫ 𝑦𝑡 (𝑥, 𝑡) 𝑑𝑥

−∞
𝑑 − 2 (𝑦𝑢) (𝑞 (𝑥0 , 𝑡) , 𝑡)
∫ 𝑦2 (𝑥, 𝑡) 𝑑𝑥
𝑑𝑡 R ∞
= −∫ (𝑦𝑥 𝑢 + 3𝑦𝑢𝑥 ) 𝑑𝑥
= −5 ∫ 𝑢𝑥 (𝑥, 𝑡) 𝑦2 (𝑥, 𝑡) 𝑑𝑥 ≤ 5𝐾 ∫ 𝑦2 (𝑥, 𝑡) 𝑑𝑥. 𝑞(𝑥0 ,𝑡)
R R
𝑞(𝑥0 ,𝑡)
(57)
+∫ (𝑦𝑥 𝑢 + 3𝑦𝑢𝑥 ) 𝑑𝑥
−∞
Thus
− 2 (𝑦𝑢) (𝑞 (𝑥0 , 𝑡) , 𝑡)
𝑑 󵄩󵄩 󵄩2 󵄩 󵄩2
󵄩𝑦 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 ≤ 5𝐾󵄩󵄩󵄩𝑦 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 . (58)
𝑑𝑡 󵄩
∞ 𝑞(𝑥0 ,𝑡)
= −2 ∫ 𝑦𝑢𝑥 𝑑𝑥 + 2 ∫ 𝑦𝑢𝑥 𝑑𝑥
𝑞(𝑥0 ,𝑡) −∞
Therefore, (56), (58), and Gronwall inequality imply that
= 𝑢2 (𝑞 (𝑥0 , 𝑡) , 𝑡) − 𝑢𝑥2 (𝑞 (𝑥0 , 𝑡) , 𝑡)
󵄨󵄨 󵄨 1 󵄩󵄩 󵄩 1 5𝐾𝑇/2 󵄩󵄩 󵄩󵄩
󵄨󵄨𝑢𝑥 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ 󵄩󵄩𝑦 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 ≤ 𝑒 󵄩󵄩𝑦0 󵄩󵄩𝐿2 . (59) ∞ 𝑞(𝑥0 ,𝑡)
2 2
=∫ 𝑒−𝜉 𝑦 (𝜉, 𝑡) 𝑑𝑥 ∫ 𝑒𝜉 𝑦 (𝜉, 𝑡) 𝑑𝑥
On the other hand, due to Propositions A.2 and A.3, 𝜆 1 (𝐷(𝑠)) 𝑞(𝑥0 ,𝑡) −∞
is Lipschitz and differentiable almost everywhere. Moreover, ≤ 0.

we have (65)
𝑑 This implies that
−4𝑀1 𝜆 1 (𝐷 (𝑠)) ≤ 𝜆 (𝐷 (𝑠)) ≤ 4𝑀1 𝜆 1 (𝐷 (𝑠)) . (60)
𝑑𝑠 1
󵄨󵄨 1 󵄩󵄩
󵄨 󵄩 1󵄩 󵄩
󵄩󵄩𝑦 (𝑥, 𝑡)󵄩󵄩󵄩𝐿1 (R) ≤ 󵄩󵄩󵄩𝑦0 (𝑥)󵄩󵄩󵄩𝐿1 (R) .
󵄨󵄨𝑢𝑥 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ (66)
Then, it follows that 2 2
Therefore, (54) follows by replacing 𝑀1 with ‖𝑦0 (𝑥)‖𝐿1 (R) /2
𝑒−4𝑀1 𝑠 𝜆 1 (𝐷 (0)) ≤ 𝜆 1 (𝐷 (𝑠)) ≤ 𝑒4𝑀1 𝑠 𝜆 1 (𝐷 (0)) (61) in (61).
with 𝜆 1 (𝐷(𝑠)) = 𝜋2 /|𝐷(𝑠)|2 . So (52) follows from (61) and

(59). Appendix
(II) If 𝑦0 ∈ 𝐻1 (R) ∩ 𝐿1 (R) does not change sign, we
conclude that solutions of (10) exist globally in time. Equality The following propositions with standard proofs are known
(56) and the conservation of ∫R 𝑦(𝑥, 𝑡)𝑑𝑥 yield in [29]; we list them here only for convenience of readers.
󵄨󵄨 󵄨 1 󵄩󵄩 󵄩 1󵄩 󵄩 Proposition A.1. Let 𝑠, 𝑠 + 𝑡 ∈ [0, 𝑇], 𝛼 ∈ 𝐷(𝑠), and 𝜓 ∈

󵄨󵄨𝑢𝑥 (𝑥, 𝑡)󵄨󵄨󵄨 ≤ 󵄩󵄩𝑦 (𝑥, 𝑡)󵄩󵄩󵄩𝐿1 (R) = 󵄩󵄩󵄩𝑦0 (𝑥)󵄩󵄩󵄩𝐿1 (R) . (62) 𝐻01 (𝐷(𝑠)); 𝑢𝑥 can be bounded by a constant 𝑀1 ; then
2 2
(a)
By similar arguments of (I), constant 𝑀1 in (61) can be
replaced by ‖𝑦0 (𝑥)‖𝐿1 (R) /2; then (54) follows. If (53) is satis- 𝑒−𝑀1 |𝑡| ≤ 𝑞𝛼 (𝑡; 𝛼, 𝑠) ≤ 𝑒𝑀1 |𝑡| , (A.1)
fied, we know that the solution of (10) exists globally in time
[21, 30]. From (53) and (48), it is easy to get (b)
󵄨󵄨 󸀠 󵄨󵄨 −𝑀1 |𝑡| 󵄨󵄨󵄨 𝑡 󸀠 󵄨󵄨
󵄨󵄨
󵄨󵄨𝜓 (𝛼)󵄨󵄨 𝑒 ≤ 󵄨
󵄨(𝜓 ) (𝑞 (𝑡; 𝛼, 𝑠)) 󵄨󵄨
𝑦 (𝑥, 𝑡) ≤ 0, 𝑥 ∈ (−∞, 𝑞 (𝑥0 , 𝑡)) , 󵄨 󵄨 󵄨
(63) (A.2)
󵄨 󵄨
𝑦 (𝑥, 𝑡) ≥ 0, 𝑥 ∈ (𝑞 (𝑥0 , 𝑡) , ∞) , ≤ 󵄨󵄨󵄨󵄨𝜓󸀠 (𝛼)󵄨󵄨󵄨󵄨 𝑒𝑀1 |𝑡| ,
(c) Combing (A.9) and (A.10) together yields

󵄩󵄩 󵄩󵄩 󵄩 󵄩
−𝑀 |𝑡|/2
󵄩󵄩𝜓󵄩󵄩𝐿2 (𝐷(𝑠)) 𝑒 1 ≤ 󵄩󵄩󵄩󵄩𝜓𝑡 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠+𝑡)) lim sup
𝜆 1 (𝐷 (𝑠 + 𝑡)) − 𝜆 1 (𝐷 (𝑠))
(A.3) 𝑡 → 0+ 𝑡
󵄩 󵄩
≤ 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐿2 (𝐷(𝑠)) 𝑒𝑀1 |𝑡|/2 . 󵄩 󵄩2 󵄩 󵄩2
𝑒4𝑀1 𝑡 󵄩󵄩󵄩󵄩𝜙1󸀠 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠)) − 󵄩󵄩󵄩󵄩𝜙1󸀠 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠)) (A.11)
Proof. (a) Differentiating (47) with respect to 𝛼, we obtain ≤ lim+ sup
𝑡→0 𝑡
𝑑𝑞𝑡 = 4𝑀1 𝜆 1 (𝐷 (𝑠)) .
= 𝑢𝑞 𝑞𝛼 . (A.4)
𝑑𝛼
The second one follows by similar arguments for 𝑡 < 0.
Since 𝑞(𝑡; ⋅, 𝑠) : R → R is an increasing diffeomorphism,
then 𝑞𝛼 > 0. Combining the bound of 𝑢𝑥 , there holds Proposition A.3. Under the hypothesis of Theorem 3, for 𝑠, 𝑠 +
−𝑀1 𝑞𝛼 ≤ 𝑞𝛼𝑡 ≤ 𝑀1 𝑞𝛼 . (A.5) 𝑡 ∈ [0, 𝑇],
This can be solved as (a). 𝜆 1 (𝐷 (𝑠 + 𝑡)) − 𝜆 1 (𝐷 (𝑠))

lim− sup ≤ 4𝑀1 𝜆 1 (𝐷 (𝑠)) ,
(b) Differentiating (49) with respect to 𝛼 to get 𝑡→0 𝑡
𝜓𝑞𝑡 𝑞𝛼 = 𝜓󸀠 (𝛼) , 𝜆 1 (𝐷 (𝑠 + 𝑡)) − 𝜆 1 (𝐷 (𝑠))

(A.6) lim inf ≥ −4𝑀1 𝜆 1 (𝐷 (𝑠)) .
𝑡 → 0+ 𝑡
then (A.2) is a direct consequence of (A.1). (A.12)
(c) Equation (49) and the definition of Sobolev norm give
that Proof. Let 𝜙1 ∈ 𝐻01 (𝐷(𝑠)) with ‖𝜙1 ‖𝐿2 (𝐷(𝑠)) = 1 be a first
󵄩󵄩 𝑡 󵄩󵄩2 𝑡 2 2
normalized eigenfunction on 𝐷(𝑠), and let 𝜙2 ∈ 𝐿2 (𝐷(𝑠)) be
󵄩󵄩𝜓 󵄩󵄩 2
󵄩 󵄩𝐿 (𝐷(𝑠+𝑡)) = ∫𝐷(𝑠+𝑡) 𝜓 (𝑥) 𝑑𝑥 = ∫𝐷(𝑠) 𝜓 (𝛼) 𝑞𝛼 𝑑𝛼, (A.7) such that its 𝑡-transport is a normalized first eigenfunction on
𝐷(𝑠 + 𝑡). For 𝑡 > 0, using the left halves of (A.1) and (A.2) and
where we have used the change of variable 𝑥 = 𝑞(𝑡; 𝛼, 𝑠). So then the right half of (A.3) we get
(A.3) follows from (A.1).
󵄩󵄩 𝑡 󸀠 󵄩󵄩2 󸀠 2
󵄩󵄩(𝜙 ) 󵄩󵄩 𝑡
󵄩󵄩 2 󵄩󵄩𝐿2 (𝐷(𝑠+𝑡)) = ∫𝐷(𝑠+𝑡) [(𝜙2 (𝑥)) ] 𝑑𝑥
Proposition A.2. Under the hypothesis of Theorem 3, for 𝑠, 𝑠 +
𝑡 ∈ [0, 𝑇], 󸀠 2
𝜆 1 (𝐷 (𝑠 + 𝑡)) − 𝜆 1 (𝐷 (𝑠)) =∫ [(𝜙2𝑡 ) ] 𝑞𝛼 𝑑𝛼
lim sup ≤ 4𝑀1 𝜆 1 (𝐷 (𝑠)) , 𝐷(𝑠)
𝑡 → 0+ 𝑡
2
𝜆 (𝐷 (𝑠 + 𝑡)) − 𝜆 1 (𝐷 (𝑠)) ≥ 𝑒−3𝑀1 𝑡 ∫ [𝜙2󸀠 (𝛼)] 𝑑𝛼
lim− inf 1 ≥ −4𝑀1 𝜆 1 (𝐷 (𝑠)) . 𝐷(𝑠)
𝑡→0 𝑡
󵄩󵄩 󸀠󵄩 󵄩󵄩2
(A.8) 󵄩󵄩 𝜙2 󵄩󵄩
−3𝑀1 𝑡 󵄩
󵄩 󵄩󵄩2 󵄩󵄩
=𝑒 󵄩󵄩𝜙2 󵄩󵄩𝐿2 (𝐷(𝑠)) 󵄩󵄩( 󵄩 󵄩2 ) 󵄩󵄩󵄩
Proof. Let 𝑡 > 0, 𝜙1 ∈ 𝐻01 (𝐷(𝑠)) with ‖𝜙1 ‖𝐿2 (𝐷(𝑠)) = 1 be a first 󵄩󵄩 󵄩󵄩𝜙 󵄩󵄩 2 󵄩
󵄩󵄩 󵄩 2 󵄩𝐿 (𝐷(𝑠)) 󵄩󵄩󵄩𝐿2 (𝐷(𝑠))
normalized eigenfunction on 𝐷(𝑠). Then, for 𝜑 ∈ 𝐻01 (𝐷(𝑠+𝑡))
󵄩 󵄩2
with ‖𝜑‖𝐿2 (𝐷(𝑠+𝑡)) = 1, we have ≥ 𝑒−4𝑀1 𝑡 󵄩󵄩󵄩󵄩𝜙2𝑡 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠+𝑡)) 𝜆 1 (𝐷 (𝑠))
󵄩 󵄩2 󵄩 󵄩2
𝜆 1 (𝐷 (𝑠 + 𝑡)) − 𝜆 1 (𝐷 (𝑠)) = inf 󵄩󵄩󵄩󵄩𝜑󸀠 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠+𝑡)) − 󵄩󵄩󵄩󵄩𝜙1󸀠 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠)) = 𝑒−4𝑀1 𝑡 𝜆 1 (𝐷 (𝑠)) .
󵄩 󵄩−2 󵄩󵄩 󸀠󵄩
(A.13)
󵄩2
≤ 󵄩󵄩󵄩󵄩𝜙1𝑡 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠+𝑡)) 󵄩󵄩󵄩(𝜙1𝑡 ) 󵄩󵄩󵄩 2
󵄩 󵄩𝐿 (𝐷(𝑠+𝑡)) Hence
󵄩 󵄩2 𝜆 1 (𝐷 (𝑠 + 𝑡)) − 𝜆 1 (𝐷 (𝑠))
− 󵄩󵄩󵄩󵄩𝜙1󸀠 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠)) . lim inf
𝑡 → 0+ 𝑡
(A.9)
𝑒−4𝑀1 𝑡 − 1 (A.14)
Furthermore ≥ lim+ inf 𝜆 1 (𝐷 (𝑠))
𝑡→0 𝑡
󵄩󵄩 𝑡 󵄩󵄩−2 󵄩󵄩 𝑡 󸀠 󵄩󵄩2
󵄩󵄩𝜙1 󵄩󵄩 2 󵄩 󵄩
󵄩 󵄩𝐿 (𝐷(𝑠+𝑡)) 󵄩󵄩󵄩(𝜙1 ) 󵄩󵄩󵄩𝐿2 (𝐷(𝑠+𝑡)) = −4𝑀1 𝜆 1 (𝐷 (𝑠)) .
󵄩 󵄩−2 󸀠 2 The other part is similar.
= 󵄩󵄩󵄩󵄩𝜙1𝑡 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠+𝑡)) ∫ [(𝜙1𝑡 ) ] 𝑞𝛼 𝑑𝛼
𝐷(𝑠)
(A.10)
󵄩 󵄩−2 󵄩 󵄩2 Acknowledgments
≤ 󵄩󵄩󵄩󵄩𝜙1𝑡 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠+𝑡)) 𝑒3𝑀1 𝑡 󵄩󵄩󵄩󵄩𝜙1󸀠 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠))
This work was partially supported by ZJNSF, under Grant
󵄩 󵄩2
𝑒4𝑀1 𝑡 󵄩󵄩󵄩󵄩𝜙1󸀠 󵄩󵄩󵄩󵄩𝐿2 (𝐷(𝑠)) .
nos. LQ12A01009 and LQ13A010008, and NSFC, under Grant
≤
nos. 11301394,11226176, and 11226172.
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http://dx.doi.org/10.1155/2013/903625
Research Article
Existence and Decay Estimate of Global Solutions to Systems of
Nonlinear Wave Equations with Damping and Source Terms
Yaojun Ye
Department of Mathematics and Information Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
Correspondence should be addressed to Yaojun Ye; yeyaojun@zust.edu.cn
Received 30 April 2013; Revised 1 September 2013; Accepted 2 September 2013
Academic Editor: T. Raja Sekhar
Copyright © 2013 Yaojun Ye. This is an open access article distributed under the Creative Commons Attribution License, which
The initial-boundary value problem for a class of nonlinear wave equations system in bounded domain is studied. The existence
of global solutions for this problem is proved by constructing a stable set and obtain the asymptotic stability of global solutions
through the use of a difference inequality.
1. Introduction When 𝑝 = 2, Medeiros and Miranda [1] proved the exis-

tence and uniqueness of global weak solutions. Cavalcanti
In this paper, we are concerned with the global solvability and et al. in [2–4] considered the asymptotic behavior for wave
decay stabilization for the following nonlinear wave equa- equation and an analogous hyperbolic-parabolic system with
tions system: boundary damping and boundary source term. In paper [5,
󵄨 󵄨𝑞−2
𝑢𝑡𝑡 − div (|∇𝑢|𝑝−2 ∇𝑢) + 󵄨󵄨󵄨𝑢𝑡 󵄨󵄨󵄨 𝑢𝑡 − Δ𝑢𝑡 6], the authors dealt with the existence, uniform decay rates,
(1) and blowup for solutions of systems of nonlinear wave equa-
= |V|𝑟+2 |𝑢|𝑟 𝑢, (𝑥, 𝑡) ∈ Ω × 𝑅+ , tions with damping and source terms.
Rammaha and Wilstein [7] and Yang [8] are concerned
󵄨 󵄨𝑞−2
V𝑡𝑡 − div (|∇V|𝑝−2 ∇V) + 󵄨󵄨󵄨V𝑡 󵄨󵄨󵄨 V𝑡 − ΔV𝑡 with the initial boundary value problem for a class of quasilin-
(2) ear evolution equations with nonlinear damping and source
= |𝑢|𝑟+2 |V|𝑟 V, (𝑥, 𝑡) ∈ Ω × 𝑅+ terms. Under appropriate conditions, by a Galerkin approx-
imation scheme combined with the potential well method,
with the initial-boundary value conditions they proved the existence and asymptotic behavior of global
1,𝑝
𝑢 (𝑥, 0) = 𝑢0 (𝑥) ∈ 𝑊0 (Ω) , 𝑢𝑡 (𝑥, 0) = 𝑢1 (𝑥) ∈ 𝐿2 (Ω) weak solutions when 𝑚 < 𝑝, where 𝑚 ≥ 0 and 𝑝 are, respec-
tively, the growth orders of the nonlinear strain terms and the
𝑥 ∈ Ω, source term.
(3) Ono [9] considers the following initial-boundary value
problem for nonlinear wave equations with nonlinear dissi-
1,𝑝
V (𝑥, 0) = V0 (𝑥) ∈ 𝑊0 (Ω) , V𝑡 (𝑥, 0) = V1 (𝑥) ∈ 𝐿2 (Ω) pative terms:
𝑥 ∈ Ω, 󵄨 󵄨𝛽
(4) 𝑢𝑡𝑡 − Δ𝑢 + 𝛿1 𝑢𝑡 + 𝛿2 󵄨󵄨󵄨𝑢𝑡 󵄨󵄨󵄨 𝑢𝑡 − 𝛿3 Δ𝑢𝑡 = |𝑢|𝛼 𝑢,
𝑢(𝑥, 𝑡) = 0, V (𝑥, 𝑡) = 0, (𝑥, 𝑡) ∈ 𝜕Ω × 𝑅+ , (5) (𝑥, 𝑡) ∈ Ω × 𝑅+ ,

(6)
where Ω is a bounded open domain in 𝑅𝑛 with a smooth 𝑢 (𝑥, 0) = 𝑢0 (𝑥) , 𝑢𝑡 (𝑥, 0) = 𝑢1 (𝑥) , 𝑥 ∈ Ω,
boundary 𝜕Ω, 𝑝, 𝑞 ≥ 2, 𝑟 > 0 and 𝑝 < 2(𝑟+2) ≤ 𝑛𝑝/(𝑛−𝑝) for
𝑛 ≥ 𝑝 and 𝑝 < 2(𝑟 + 2) < +∞ for 𝑛 < 𝑝. 𝑢 (𝑥, 𝑡) = 0, 𝑥 ∈ 𝜕Ω, 𝑡 ≥ 0,
1,𝑝
where 𝛿𝑖 ≥ 0, 𝑖 = 1, 2, 3, and 𝛼, 𝛽 > 0 are constants. The 𝑢𝑡 , V𝑡 ∈ 𝐶([0, 𝑇], 𝐿2 (Ω)), [𝑢(0), V(0)] = [𝑢0 , V0 ] ∈ 𝑊0 (Ω) ×
1,𝑝
author mainly investigates on the blowup phenomenon to 𝑊0 (Ω), [𝑢𝑡 (0), V𝑡 (0)] = [𝑢1 , V1 ] ∈ 𝐿2 (Ω) × 𝐿2 (Ω), and [𝑢, V]
problem (6). On the other hand, in the case of 𝛿1 +𝛿2 +𝛿3 > 0, satisfies
he shows that the problem (6) admits a unique global solu-
⟨𝑢𝑡 (𝑡) , 𝜙⟩𝐿2 (Ω) − ⟨𝑢1 , 𝜙⟩𝐿2 (Ω)
tion, and its energy has some decay properties under some
assumptions on 𝑢0 and initial energy 𝐸(0) ≡ 𝐸(𝑢0 , 𝑢1 ). In par- 𝑡
ticular, when 𝛿2 > 0 and 𝛿1 + 𝛿3 > 0 in (6), the energy 𝐸(𝑡) ≡ + ∫ ⟨(|∇𝑢|𝑝−2 ∇𝑢) , ∇𝜙⟩𝐿2 (Ω) 𝑑𝜏
0
𝐸(𝑢(𝑡), 𝑢𝑡 (𝑡)) has some polynomial and exponential decay
𝑡 𝑡
rates, respectively. 󵄨 󵄨𝑞−2
+ ∫ ⟨󵄨󵄨󵄨𝑢𝑡 󵄨󵄨󵄨 𝑢𝑡 , 𝜙⟩𝐿2 (Ω) 𝑑𝜏 + ∫ ⟨∇𝑢𝑡 , ∇𝜙⟩𝐿2 (Ω)
For the following strongly damped nonlinear wave equa- 0 0
tion 𝑡
= ∫ ⟨|V|𝑟+2 |𝑢|𝑟 𝑢, 𝜙⟩𝐿2 (Ω) 𝑑𝜏,
0
𝑢𝑡𝑡 − Δ𝑢𝑡 − Δ𝑢 + 𝑓 (𝑢𝑡 ) + 𝑔 (𝑢) = ℎ, (7) (9)
⟨V𝑡 (𝑡) , 𝜓⟩𝐿2 (Ω) − ⟨V1 , 𝜓⟩𝐿2 (Ω)
Dell’Oro and Pata [10] obtain the long-time behavior of the 𝑡
related solution semigroup, which is shown to possess the + ∫ ⟨(|∇V|𝑝−2 ∇V) , ∇𝜓⟩𝐿2 (Ω) 𝑑𝜏
0
global attractor in the natural weak energy space. In addition,
𝑡 𝑡
󵄨 󵄨𝑞−2
the existence of global and local solutions, decay estimates, + ∫ ⟨󵄨󵄨󵄨V𝑡 󵄨󵄨󵄨 V𝑡 , 𝜓⟩𝐿2 (Ω) 𝑑𝜏 + ∫ ⟨∇V𝑡 , ∇𝜓⟩𝐿2 (Ω)
and blowup for solutions of nonlinear wave equation with 0 0
source and damping terms and exponential nonlinearities are 𝑡
studied in [11–14]. = ∫ ⟨|𝑢|𝑟+2 |V|𝑟 V, 𝜓⟩𝐿2 (Ω) 𝑑𝜏,
0
In this paper, we prove the global existence for the prob-
1,𝑝
lem (1)–(5) by applying the potential well theory introduced for all test functions 𝜙, 𝜓 ∈ 𝑊0 (Ω) and for almost all 𝑡 ∈
by Sattinger [15] and Payne and Sattinger [16]. Meanwhile, [0, 𝑇].
we obtain the asymptotic stabilization of global solutions by
using a difference inequality [17]. The local existence and uniqueness of solutions for prob-
For simplicity of notations, hereafter we denote by ‖ ⋅ ‖𝑝 lem (1)–(5) can be proved through the use of Galerkin
method. The result reads as follows.
the norm of 𝐿𝑝 (Ω); ‖ ⋅ ‖ denotes 𝐿2 (Ω) norm, and we write
1,𝑝
equivalent norm ‖ ⋅ ∇‖𝑝 instead of 𝑊0 (Ω) norm ‖ ⋅ ‖𝑊1,𝑝 (Ω) . Theorem 4 (local solution). Supposed that [𝑢0 , V0 ] ∈
0
Moreover, 𝐶 denotes various positive constants depending on 1,𝑝 1,𝑝
𝑊0 (Ω) × 𝑊0 (Ω), [𝑢1 , V1 ] ∈ 𝐿2 (Ω) × 𝐿2 (Ω), and 𝑝 < 2(𝑟 +
the known constants and may be different at each appearance. 2) ≤ 𝑛𝑝/(𝑛−𝑝) if 𝑛 ≥ 𝑝 and 𝑝 < 2(𝑟+2) < +∞ for 𝑛 < 𝑝, then
there exists 𝑇 > 0 such that the problem (1)–(5) has a unique
local solution [𝑢(𝑡), V(𝑡)] satisfying
2. Local Existence
1,𝑝 1,𝑝
[𝑢, V] ∈ 𝐿∞ ([0, 𝑇) ; 𝑊0 (Ω) × 𝑊0 (Ω)) ;
In this section, we investigate the local existence and unique- (10)
ness of the solutions of the problem (1)–(5). For this purpose, [𝑢𝑡 , V𝑡 ] ∈ 𝐿∞ ([0, 𝑇) ; 𝐿2 (Ω) × 𝐿2 (Ω)) ,
we list up two useful lemmas which will be used later and give
the definition of weak solutions. 𝑡
󵄩 󵄩2 󵄩 󵄩2
𝐸 (𝑡) + ∫ (󵄩󵄩󵄩∇𝑢𝜏 (𝜏)󵄩󵄩󵄩 + 󵄩󵄩󵄩∇V𝜏 (𝜏)󵄩󵄩󵄩
1,𝑝 0
Lemma 1. Let 𝑢 ∈ 𝑊0 (Ω), then 𝑢 ∈ 𝐿𝑠 (Ω); and the inequal- (11)
ity ‖𝑢‖𝑠 ≤ 𝐶‖𝑢‖𝑊1,𝑝 (Ω) holds with a constant 𝐶 > 0 depending
0
+‖𝑢 (𝜏)‖𝑞𝑞 + ‖V (𝜏)‖𝑞𝑞 ) 𝑑𝜏 = 𝐸 (0) ,
on Ω, 𝑝, and 𝑠, provided that 2 ≤ 𝑠 < +∞, 2 ≤ 𝑛 ≤ 𝑝 and
2 ≤ 𝑠 ≤ 𝑛𝑝/(𝑛 − 𝑝), 2 < 𝑝 < 𝑛. where
1 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 1
𝐸 (𝑡) = (󵄩𝑢 󵄩 + 󵄩V 󵄩 ) + (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 )
Lemma 2 (Young inequality). Let 𝑎, 𝑏 ≥ 0 and 1/𝑝 + 1/𝑞 = 1 2 󵄩 𝑡󵄩 󵄩 𝑡󵄩 𝑝
for 1 < 𝑝, 𝑞 < +∞; then one has the inequality (12)
1
− ‖𝑢V‖𝑟+2
𝑟+2 .
𝑟+2
𝑎𝑏 ≤ 𝛿𝑎𝑝 + 𝐶 (𝛿) 𝑏𝑞 , (8)
1,𝑝
Proof. Let {𝜔𝑖 }∞
𝑖=1 be a basis for 𝑊0 (Ω). Supposed that 𝑉𝑘 is
1,𝑝
the subspace of 𝑊0 (Ω) generated by {𝜔1 , 𝜔2 , . . . , 𝜔𝑘 }, 𝑘 ∈ 𝑁.
where 𝛿 > 0 is an arbitrary constant, and 𝐶(𝛿) is a positive con-
We are going to look for the approximate solution
stant depending on 𝛿.
𝑘 𝑘
Definition 3. A pair of functions (𝑢, V) is said to be a weak 𝑢𝑘 (𝑡) = ∑𝑔𝑖𝑘 (𝑡) 𝜔𝑖 , V𝑘 (𝑡) = ∑ℎ𝑖𝑘 (𝑡) 𝜔𝑖 (13)
1,𝑝
solution of (1)–(5) on [0, 𝑇] if 𝑢, V ∈ 𝐶([0, 𝑇], 𝑊0 (Ω)), 𝑖=1 𝑖=1
which satisfies the following Cauchy problem: We estimate the right-hand terms of (22) as follows: we get
󵄨 󵄨𝑞−2 from Hölder inequality and Lemmas 1 and 2 that
󵄨 󵄨𝑝−2
∫ (𝑢𝑘󸀠󸀠 − div (󵄨󵄨󵄨∇𝑢𝑘 󵄨󵄨󵄨 ∇𝑢𝑘 ) + 󵄨󵄨󵄨󵄨𝑢𝑘󸀠 󵄨󵄨󵄨󵄨 𝑢𝑘󸀠 − Δ𝑢𝑘󸀠 ) 𝜔𝑖 𝑑𝑥
Ω 󵄨󵄨 𝑡 󵄨󵄨
(14) 󵄨󵄨 󵄨 󵄨𝑟+2 󵄨 󵄨𝑟 󵄨 󵄨𝑟+2 󵄨 󵄨𝑟 󸀠 󵄨
󵄨󵄨∫ ∫ (󵄨󵄨󵄨V𝑘 󵄨󵄨󵄨 󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 𝑢𝑘 𝑢𝑘 + 󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 󵄨󵄨󵄨V𝑘 󵄨󵄨󵄨 V𝑘 V𝑘 ) 𝑑𝑥 𝑑𝜏󵄨󵄨󵄨
󸀠
󵄨 󵄨𝑟+2 󵄨 󵄨𝑟
= ∫ 󵄨󵄨󵄨V𝑘 󵄨󵄨󵄨 󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 𝑢𝑘 𝜔𝑖 𝑑𝑥, 󵄨󵄨 0 Ω 󵄨󵄨
Ω
𝑡
󵄩 󵄩2 󵄩 󵄩2
∫ (V𝑘󸀠󸀠
󵄨 󵄨𝑝−2 󵄨 󵄨𝑞−2
− div (󵄨󵄨󵄨∇V𝑘 󵄨󵄨󵄨 ∇V𝑘 ) + 󵄨󵄨󵄨󵄨V𝑘󸀠 󵄨󵄨󵄨󵄨 V𝑘󸀠 − ΔV𝑘󸀠 ) 𝜔𝑖 𝑑𝑥 ≤ ∫ (󵄩󵄩󵄩󵄩𝑢𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩V𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩 ) 𝑑𝜏
Ω 0
(15) 𝑡
󵄨 󵄨𝑟+2 󵄨 󵄨𝑟 󵄨 󵄨2(𝑟+1) 󵄨󵄨 󵄨󵄨2 󵄨󵄨 󵄨󵄨2
= ∫ 󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 󵄨󵄨󵄨V𝑘 󵄨󵄨󵄨 V𝑘 𝜔𝑖 𝑑𝑥, + ∫ ∫ 󵄨󵄨󵄨𝑢𝑘 V𝑘 󵄨󵄨󵄨 (󵄨󵄨𝑢𝑘 󵄨󵄨 + 󵄨󵄨V𝑘 󵄨󵄨 ) 𝑑𝑥 𝑑𝜏
Ω 0 Ω
𝑘 𝑡
1,𝑝 󵄩 󵄩2 󵄩 󵄩2
𝑢𝑘 (0) = 𝑢0𝑘 = ∑ (𝑢0 , 𝜔𝑖 ) 𝜔𝑖 󳨀→ 𝑢0 , in 𝑊0 (Ω) , ≤ ∫ (󵄩󵄩󵄩󵄩𝑢𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩V𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩 ) 𝑑𝜏
𝑖=1 (16) 0
𝑘 󳨀→ ∞, 𝑡
󵄩 󵄩2(𝑟+2) 󵄩 󵄩2(𝑟+2) (23)
+ 𝐶 ∫ (󵄩󵄩󵄩𝑢𝑘 󵄩󵄩󵄩2(𝑟+2) + 󵄩󵄩󵄩V𝑘 󵄩󵄩󵄩2(𝑟+2) ) 𝑑𝜏
𝑘 0
1,𝑝
V𝑘 (0) = V0𝑘 = ∑ (V0 , 𝜔𝑖 ) 𝜔𝑖 󳨀→ V0 in 𝑊0 (Ω) , 𝑡
(17) 󵄩 󵄩2 󵄩 󵄩2
𝑖=1
≤ 𝐶 ∫ (󵄩󵄩󵄩󵄩𝑢𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩V𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩
𝑘 󳨀→ ∞, 0
󵄩 󵄩2(𝑟+2) 󵄩󵄩 󵄩󵄩2(𝑟+2)
𝑘
+󵄩󵄩󵄩∇𝑢𝑘 󵄩󵄩󵄩𝑝 + 󵄩󵄩∇V𝑘 󵄩󵄩𝑝 ) 𝑑𝜏
𝑢𝑘󸀠 (0) = 𝑢1𝑘 = ∑ (𝑢1 , 𝜔𝑖 ) 𝜔𝑖 󳨀→ 𝑢1 in 𝐿2 (Ω) ,
𝑖=1 (18) 𝑡
󵄩 󵄩2 󵄩 󵄩2
𝑘 󳨀→ ∞, ≤ 𝐶 ∫ (󵄩󵄩󵄩󵄩𝑢𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩V𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩
0
𝑘
V𝑘󸀠 (0) = V1𝑘 = ∑ (V1 , 𝜔𝑖 ) 𝜔𝑖 󳨀→ V1 in 𝐿2 (Ω) , 󵄩 󵄩𝑝 󵄩 󵄩𝑝 2(𝑟+2)/𝑝
+󵄩󵄩󵄩∇𝑢𝑘 󵄩󵄩󵄩𝑝 + 󵄩󵄩󵄩∇V𝑘 󵄩󵄩󵄩𝑝 ) 𝑑𝜏.
𝑖=1 (19)
𝑘 󳨀→ ∞. It follows from (22) and (23) that
Note that, we can solve the problem (14)–(19) by a Picard’s
󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩
iteration method in ordinary differential equations. Hence, 󵄩󵄩𝑢𝑘 (𝑡)󵄩󵄩 + 󵄩󵄩V𝑘 (𝑡)󵄩󵄩 + 󵄩󵄩∇𝑢𝑘 󵄩󵄩󵄩󵄩𝑝𝑝 + 󵄩󵄩󵄩󵄩∇V𝑘 󵄩󵄩󵄩󵄩𝑝𝑝
󵄩 󵄩 󵄩 󵄩
there exists a solution in [0, 𝑇𝑘 ) for some 𝑇𝑘 > 0, and we can
𝑡
extend this solution to the whole interval [0, 𝑇] for any given 󵄩 󵄩𝑞 󵄩 󵄩𝑞
𝑇 > 0 by making use of the a priori estimates below. + 2 ∫ (󵄩󵄩󵄩󵄩𝑢𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩𝑞 ‖ +󵄩󵄩󵄩󵄩V𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩𝑞
0
󸀠 󸀠
Multiplying (14) by 𝑔𝑖𝑘 (𝑡) and (15) by ℎ𝑖𝑘 (𝑡) and summing
󵄩 󵄩2 󵄩 󵄩2
over 𝑖 from 1 to 𝑘, we obtain +󵄩󵄩󵄩󵄩∇𝑢𝑘󸀠 (𝑡)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩∇V𝑘󸀠 (𝑡)󵄩󵄩󵄩󵄩 ) 𝑑𝜏 (24)
1 𝑑 󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩 󵄩𝑝 󵄩 󵄩𝑞 󵄩 󵄩2
(󵄩󵄩󵄩𝑢𝑘 (𝑡)󵄩󵄩󵄩 + 󵄩󵄩∇𝑢𝑘 󵄩󵄩󵄩𝑝 ) + 󵄩󵄩󵄩󵄩𝑢𝑘󸀠 (𝑡)󵄩󵄩󵄩󵄩𝑞 + 󵄩󵄩󵄩󵄩∇𝑢𝑘󸀠 (𝑡)󵄩󵄩󵄩󵄩 𝑡
󵄩 󵄩2 󵄩 󵄩2
2 𝑑𝑡
(20) ≤ 2𝐶0 + 𝐶 ∫ (󵄩󵄩󵄩󵄩𝑢𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩V𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩
0
󵄨 󵄨𝑟+2 󵄨 󵄨𝑟
= ∫ 󵄨󵄨󵄨V𝑘 󵄨󵄨󵄨 󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 𝑢𝑘 𝑢𝑘󸀠 𝑑𝑥,
Ω 󵄩 󵄩𝑝 󵄩 󵄩𝑝 2(𝑟+2)/𝑝
+󵄩󵄩󵄩∇𝑢𝑘 󵄩󵄩󵄩𝑝 + 󵄩󵄩󵄩∇V𝑘 󵄩󵄩󵄩𝑝 ) 𝑑𝜏,
1 𝑑 󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩 󵄩󵄩𝑝 󵄩 󵄩𝑞 󵄩 󵄩2
(󵄩󵄩V (𝑡)󵄩󵄩 + 󵄩∇V 󵄩 ) + 󵄩󵄩󵄩󵄩V𝑘󸀠 (𝑡)󵄩󵄩󵄩󵄩𝑞 + 󵄩󵄩󵄩󵄩∇V𝑘󸀠 (𝑡)󵄩󵄩󵄩󵄩
2 𝑑𝑡 󵄩 𝑘 󵄩 󵄩 𝑘 󵄩𝑝 which implies that
(21)
󵄨 󵄨𝑟+2 󵄨 󵄨𝑟
= ∫ 󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 󵄨󵄨󵄨V𝑘 󵄨󵄨󵄨 V𝑘 V𝑘󸀠 𝑑𝑥.
󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩
Ω 󵄩󵄩𝑢𝑘 (𝑡)󵄩󵄩 + 󵄩󵄩V𝑘 (𝑡)󵄩󵄩 + 󵄩󵄩∇𝑢𝑘 󵄩󵄩󵄩󵄩𝑝𝑝 + 󵄩󵄩󵄩󵄩∇V𝑘 󵄩󵄩󵄩󵄩𝑝𝑝
󵄩 󵄩 󵄩 󵄩
By summing (20) and (21) and integrating the resulting iden-
tity over [0, 𝑡], we have 𝑡
󵄩 󵄩2 󵄩 󵄩2
≤ 2𝐶0 + 𝐶 ∫ (󵄩󵄩󵄩󵄩𝑢𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩V𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩 (25)
1 󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩 󵄩𝑝 󵄩 󵄩𝑝 0
(󵄩󵄩𝑢 (𝑡)󵄩󵄩 + 󵄩󵄩V (𝑡)󵄩󵄩 + 󵄩∇𝑢 󵄩󵄩 + 󵄩󵄩∇V 󵄩󵄩 )
2 󵄩 𝑘 󵄩 󵄩 𝑘 󵄩 󵄩 𝑘 󵄩𝑝 󵄩 𝑘 󵄩𝑝 󵄩 󵄩𝑝 󵄩 󵄩𝑝 2(𝑟+2)/𝑝
𝑡 +󵄩󵄩󵄩∇𝑢𝑘 󵄩󵄩󵄩𝑝 + 󵄩󵄩󵄩∇V𝑘 󵄩󵄩󵄩𝑝 ) 𝑑𝜏.
󵄩 󵄩2 󵄩 󵄩2
+ ∫ (󵄩󵄩󵄩󵄩∇𝑢𝑘󸀠 (𝑡)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩∇V𝑘󸀠 (𝑡)󵄩󵄩󵄩󵄩
0
󵄩 󵄩𝑞 󵄩 󵄩𝑞 We get from (25) and Gronwall type inequality that
+󵄩󵄩󵄩󵄩𝑢𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩𝑞 + 󵄩󵄩󵄩󵄩V𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩𝑞 ) 𝑑𝜏 (22)
󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩
𝑡 󵄩󵄩𝑢𝑘 (𝑡)󵄩󵄩 + 󵄩󵄩V𝑘 (𝑡)󵄩󵄩 + 󵄩󵄩∇𝑢𝑘 󵄩󵄩󵄩󵄩𝑝𝑝 + 󵄩󵄩󵄩󵄩∇V𝑘 󵄩󵄩󵄩󵄩𝑝𝑝
󵄨 󵄨𝑟+2 󵄨 󵄨𝑟 󵄩 󵄩 󵄩 󵄩
≤ 𝐶0 + ∫ ∫ (󵄨󵄨󵄨V𝑘 󵄨󵄨󵄨 󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 𝑢𝑘 𝑢𝑘󸀠
0 Ω −𝑝/(2(𝑟+2)−𝑝) (26)
2 (𝑟 + 2) − 𝑝
󵄨 󵄨𝑟+2 󵄨 󵄨𝑟 ≤ [2𝐶0 − 𝐶𝑡] .
+󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 󵄨󵄨󵄨V𝑘 󵄨󵄨󵄨 V𝑘 V𝑘󸀠 ) 𝑑𝑥 𝑑𝜏. 𝑝
Thus, we deduce from (26) that there exists a time 𝑇 > 0 such Lemma 5. Let [𝑢, V] be a solution to problem (1)–(5); then,
that 𝐸(𝑡) is a nonincreasing function for 𝑡 > 0 and
󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩 󸀠 󵄩󵄩2 󵄩󵄩 󵄩𝑝 󵄩 󵄩𝑝
󵄩󵄩󵄩𝑢𝑘 (𝑡)󵄩󵄩󵄩 + 󵄩󵄩󵄩V𝑘 (𝑡)󵄩󵄩󵄩 + 󵄩󵄩∇𝑢𝑘 󵄩󵄩󵄩𝑝 + 󵄩󵄩󵄩∇V𝑘 󵄩󵄩󵄩𝑝 ≤ 𝐶1 , ∀𝑡 ∈ [0, 𝑇] , 𝑑 󵄩 󵄩𝑞 󵄩 󵄩𝑞 󵄩 󵄩2 󵄩 󵄩2
(27) 𝐸 (𝑡) = − (󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝑞 + 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩𝑞 + 󵄩󵄩󵄩∇𝑢𝑡 󵄩󵄩󵄩2 + 󵄩󵄩󵄩∇V𝑡 󵄩󵄩󵄩2 ) . (35)
𝑑𝑡
where 𝐶1 is a positive constant independent of 𝑘. We have from (11) that 𝐸(𝑡) is the primitive of an inte-
We have from (24) and (26) that grable function. Therefore, 𝐸(𝑡) is absolutely continuous, and
𝑡
󵄩 󵄩𝑞 󵄩 󵄩𝑞 equality (35) is satisfied.
2 ∫ (󵄩󵄩󵄩󵄩𝑢𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩𝑞 + 󵄩󵄩󵄩󵄩V𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩𝑞
0 1,𝑝 1,𝑝
Lemma 6. Supposed that [𝑢, V] ∈ 𝑊0 (Ω) × 𝑊0 (Ω), and
󵄩 󵄩2 󵄩 󵄩2
+󵄩󵄩󵄩󵄩∇𝑢𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩∇V𝑘󸀠 (𝜏)󵄩󵄩󵄩󵄩 ) 𝑑𝜏 ≤ 𝐶2 , ∀𝑡 ∈ [0, 𝑇] . 𝑝 < 2(𝑟 + 2) ≤ 𝑛𝑝/(𝑛 − 𝑝) if 𝑛 ≥ 𝑝; 𝑝 < 2(𝑟 + 2) < +∞ if
𝑛 < 𝑝, then 𝑑 > 0.
(28)
It follows from (27) and (28) that Proof. Since
󵄩󵄩󵄩𝑢󸀠 (𝑡)󵄩󵄩󵄩2 ≤ 𝐶 , 󵄩󵄩󵄩V󸀠 (𝑡)󵄩󵄩󵄩2 ≤ 𝐶 ,
󵄩󵄩 𝑘 󵄩󵄩 1 󵄩󵄩 𝑘 󵄩󵄩 1 𝜆𝑝 𝜆2(𝑟+2)
𝐽 (𝜆 [𝑢, V]) = (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) − ‖𝑢V‖𝑟+2
𝑟+2 , (36)
󵄩󵄩 󵄩󵄩𝑝 󵄩󵄩 󵄩󵄩𝑝 𝑝 𝑟+2
󵄩󵄩∇𝑢𝑘 󵄩󵄩𝑝 ≤ 𝐶1 , 󵄩󵄩∇V𝑘 󵄩󵄩𝑝 ≤ 𝐶1 .
(29) so we get
𝑢𝑘󸀠 (𝑡) and V𝑘󸀠 2
(𝑡) are bounded in 𝐿 ([0, 𝑇] ; 𝐿 (Ω)) 𝑞
𝑑
𝐽 (𝜆 [𝑢, V]) = 𝜆𝑝−1 (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) − 2𝜆2𝑟+3 ‖𝑢V‖𝑟+2
𝑟+2 .
and 𝐿2 ([0, 𝑇] ; 𝐻01 (Ω)) . 𝑑𝜆
(37)
Using the same process as the proof of Theorem 2.1 in paper
[18], we derive that [𝑢(𝑡), V(𝑡)] is a local solution of the pro- In case 𝑢V ≠0, let (𝑑/𝑑𝜆)𝐽(𝜆[𝑢, V]) = 0, which implies that
blem (1)–(5). By (20) and (21), we conclude that (11) is valid.
𝑝 𝑝 1/(2𝑟−𝑝+4)
‖∇𝑢‖𝑝 + ‖∇V‖𝑝
𝜆1 = ( ) . (38)
2‖𝑢V‖𝑟+2
𝑟+2
3. Global Existence
As 𝜆 = 𝜆 1 , an elementary calculation shows that
In order to state our main results, we first introduce the fol-
lowing functionals: (𝑑2 /𝑑𝜆2 )𝐽(𝜆[𝑢, V])|𝜆=𝜆 1 < 0. Therefore, we have that
1 1 sup𝐽 (𝜆 [𝑢, V])

𝐽 ([𝑢, V]) = (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) − ‖𝑢V‖𝑟+2
𝑟+2 , (30)
𝑝 𝑟+2 𝜆≥0
𝐾 ([𝑢, V]) = (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) − 2‖𝑢V‖𝑟+2 (31) = 𝐽 (𝜆 1 [𝑢, V])

𝑟+2 (39)
1,𝑝 1,𝑝 𝑝 (2𝑟+4)/(2𝑟−𝑝+4)
𝑝
for [𝑢, V] ∈ 𝑊0 (Ω) × 𝑊0 (Ω). 2 (𝑟 + 2) − 𝑝 + ‖∇V‖𝑝
‖∇𝑢‖𝑝
We put that = ( ) .
2𝑝 (𝑟 + 2) 2𝑝/(2𝑟+4) ‖𝑢V‖𝑝/2
𝑟+2
1,𝑝
𝑑 = inf {sup𝐽 (𝜆 [𝑢, V]) : [𝑢, V] ∈ 𝑊0 (Ω) It follows from Hölder inequality and Lemma 1 that
𝜆≥0
(32) 𝑝/2 𝑝/2 𝑝/2
1,𝑝
‖𝑢V‖𝑟+2 ≤ ‖𝑢‖2(𝑟+2) ‖V‖2(𝑟+2)
×𝑊0 (Ω) / {[0, 0]} } .
1 𝑝 𝑝
≤ (‖𝑢‖2(𝑟+2) + ‖V‖2(𝑟+2) ) (40)
Then, we are able to define the stable set as follows for prob- 2
lem (1)–(5):
≤ 𝐶 (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) .
1,𝑝 1,𝑝
𝑊 = {[𝑢, V] ∈ 𝑊0 (Ω) × 𝑊0 (Ω) | 𝐾 ([𝑢, V]) > 0,
(33) We get from (39) and (40) that
𝐽 ([𝑢, V]) < 𝑑} ∪ {[0, 0]} .
2 (𝑟 + 2) − 𝑝 𝑝/(2𝑟+4) −(2𝑟+4)/(2𝑟−𝑝+4)
We denote the total energy related to (1) and (2) by (12), and sup𝐽 (𝜆 [𝑢, V]) ≥ (2 𝐶) > 0.
𝜆≥0 2𝑝 (𝑟 + 2)
1 󵄩 󵄩2 󵄩 󵄩2 1 󵄩 󵄩𝑝 󵄩 󵄩𝑝 (41)
𝐸 (0) = (󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩 + 󵄩󵄩󵄩V1 󵄩󵄩󵄩 ) + (󵄩󵄩󵄩∇𝑢0 󵄩󵄩󵄩𝑝 + 󵄩󵄩󵄩∇V0 󵄩󵄩󵄩𝑝 )
2 𝑝
(34) In case 𝑢V = 0 and 𝑢 = 0 or V = 0, then
1 󵄩󵄩 󵄩𝑟+2
− 󵄩𝑢 V 󵄩󵄩 𝜆𝑝
𝑟 + 2 󵄩 0 0 󵄩𝑟+2 𝐽 (𝜆 [𝑢, V]) = (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) . (42)
is the total energy of the initial data. 𝑝
Therefore, we have Proof. It suffices to show that ‖𝑢𝑡 ‖2 + ‖V𝑡 ‖2 + ‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 is
bounded uniformly with respect to 𝑡. Under the hypotheses in
𝐽 (𝜆 [𝑢, V]) = +∞. (43) Theorem 8, we get from Lemma 7 that [𝑢, V] ∈ 𝑊 on [0, 𝑇). So
the following formula holds on [0, 𝑇):
We conclude from (41) and (43) that
1 1
2 (𝑟 + 2) − 𝑝 𝑝/(2𝑟+4) −(2𝑟+4)/(2𝑟−𝑝+4) 𝐽 ([𝑢, V]) = (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) − ‖𝑢V‖𝑟+2
𝑟+2
𝑑≥ (2 𝐶) > 0. (44) 𝑝 𝑟+2
2𝑝 (𝑟 + 2) (51)
2 (𝑟 + 2) − 𝑝
≥ (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) .
Thus, we complete the proof of Lemma 6. 2𝑝 (𝑟 + 2)
Lemma 7. Supposed that 𝑝 < 2(𝑟 + 2) ≤ 𝑛𝑝/(𝑛 − 𝑝) for 𝑛 ≥ 𝑝 We have from (51) that
and 𝑝 < 2(𝑟 + 2) < +∞ for 𝑛 < 𝑝, if [𝑢0 , V0 ] ∈ 𝑊, [𝑢1 , V1 ] ∈ 1 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 2 (𝑟 + 2) − 𝑝
𝐿2 (Ω) × 𝐿2 (Ω) and 𝐸(0) < 𝑑, then [𝑢, V] ∈ 𝑊 for ∀𝑡 ∈ [0, 𝑇). (󵄩󵄩𝑢𝑡 󵄩󵄩 + 󵄩󵄩V𝑡 󵄩󵄩 ) + (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 )
2 2𝑝 (𝑟 + 2)
Proof. Assume that there exists a number 𝑡∗ ∈ (0, 𝑇) such that 1 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 (52)
[𝑢(𝑡), V(𝑡)] ∈ 𝑊 on [0, 𝑡∗ ) and 𝑢(𝑡∗ ) ∉ 𝑊. Then, in virtue of ≤ (󵄩𝑢 󵄩 + 󵄩V 󵄩 ) + 𝐽 ([𝑢 (𝑡) , V (𝑡)])
2 󵄩 𝑡󵄩 󵄩 𝑡󵄩
the continuity of 𝑢(𝑡), we see 𝑢(𝑡∗ ) ∈ 𝜕𝑊, where 𝜕𝑊 denotes
the boundary of domain 𝑊. From the definition of 𝑊 and the = 𝐸 (𝑡) ≤ 𝐸 (0) < 𝑑.
continuity of 𝐽([𝑢(𝑡), V(𝑡)]) and 𝐾([𝑢(𝑡), V(𝑡)]) in 𝑡, we have
Hence, we get
either
󵄩 󵄩2 󵄩 󵄩2
(󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 ) + (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 )
𝐽 ([𝑢 (𝑡∗ ) , V (𝑡∗ )]) = 𝑑 (45)
2𝑝 (𝑟 + 2) (53)
or ≤ max (2, ) 𝑑 < +∞.
2 (𝑟 + 2) − 𝑝
𝐾 ([𝑢 (𝑡∗ ) , V (𝑡∗ )]) = 0. (46) The above inequality and the continuation principle lead to
the global existence of the solution [𝑢, V] for problem (1)–(5).
It follows from (12) and (30) that
𝐽 ([𝑢 (𝑡∗ ) , V (𝑡∗ )]) ≤ 𝐸 (𝑡∗ ) ≤ 𝐸 (0) < 𝑑. (47)

4. Asymptotic Behavior of Global Solutions
So, case (45) is impossible.
The following lemma plays an important role in studying the
Assume that (46) holds; then, we get that
decay estimate of global solutions for the problem (1)–(5).
𝑑
𝐽 (𝜆 [𝑢 (𝑡∗ ) , V (𝑡∗ )]) Lemma 9 (see [9]). Suppose that 𝜑(𝑡) is a nonincreasing non-
𝑑𝜆 (48) negative function on [0, +∞) and satisfies
= 𝜆𝑝−1 (1 − 𝜆2𝑟−𝑝+4 ) (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) .
𝜑(𝑡)𝑟+1 ≤ 𝑘 (𝜑 (𝑡) − 𝜑 (𝑡 + 1)) , ∀𝑡 ≥ 0. (54)
We obtain from (𝑑/𝑑𝜆)𝐽(𝜆[𝑢(𝑡∗ ), V(𝑡∗ )]) = 0 that 𝜆 = 1. Since Then, 𝜑(𝑡) has the decay property
󵄨󵄨
𝑑2 󵄨󵄨 𝑟 −1/𝑟
𝐽 (𝜆 [𝑢 (𝑡∗
) , V (𝑡∗
)]) 󵄨󵄨 𝜑 (𝑡) ≤ [ (𝑡 − 1) + 𝑀−𝑟 ] , ∀𝑡 ≥ 1, (55)
𝑑𝜆2 󵄨󵄨 𝑘
󵄨𝜆=1 (49)
where 𝑘, 𝑟 > 0 are constants and 𝑀 = max𝑡∈[0,1] 𝜑(𝑡).
= − [(2 (𝑟 + 2) − 𝑝) + (2𝑟 + 3)] < 0.
Lemma 10. Under the assumptions of Theorem 8, if initial
Consequently, we get from (47) that
value [𝑢0 , V0 ] ∈ 𝑊 and [𝑢1 , V1 ] ∈ 𝐿2 (Ω)×𝐿2 (Ω) are sufficiently
sup𝐽 (𝜆 [𝑢 (𝑡∗ ) , V (𝑡∗ )]) = 𝐽 ([𝑢 (𝑡∗ ) , V (𝑡∗ )]) < 𝑑, (50)
small such that
𝜆≥0 (2(𝑟+2)−𝑝)/𝑝
2𝑝 (𝑟 + 2)
𝐶2(𝑟+2) ( 𝐸 (0)) < 1, (56)
which contradicts the definition of 𝑑. Hence, case (46) is 2𝑝 (𝑟 + 2) − 𝑝
impossible as well. Thus we conclude that [𝑢(𝑡), V(𝑡)] ∈ 𝑊
then
on [0, 𝑇).
1
Theorem 8 (global solution). Supposed that 𝑝 < 2(𝑟 + 2) ≤ (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) ≤ 𝐾 ([𝑢, V]) , (57)
𝜃
𝑛𝑝/(𝑛 − 𝑝) as 𝑛 ≥ 𝑝 and 𝑝 < 2(𝑟 + 2) < +∞ as 𝑛 < 𝑝,
and [𝑢(𝑡), V(𝑡)] is a local solution of problem (1)–(5) on [0, 𝑇). where 𝜃 = 1 − 𝐶2(𝑟+2) ((2𝑝(𝑟 + 2)/(2𝑝(𝑟 + 2) −
If [𝑢0 , V0 ] ∈ 𝑊, [𝑢1 , V1 ] ∈ 𝐿2 (Ω) × 𝐿2 (Ω) and 𝐸(0) < 𝑑, then 𝑝))𝐸(0))(2(𝑟+2)−𝑝)/𝑝 > 0 is a positive constant and 𝐶 is the
1,𝑝
[𝑢(𝑡), V(𝑡)] is a global solution of problem (1)–(5). optimal Sobolev’s constant from 𝑊0 (Ω) to 𝐿2(𝑟+2) (Ω).
Proof. We have from Lemma 1 and (52) that On the other hand, we multiply (1) by 𝑢 and (2) by V and
integrate over Ω × [𝑡1 , 𝑡2 ]. Adding them together, we obtain
2‖𝑢V‖𝑟+2 𝑟+2 𝑟+2
𝑟+2 ≤ 2‖𝑢‖2(𝑟+2) ‖V‖2(𝑟+2)
𝑡 𝑡 𝑡
2 2
󵄩 󵄩2 2
󵄩 󵄩2
∫ 𝐾 ([𝑢, V]) 𝑑𝑠 = ∫ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 𝑑𝑠 + ∫ 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 𝑑𝑠
≤ ‖𝑢‖2(𝑟+2)
2(𝑟+2) + ‖V‖2(𝑟+2)
2(𝑟+2) 𝑡 1 𝑡 𝑡1 1
≤ 𝐶2(𝑟+2) (‖∇𝑢‖2(𝑟+2)
𝑝 + ‖∇V‖2(𝑟+2)
𝑝 ) + (𝑢𝑡 (𝑡1 ) , 𝑢 (𝑡1 )) − (𝑢𝑡 (𝑡2 ) , 𝑢 (𝑡2 ))
≤ 𝐶2(𝑟+2) (‖∇𝑢‖2(𝑟+2)−𝑝 ‖∇𝑢‖𝑝𝑝 + (V𝑡 (𝑡1 ) , V (𝑡2 )) − (V𝑡 (𝑡2 ) V (𝑡2 ))

𝑝 (58)
𝑡
2
󵄨 󵄨𝑞−2
+‖∇V‖2(𝑟+2)−𝑝
𝑝 ‖∇V‖𝑝𝑝 ) − (∫ ∫ 󵄨󵄨󵄨𝑢𝑡 󵄨󵄨󵄨 𝑢𝑡 𝑢 𝑑𝑥 𝑑𝑠
𝑡 Ω 1
(2(𝑟+2)−𝑝)/𝑝
2𝑝 (𝑟 + 2) 𝑡
󵄨 󵄨𝑞−2
≤ 𝐶2(𝑟+2) (
2
2𝑝 (𝑟 + 2) − 𝑝
𝐸 (0)) + ∫ ∫ 󵄨󵄨󵄨V𝑡 󵄨󵄨󵄨 V𝑡 V 𝑑𝑥 𝑑𝑠)
𝑡 Ω1
× (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) . 𝑡2 𝑡2
−∫ ∫ ∇𝑢𝑡 ∇𝑢 𝑑𝑥 𝑑𝑠−∫ ∫ ∇V𝑡 ∇V 𝑑𝑥 𝑑𝑠.
𝑡1 Ω 𝑡1 Ω
Therefore, we get from (58) and (31) that (65)
(2(𝑟+2)−𝑝)/𝑝
2𝑝 (𝑟 + 2)
[1 − 𝐶2(𝑟+2) ( 𝐸 (0)) ] From (63), Sobolev inequality, and Hölder inequality, we have
2𝑝 (𝑟 + 2) − 𝑝 (59) 𝑡 𝑡
2
󵄩 󵄩2 2
󵄩 󵄩2
× (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) ≤ 𝐾 ([𝑢, V]) . ∫ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 𝑑𝑠 ≤ 𝐶 ∫ 󵄩󵄩󵄩∇𝑢𝑡 󵄩󵄩󵄩 𝑑𝑠 ≤ 𝐶 (𝐸 (𝑡) − 𝐸 (𝑡 + 1)) ,
𝑡 1 𝑡 1
(66)
Let 𝑡2 𝑡
󵄩 󵄩2 2
󵄩 󵄩2
∫ 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 𝑑𝑠 ≤ 𝐶 ∫ 󵄩󵄩󵄩∇V𝑡 󵄩󵄩󵄩 𝑑𝑠 ≤ 𝐶 (𝐸 (𝑡) − 𝐸 (𝑡 + 1)) .
(2(𝑟+2)−𝑝)/𝑝 𝑡 𝑡
2𝑝 (𝑟 + 2) 1 1
𝜃 = 1 − 𝐶2(𝑟+2) ( 𝐸 (0)) > 0; (60)
2𝑝 (𝑟 + 2) − 𝑝 We get from (52), (64), and Lemmas 1 and 2 that
󵄨󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩
then, we have from (59) that 󵄨󵄨𝑢𝑡 (𝑡𝑖 ) , 𝑢 (𝑡𝑖 )󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝑢𝑡 (𝑡𝑖 )󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩𝑢 (𝑡𝑖 )󵄩󵄩󵄩 ≤ 𝐶 󵄩󵄩󵄩∇𝑢𝑡 (𝑡𝑖 )󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩∇𝑢(𝑡𝑖 )󵄩󵄩󵄩𝑝
1
‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ≤ 𝐾 ([𝑢, V]) . (61) ≤ 𝐶(𝐸 (𝑡) − 𝐸 (𝑡 + 1))1/2 sup 𝐸(𝑠)1/𝑝
𝜃 𝑡≤𝑠≤𝑡+1
≤ 𝐶 (𝜀) (𝐸 (𝑡) − 𝐸 (𝑡 + 1))𝑝/2(𝑝−1)

Theorem 11. Under the assumptions of Theorem 8, if 𝑝 < 𝑞 < + 𝜀 sup 𝐸 (𝑠) , 𝑖 = 1, 2,
𝑟 + 2 and (56) hold, then the global solution [𝑢, V] in 𝑊 of the 𝑡≤𝑠≤𝑡+1
problem (1)–(5) has the following decay property: 󵄨󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩
󵄨󵄨(V𝑡 (𝑡𝑖 ) , V (𝑡𝑖 ))󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩V𝑡 (𝑡𝑖 )󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩V (𝑡𝑖 )󵄩󵄩󵄩 ≤ 𝐶 󵄩󵄩󵄩∇V𝑡 (𝑡𝑖 )󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩∇V(𝑡𝑖 )󵄩󵄩󵄩𝑝
𝑝/(𝑝+𝑞−𝑝𝑞)
𝑝−2
𝐸 (𝑡) ≤ [ (𝑡 − 1) + 𝑀(𝑝+𝑞−𝑝𝑞)/𝑝 ] , ∀𝑡 > 1, ≤ 𝐶(𝐸 (𝑡) − 𝐸 (𝑡 + 1))1/2 sup 𝐸(𝑠)1/𝑝
𝑝𝐶
𝑡≤𝑠≤𝑡+1
(62)
≤ 𝐶 (𝜀) (𝐸 (𝑡) − 𝐸 (𝑡 + 1))𝑝/2(𝑝−1)
where 𝑀 = max𝑡∈[0,1] 𝐸(𝑡) > 0 is some constant depending only
on [𝑢0 , V0 ] and [𝑢1 , V1 ]. + 𝜀 sup 𝐸 (𝑠) , 𝑖 = 1, 2.
𝑡≤𝑠≤𝑡+1
Proof. Multiplying (1) by 𝑢𝑡 and (2) by V𝑡 and integrating over (67)
Ω × [𝑡, 𝑡 + 1], and summing up together, we get
From Hölder inequality and Lemma 2,we get
𝑡+1
󵄩 󵄩𝑞 󵄩 󵄩𝑞 󵄩 󵄩2
∫ (󵄩󵄩󵄩𝑢𝑡 (𝑠)󵄩󵄩󵄩𝑞 + 󵄩󵄩󵄩V𝑡 (𝑠)󵄩󵄩󵄩𝑞 + 󵄩󵄩󵄩∇𝑢𝑡 (𝑠)󵄩󵄩󵄩2 󵄨󵄨 𝑡2
󵄨󵄨
󵄨󵄨
󵄨 𝑡2
𝑡 (63) 󵄨󵄨∫ ∫ 󵄨󵄨󵄨󵄨𝑢𝑡 󵄨󵄨󵄨󵄨𝑞−2 𝑢𝑡 𝑢 𝑑𝑥 𝑑𝑠󵄨󵄨󵄨 ≤ ∫ 󵄩󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩󵄩𝑞−1
𝑞 ‖𝑢‖𝑞 𝑑𝑠
󵄨󵄨 𝑡 Ω 󵄨󵄨
󵄩 󵄩2 󵄨 1 󵄨 𝑡1
+󵄩󵄩󵄩∇V𝑡 (𝑠)󵄩󵄩󵄩2 ) 𝑑𝑠 = 𝐸 (𝑡) − 𝐸 (𝑡 + 1) .
𝑡 (𝑞−1)/𝑞 𝑡2 1/𝑞
2
󵄩 󵄩𝑞
Thus, there exists 𝑡1 ∈ [𝑡, 𝑡 + 1/4], 𝑡2 ∈ [𝑡 + 3/4, 𝑡 + 1] such that ≤ (∫ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝑞 𝑑𝑠) (∫ ‖𝑢‖𝑞𝑞 𝑑𝑠)
𝑡 1 𝑡1
󵄩 󵄩𝑞 󵄩 󵄩𝑞 󵄩 󵄩2 󵄩 󵄩2
4 (󵄩󵄩󵄩𝑢𝑡 (𝑡𝑖 )󵄩󵄩󵄩𝑞 + 󵄩󵄩󵄩V𝑡 (𝑡𝑖 )󵄩󵄩󵄩𝑞 + 󵄩󵄩󵄩∇𝑢𝑡 (𝑡𝑖 )󵄩󵄩󵄩2 + 󵄩󵄩󵄩∇V𝑡 (𝑡𝑖 )󵄩󵄩󵄩2 ) 2
󵄩 󵄩𝑞
𝑡 2 𝑡
(64) ≤ 𝐶 (𝜀) ∫ 󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩𝑞 𝑑𝑠 + 𝜀 ∫ ‖𝑢‖𝑞𝑞 𝑑𝑠,
𝑡 1 𝑡 1
= 𝐸 (𝑡) − 𝐸 (𝑡 + 1) , 𝑖 = 1, 2. (68)
󵄨󵄨 𝑡2 󵄨󵄨 𝑡2 1/2
󵄨󵄨 󵄨 𝑡2
󵄨󵄨∫ ∫ 󵄨󵄨󵄨󵄨V𝑡 󵄨󵄨󵄨󵄨𝑞−2 V𝑡 V 𝑑𝑥 𝑑𝑠󵄨󵄨󵄨 ≤ ∫ 󵄩󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩󵄩𝑞−1
𝑞 ‖V‖𝑞 𝑑𝑠 ≤ 𝐶(𝐸 (𝑡) − 𝐸 (𝑡 + 1))1/2 (∫ ‖∇V‖2𝑝 𝑑𝑠)
󵄨󵄨 𝑡 Ω 󵄨󵄨
󵄨 1 󵄨 𝑡1 𝑡1
𝑡2 (𝑞−1)/𝑞 𝑡2 1/𝑞 1/𝑝

󵄩 󵄩𝑞 𝑡2
≤ (∫ 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩𝑞 𝑑𝑠) (∫ ‖V‖𝑞𝑞 𝑑𝑠) ≤ 𝐶(𝐸 (𝑡) − 𝐸 (𝑡 + 1))1/2 (∫ ‖∇V‖𝑝𝑝 𝑑𝑠)
𝑡 1 𝑡1 𝑡1
𝑡 𝑡
2
󵄩 󵄩𝑞 2
≤ 𝐶(𝐸 (𝑡) − 𝐸 (𝑡 + 1))𝑝/2(𝑝−1)
≤ 𝐶 (𝜀) ∫ 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩𝑞 𝑑𝑠 + 𝜀 ∫ ‖V‖𝑞𝑞 𝑑𝑠.
𝑡 1 𝑡 1
𝑡2
(69)
+ 𝜀 ∫ ‖∇V‖𝑝𝑝 𝑑𝑠.
𝑡1
Since 𝑝 < 𝑞 < 𝑟 + 2 and the property of the function
𝑓(𝑥) = 𝛼𝑥 /𝑥, 𝛼 ≥ 0, 𝑥 > 0, we obtain (74)
𝑝 𝑝
‖𝑢‖𝑞𝑞 ‖𝑢‖𝑝 ‖𝑢‖𝑟+2
𝑟+2
‖V‖𝑞𝑞 ‖V‖𝑝 ‖V‖𝑟+2
𝑟+2 We get from (57), (73), and (74) that
≤𝐶 +𝐶 , ≤𝐶 +𝐶 .
𝑞 𝑝 𝑟+2 𝑞 𝑝 𝑟+2 󵄨󵄨 𝑡2 󵄨󵄨
󵄨󵄨 󵄨
(70) 󵄨󵄨∫ ∫ (∇𝑢𝑡 ∇𝑢 + ∇V𝑡 ∇V) 𝑑𝑠󵄨󵄨󵄨
󵄨󵄨 𝑡 Ω 󵄨󵄨
We conclude from (69), (70), [𝑢, V] ∈ 𝑊, and Lemma 1 󵄨 1 󵄨
that 𝑡2
≤ 𝐶(𝐸 (𝑡) − 𝐸 (𝑡 + 1))𝑝/2(𝑝−1) + 𝜀 ∫ (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) 𝑑𝑠
‖𝑢‖𝑞𝑞 + ‖V‖𝑞𝑞 ≤ 𝐶 (‖𝑢‖𝑝𝑝 + ‖𝑢‖𝑟+2 𝑝 𝑟+2
𝑟+2 + ‖V‖𝑝 + ‖V‖𝑟+2 )
𝑡1
𝜀 𝑡2
≤ 𝐶 (‖𝑢‖𝑝𝑝 + ‖∇𝑢‖𝑝𝑝 + ‖V‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) (71) ≤ 𝐶(𝐸 (𝑡) − 𝐸 (𝑡 + 1))𝑝/2(𝑝−1) + ∫ 𝐾 ([𝑢, V]) 𝑑𝑠.
𝜃 𝑡1
≤ 𝐶 (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) ≤ 𝐶𝐸 (𝑡) . (75)
It follows from (63), (68), (69), and (71) that Choosing small enough 𝜀, we have from (65), (66), (67),
󵄨󵄨 𝑡2 𝑡2 󵄨󵄨 (72), and (75) that
󵄨󵄨󵄨− (∫ ∫ 󵄨󵄨󵄨𝑢 󵄨󵄨󵄨𝑞−2 𝑢 𝑢 𝑑𝑥 𝑑𝑠+∫ ∫ 󵄨󵄨󵄨V 󵄨󵄨󵄨𝑞−2 V V 𝑑𝑥 𝑑𝑠)󵄨󵄨󵄨
󵄨󵄨 󵄨 𝑡󵄨 𝑡 󵄨 𝑡󵄨 𝑡 󵄨󵄨
󵄨󵄨 𝑡1 Ω 𝑡1 Ω 󵄨󵄨 𝑡2
(72) ∫ 𝐾 ([𝑢, V]) 𝑑𝑠 ≤ 𝐶 [ (𝐸 (𝑡) − 𝐸 (𝑡 + 1))
𝑡2 𝑡1
≤ 𝐶 (𝜀) (𝐸 (𝑡) − 𝐸 (𝑡 + 1)) + 𝜀𝐶 ∫ 𝐸 (𝑠) 𝑑𝑠,
𝑡1 +(𝐸 (𝑡) − 𝐸 (𝑡 + 1))𝑝/2(𝑝−1) ] (76)
and we obtain from (63), Sobolev inequality, Hölder inequal- 𝑡2
ity, and Lemma 2 that + 𝜀 sup 𝐸 (𝑠) + 𝜀 ∫ 𝐸 (𝑠) 𝑑𝑠.
󵄨󵄨󵄨 𝑡2 󵄨󵄨 𝑡2 𝑡≤𝑠≤𝑡+1 𝑡1
󵄨󵄨− ∫ ∫ ∇𝑢 ∇𝑢𝑑𝑠󵄨󵄨󵄨 ≤ ∫ 󵄩󵄩󵄩∇𝑢 󵄩󵄩󵄩 ⋅ ‖∇𝑢‖ 𝑑𝑠
󵄨󵄨 𝑡 󵄨󵄨 󵄩 𝑡󵄩
󵄨󵄨 𝑡1 Ω 󵄨󵄨 𝑡1 It follows from (30) and (31) that
1/2 1/2
2 𝑡
󵄩 󵄩2
𝑡2 2 (𝑟 + 2) − 𝑝
≤ (∫ 󵄩󵄩󵄩∇𝑢𝑡 󵄩󵄩󵄩 𝑑𝑠) (∫ ‖∇𝑢‖2 𝑑𝑠) 𝐽 ([𝑢, V]) = (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 )
𝑡 1 𝑡1 2𝑝 (𝑟 + 2)
(77)
𝑡2 1/2 1
+ 𝐾 ([𝑢, V]) .
≤ 𝐶(𝐸 (𝑡)−𝐸 (𝑡 + 1))1/2 (∫ ‖∇𝑢‖2𝑝 𝑑𝑠) 2 (𝑟 + 2)
𝑡1
𝑡2 1/𝑝 On the other hand, from (12) and using (57) and (77), we
≤ 𝐶(𝐸 (𝑡)−𝐸 (𝑡 + 1))1/2 (∫ ‖∇𝑢‖𝑝𝑝 𝑑𝑠) deduce that
𝑡1
1 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2
𝐸 (𝑡) = (󵄩𝑢 󵄩 + 󵄩V 󵄩 ) + 𝐽 ([𝑢, V])
≤ 𝐶(𝐸 (𝑡) − 𝐸 (𝑡 + 1))𝑝/2(𝑝−1) 2 󵄩 𝑡󵄩 󵄩 𝑡󵄩
𝑡2 1 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 2 (𝑟 + 2) − 𝑝
+ 𝜀 ∫ ‖∇𝑢‖𝑝𝑝 𝑑𝑠. = (󵄩𝑢 󵄩 + 󵄩V 󵄩 ) +
𝑡1
2 󵄩 𝑡󵄩 󵄩 𝑡󵄩 2𝑝 (𝑟 + 2)
(73) 1
× (‖∇𝑢‖𝑝𝑝 + ‖∇V‖𝑝𝑝 ) + 𝐾 ([𝑢, V])
Similarly, we have the following formula: 2 (𝑟 + 2)
󵄨󵄨 𝑡2 󵄨󵄨 𝑡2 1 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 2 (𝑟 + 2) − 𝑝 1
󵄨󵄨 󵄨
󵄨󵄨− ∫ ∫ ∇V𝑡 ∇V𝑑𝑠󵄨󵄨󵄨 ≤ ∫ 󵄩󵄩󵄩󵄩∇V𝑡 󵄩󵄩󵄩󵄩 ⋅ ‖∇V‖ 𝑑𝑠 ≤ (󵄩󵄩𝑢𝑡 󵄩󵄩 + 󵄩󵄩V𝑡 󵄩󵄩 ) + ( + )
󵄨󵄨 𝑡 Ω 󵄨󵄨 2 2𝜃𝑝 (𝑟 + 2) 2 (𝑟 + 2)
󵄨 1 󵄨 𝑡1
2𝑡
󵄩 󵄩2
1/2 𝑡2 1/2 × 𝐾 ([𝑢, V]) .
≤ (∫ 󵄩󵄩󵄩∇V𝑡 󵄩󵄩󵄩 𝑑𝑠) (∫ ‖∇V‖2 𝑑𝑠) (78)
𝑡 1 𝑡1
By integrating (78) over [𝑡1 , 𝑡2 ], we obtain Acknowledgments

𝑡2 𝑡2
1 󵄩 󵄩2 󵄩 󵄩2
∫ 𝐸 (𝑠) 𝑑𝑠 ≤ ∫ (󵄩󵄩󵄩𝑢𝑡 󵄩󵄩󵄩 + 󵄩󵄩󵄩V𝑡 󵄩󵄩󵄩 ) 𝑑𝑠 This research was supported by the National Natural Sci-
𝑡1 2 𝑡1 ence Foundation of China (no. 61273016), The Natural Sci-
𝑡2
ence Foundation of Zhejiang Province (no. Y6100016), The
2 (𝑟 + 2) − 𝑝 1 Middle-aged and Young Leader in Zhejiang University of Sci-
+( + ) ∫ 𝐾 ([𝑢, V]) 𝑑𝑠.
2𝜃𝑝 (𝑟 + 2) 2 (𝑟 + 2) 𝑡1 ence and Technology (2008–2012), and the Interdisciplinary
(79) Pre-research Project of Zhejiang University of Science and
Technology (2010–2012).
For small enough 𝜀, we have from (76) and (79) that
𝑡2
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http://dx.doi.org/10.1155/2013/262010
Research Article
A Class of Spectral Element Methods and
Its A Priori/A Posteriori Error Estimates for 2nd-Order
Elliptic Eigenvalue Problems
Jiayu Han and Yidu Yang

School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China
Correspondence should be addressed to Yidu Yang; ydyang@gznu.edu.cn
Received 24 May 2013; Accepted 1 September 2013
Copyright © 2013 J. Han and Y. Yang. This is an open access article distributed under the Creative Commons Attribution License,
This paper discusses spectral and spectral element methods with Legendre-Gauss-Lobatto nodal basis for general 2nd-order elliptic
eigenvalue problems. The special work of this paper is as follows. (1) We prove a priori and a posteriori error estimates for spectral
and spectral element methods. (2) We compare between spectral methods, spectral element methods, finite element methods and
their derived p-version, h-version, and ℎ𝑝-version methods from accuracy, degree of freedom, and stability and verify that spectral
methods and spectral element methods are highly efficient computational methods.
1. Introduction nonsymmetric elliptic eigenvalue problems. This paper will

mainly perform the following work.
As we know, finite element methods are local numerical
methods for partial differential equations and particularly (1) We prove a priori and a posteriori error esti-
well suitable for problems in complex geometries, whereas mates of spectral and spectral element methods with
spectral methods can provide a superior accuracy, at the Legendre-Gauss-Lobatto nodal basis, respectively, for
expense of domain flexibility. Spectral element methods the general 2nd-order elliptic eigenvalue problems.
combine the advantages of the above methods (see [1]). So
far, spectral and spectral element methods are widely applied (2) We compare between spectral methods, spectral ele-
to boundary value problems (see [1, 2]), as well as applied ment methods with Legendre-Gauss-Lobatto nodal
to symmetric eigenvalue problems (see [3]). However, it is basis, finite element methods, and their derived 𝑝-
still a new subject to apply them to nonsymmetric elliptic version, ℎ-version, and ℎ𝑝-version methods from
eigenvalue problems. accuracy, degree of freedom, and stability and verify
A posteriorii error estimates and highly efficient compu- that spectral methods and spectral element methods
tational methods for finite elements of eigenvalue problems are highly efficient computational methods for non-
are the subjects focused on by the academia these years; symmetric 2nd-order elliptic eigenvalue problems.
see [3–16], and among them, for nonsymmetric 2nd-order
elliptic eigenvalue problems, [5, 15] provide a posteriori This paper is organized as follows. Section 2 introduces
error estimates and adaptive algorithms, [9] the function basic knowledge of second elliptic eigenvalue problems.
value recovery techniques and [8, 10] two-level discretization Sections 3 and 4 are devoted to a priori and a posteriori
schemes. error estimates of spectral and spectral element methods,
Based on the work mentioned above, this paper shall respectively. In Section 5, some numerical experiments are
further apply spectral and spectral element methods to performed by the methods mentioned above.
2. Preliminaries Ω = ⋃ 𝜅. We associate with the partition a polynomial degree

vector N = {𝑁𝜅 }, where 𝑁𝜅 is the polynomial degree in 𝜅. Let
Consider the 2nd-order elliptic boundary value problem ℎ𝜅 be the diameter of the element 𝜅, and let ℎ = max𝜅∈𝐾ℎ ℎ𝜅 .
We define spectral and spectral element spaces as follows:
𝐿𝑤 = −∇ ⋅ (𝐷∇𝑤) + b ⋅ ∇𝑤 + 𝑐𝑤 = 𝑓, in Ω,
(1)
𝑤 = 0, on 𝜕Ω, 𝑆𝑁 (Ω) = {V ∈ 𝑃𝑁 (Ω) , V|𝜕Ω = 0} ,
where Ω ⊂ 𝑅𝑑 (𝑑 = 2, 3) is a bounded domain, b and 𝑐 𝑆𝑁,ℎ (Ω) = {V ∈ 𝐶 (Ω) : V|𝜅 ∈ 𝑃𝑁𝜅 (𝜅) , ∀𝜅 ∈ 𝐾ℎ , V|𝜕Ω = 0} ,
are a real-valued vector function and a real-valued function, (9)
respectively, and 𝐷 is a positive scalar function with 𝐷(𝑥) ≥
𝐷0 > 0 (∀𝑥 ∈ Ω). where 𝑃𝑁(Ω) and 𝑃𝑁𝜅 (𝜅) are polynomial spaces of degree 𝑁
We denote the complex Sobolev spaces with norm ‖ ⋅ ‖𝑚 (resp. degree 𝑁 in every direction) in Ω and degree 𝑁𝜅 (resp.
by 𝐻𝑚 (Ω), 𝐻01 (Ω) = {V ∈ 𝐻1 (Ω), V|𝜕Ω = 0}. Let (⋅, ⋅) and degree 𝑁𝜅 in every direction) in the element 𝜅, respectively.
‖ ⋅ ‖0,Ω be a inner product and a norm in the complex space The spectral approximation of (5) is as follows: find 𝑤𝑁 ∈
𝐿2 (Ω), respectively. 𝑆𝑁(Ω), such that
In this paper, 𝐶 denotes a generic positive constant
independent of the polynomial degrees and mesh scales, 𝑎 (𝑤𝑁, V) = (𝑓, V) , ∀V ∈ 𝑆𝑁 (Ω) . (10)
which may not be the same at different occurrences.
Define the bilinear form 𝑎(⋅, ⋅) as follows: The spectral element approximation of (5) is as follows:
find 𝑤𝑁,ℎ ∈ 𝑆𝑁,ℎ (Ω), such that
𝑎 (𝑤, V) = ∫ 𝐷∇𝑤∇V + (b ⋅ ∇𝑤) V
Ω (2) 𝑎 (𝑤𝑁,ℎ , V) = (𝑓, V) , ∀V ∈ 𝑆𝑁,ℎ (Ω) . (11)
+ 𝑐𝑤V, ∀𝑤, V ∈ 𝐻01 (Ω) .
We assume that 𝑓 ∈ 𝐿2 (Ω) and derive from Lax-Milgram
We assume that 𝑓 ∈ 𝐿2 (Ω), 𝐷, b, and 𝑐 are bounded func- theorem that the variational formations (5), (6), (10), and (11)
tions on Ω, namely 𝐷, 𝑐 ∈ 𝐿∞ (Ω), b ∈ (𝐿∞ (Ω))𝑑 . Further have a unique solution, respectively.
more, we assume that ∇ ⋅ b exists and satisfies Define the interpolation operators
1
− ∇ ⋅ b + 𝑐 ≥ 0, in Ω. (3) 𝐼𝑁𝜅 ,ℎ𝜅 : 𝐶 (𝜅) 󳨀→ 𝑃𝑁𝜅 (𝜅) ,
2 (12)
Under these assumptions, the bilinear form 𝑎(⋅, ⋅) is 𝐼𝑁 : 𝐶 (Ω) 󳨀→ 𝑆𝑁 (Ω) ,
continuous in 𝐻01 (Ω) and 𝐻01 (Ω)-elliptic; that is, there exist
two constants 𝐴, 𝐵 > 0 independent of 𝑤, V such that as the interpolations in the element 𝜅 and the domain
Ω, respectively, with the tensorial Legendre-Gauss-Lobatto
|𝑎 (𝑤, V)| ≤ 𝐴‖𝑤‖1,Ω ‖V‖1,Ω , ∀𝑤, V ∈ 𝐻01 (Ω) , (LGL) interpolation nodes.
(4) Define the interpolation operator
Re 𝑎 (V, V) ≥ 𝐵‖V‖21,Ω , ∀V ∈ 𝐻01 (Ω) .
𝐼𝑁,ℎ : 𝐶 (Ω) 󳨀→ 𝑆𝑁,ℎ (Ω) , satisfying 𝐼𝑁,ℎ |𝜅 = 𝐼𝑁𝜅 ,ℎ𝜅 . (13)
The corresponding variational formulation of (1) is given
as follows: find 𝑤 ∈ 𝐻01 (Ω), such that We quote from [2] (see (5.8.27) therein) the interpolation
estimates for spectral and spectral element methods with LGL
𝑎 (𝑤, V) = (𝑓, V) , ∀V ∈ 𝐻01 (Ω) . (5) Nodal-basis as follows.
For all V ∈ 𝐻𝑠𝜅 (𝜅), 𝑠𝜅 ≥ (𝑑 + 1)/2,
The adjoint problem of (5) is as follows: find 𝑤∗ ∈ 𝐻01 (Ω),
󵄩󵄩 󵄩
such that 󵄩󵄩V − 𝐼𝑁𝜅 ,ℎ𝜅 V󵄩󵄩󵄩 ≤ 𝐶 (𝑠𝜅 ) ℎ𝜅min(𝑁𝜅 +1,𝑠𝜅 )−1 𝑁𝜅−𝑠𝜅 +1 ‖V‖𝑠𝜅 ,𝜅 , (14)
󵄩 󵄩1,𝜅
∗
𝑎 (V, 𝑤 ) = (V, 𝑓) , ∀V ∈ 𝐻01 (Ω) . (6) 󵄩󵄩 󵄩
󵄩󵄩V − 𝐼𝑁𝜅 ,ℎ𝜅 V󵄩󵄩󵄩 ≤ 𝐶 (𝑠𝜅 ) ℎ𝜅min(𝑁𝜅 +1,𝑠𝜅 ) 𝑁𝜅−𝑠𝜅 ‖V‖𝑠𝜅 ,𝜅 . (15)
󵄩 󵄩0,𝜅
As the general 2nd-order elliptic boundary value prob-
lems, we assume that the regularity estimates for problem (5) For all V ∈ 𝐻𝑠 (Ω), 𝑠 ≥ (𝑑 + 1)/2,
and its adjoint problem (6) hold, respectively. Namely
󵄩󵄩 󵄩
󵄩󵄩V − 𝐼𝑁V󵄩󵄩󵄩1,Ω ≤ 𝐶 (𝑠) 𝑁
−𝑠+1
󵄩 󵄩 ‖V‖𝑠,Ω ,
‖𝑤‖𝑟1 +1,Ω ≤ 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩0,Ω , 0 < 𝑟1 ≤ 1, (7) (16)
󵄩󵄩 󵄩
󵄩󵄩V − 𝐼𝑁V󵄩󵄩󵄩0,Ω ≤ 𝐶 (𝑠) 𝑁 ‖V‖𝑠,Ω .
󵄩󵄩 ∗ 󵄩󵄩 󵄩 󵄩 −𝑠
󵄩󵄩𝑤 󵄩󵄩𝑟2 +1,Ω ≤ 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩0,Ω , 0 < 𝑟2 ≤ 1. (8) (17)
We assume that 𝐾ℎ = {𝜅} is a regular rectangle (resp. We assume that the solution of boundary value problem
cuboid) or simplex partition of the domain Ω and satisfies (5) 𝑤 ∈ 𝐻01 (Ω) ∩ 𝐻𝑚 (Ω) (𝑚 > 1), that 𝑤𝑁 and 𝑤𝑁,ℎ are the
solutions of (10) and (11), respectively; then we derive from Define the solution operators 𝑇 : 𝐿2 (Ω) → 𝐻01 (Ω), 𝑇𝑁 :
Céa lemma and the interpolation estimates that 𝐿 (Ω) → 𝑆𝑁(Ω), and 𝑇𝑁,ℎ : 𝐿2 (Ω) → 𝑆𝑁,ℎ (Ω) by
2
󵄩󵄩 󵄩 ∀𝑓 ∈ 𝐿2 (Ω) , ∀V ∈ 𝐻01 (Ω) ,

󵄩󵄩𝑤𝑁 − 𝑤󵄩󵄩󵄩1,Ω ≤ 𝐶 (𝑚) 𝑁
−𝑚+1
‖𝑤‖𝑚,Ω , (18) 𝑎 (𝑇𝑓, V) = (𝑓, V) ,
󵄩󵄩 󵄩 𝑎 (𝑇𝑁𝑓, V𝑁) = (𝑓, V𝑁) , ∀𝑓 ∈ 𝐿2 (Ω) ,

󵄩󵄩𝑤𝑁,ℎ − 𝑤󵄩󵄩󵄩1,Ω ≤ {∑𝐶 (𝑠𝜅 ) ℎ𝜅
2{min(𝑁𝜅 +1,𝑠𝜅 )−1}
𝜅 ∀V𝑁 ∈ 𝑆𝑁 (Ω) , (27)
(19)
1/2
× 𝑁𝜅2(−𝑠𝜅 +1) ‖𝑤‖2𝑠𝜅 ,𝜅 } , 𝑎 (𝑇𝑁,ℎ 𝑓, V𝑁,ℎ ) = (𝑓, V𝑁,ℎ ) , ∀𝑓 ∈ 𝐿2 (Ω) ,
∀V𝑁,ℎ ∈ 𝑆𝑁,ℎ (Ω) .
where 𝑠𝜅 ≥ (𝑑 + 1)/2, ∀𝜅 ∈ 𝐾ℎ .
Obviously (see [17]), the equivalent operator forms for (24)
Particularly, if 𝑁𝜅 = 𝑁, ∀𝜅 ∈ 𝐾ℎ , then we have
and (26) are the following.
󵄩󵄩 󵄩 Find 𝜆 ∈ C, 0 ≠ 𝑢 ∈ 𝐻01 (Ω), such that
󵄩󵄩𝑤𝑁,ℎ − 𝑤󵄩󵄩󵄩1,Ω ≤ 𝐶 (𝑚) ℎ
min(𝑁+1,𝑚)−1
(20)
𝑇𝑢 = 𝜆−1 𝑢. (28)
× 𝑁1−𝑚 ‖𝑤‖𝑚,Ω .
Find 𝜆 𝑁,ℎ ∈ C, 0 ≠
𝑢𝑁,ℎ ∈ 𝑆𝑁,ℎ (Ω), such that
Note that (18) is also suited to spectral methods with
modal basis (see [1, 2]). 𝑇𝑁,ℎ 𝑢𝑁,ℎ = 𝜆−1
𝑁,ℎ 𝑢𝑁,ℎ . (29)
Using Aubin-Nitsche technique, we deduce from the
regularity estimate (8) and the estimates (18)–(20) the priori The adjoint problem of the eigenvalue problem (23) is
estimates of boundary value problem (5) for spectral and
spectral element methods; that is, 𝐿∗ 𝑢∗ = 𝜆∗ 𝑢∗ , in Ω,
(30)
󵄩󵄩 󵄩 𝑢∗ = 0, on 𝜕Ω,
󵄩󵄩𝑤𝑁 − 𝑤󵄩󵄩󵄩0,Ω ≤ 𝐶𝑁
−𝑚−𝑟2 +1
‖𝑤‖𝑚,Ω , (21)
󵄩󵄩 󵄩 where 𝐿∗ 𝑢∗ = −∇ ⋅ (𝐷∇𝑢∗ ) − b ⋅ ∇𝑢∗ + (𝑐 − ∇ ⋅ b)𝑢∗ .
󵄩󵄩𝑤𝑁,ℎ − 𝑤󵄩󵄩󵄩0,Ω ≤ 𝐶𝑁
−𝑚−𝑟2 +1 𝑟2 +min(𝑁+1,𝑚)−1
ℎ ‖𝑤‖𝑚,Ω . (22) The variational formation of (30) is given by the follow-
ing: find 𝜆∗ ∈ C, 0 ≠
𝑢∗ ∈ 𝐻01 (Ω), such that
3. Spectral and Spectral-Element
Approximations and Error Estimates for 𝑎∗ (𝑢∗ , V) := 𝑎 (V, 𝑢∗ ) = 𝜆∗ (𝑢∗ , V) , ∀V ∈ 𝐻01 (Ω) . (31)
Eigenvalue Problems The spectral element approximation scheme of (31) is
given by the following: find 𝜆∗𝑁,ℎ ∈ C, 0 ≠∗
𝑢𝑁,ℎ ∈ 𝑆𝑁,ℎ (Ω),
3.1. Spectral and Spectral-Element Approximations for Eigen- such that
value Problems. Consider the following eigenvalue problem
corresponding to the boundary value problem (1): 𝑎∗ (𝑢𝑁,ℎ
∗
, V𝑁,ℎ ) = 𝜆∗𝑁,ℎ (𝑢𝑁,ℎ
∗
, V𝑁,ℎ ) , ∀V𝑁,ℎ ∈ 𝑆𝑁,ℎ (Ω) .
(32)
𝐿𝑢 = 𝜆𝑢, in Ω,
(23) We can likewise define the equivalent operator forms for
𝑢 = 0, on 𝜕Ω.
the eigenvalue problems (31) and (32) as
The variational formation of (23) is given by the follow- 𝑇∗ 𝑢∗ = 𝜆∗−1 𝑢∗ ,
𝑢 ∈ 𝐻01 (Ω), such that
ing: find 𝜆 ∈ C, 0 ≠ (33)
∗ ∗
𝑇𝑁,ℎ 𝑢𝑁,ℎ = 𝜆∗−1 ∗
𝑁,ℎ 𝑢𝑁,ℎ .
𝑎 (𝑢, V) = 𝜆 (𝑢, V) , ∀V ∈ 𝐻01 (Ω) . (24)
Let 𝜆 be an eigenvalue of (23). There exists a smallest
The spectral approximation scheme of (24) is given by the integer 𝜇, called the ascent of 𝜆, such that ker((𝜆−1 − 𝑇)𝜇 ) =
following: find 𝜆 𝑁 ∈ C, 0 ≠
𝑢𝑁 ∈ 𝑆𝑁(Ω), such that ker((𝜆−1 − 𝑇)𝜇+1 ). 𝑞 = dim ker((𝜆−1 − 𝑇)𝜇 ) is called the
algebraic multiplicity of 𝜆. The functions in ker((𝜆−1 − 𝑇)𝜇 )
𝑎 (𝑢𝑁, V𝑁) = 𝜆 𝑁 (𝑢𝑁, V𝑁) , ∀V𝑁 ∈ 𝑆𝑁 (Ω) . (25) are called generalized eigenfunctions of 𝑇 corresponding to
𝜆. Likewise the ascent, algebraic multiplicity and generalized
The spectral element approximation scheme of (24) is eigenfunctions of 𝜆 𝑁,ℎ , 𝜆∗ and 𝜆∗𝑁,ℎ can be defined.
given by the following: find 𝜆 𝑁,ℎ ∈ C, 0 ≠
𝑢𝑁,ℎ ∈ 𝑆𝑁,ℎ (Ω), Let 𝜆 be an eigenvalue of (23) with the algebraic multiplic-
such that ity 𝑞 and the ascent 𝜇. Assume ‖𝑇𝑁,ℎ − 𝑇‖1,Ω → 0 (𝑁 → ∞,
ℎ → 0); then there are eigenvalues 𝜆 𝑗,𝑁,ℎ (𝑗 = 1, 2, . . . , 𝑞)
𝑎 (𝑢𝑁,ℎ , V𝑁,ℎ ) = 𝜆 𝑁,ℎ (𝑢𝑁,ℎ , V𝑁,ℎ ) , ∀V𝑁,ℎ ∈ 𝑆𝑁,ℎ (Ω) , of (26) which converge to 𝜆. Let 𝑀(𝜆) be the space spanned
(26) by all generalized eigenfunctions corresponding to 𝜆 of 𝑇,
and let 𝑀𝑁,ℎ (𝜆) be the space spanned by all generalized We assume that in this section, for the sake of simplicity,
eigenfunctions corresponding to all eigenvalues of 𝑇𝑁,ℎ that 𝑁𝜅 = 𝑁, ∀𝜅 ∈ 𝐾ℎ .
converge to 𝜆.
In view of adjoint problems (31) and (32), the definitions Theorem 5. If 𝑀(𝜆) ⊂ 𝐻𝑡1 (Ω) and 𝑀∗ (𝜆∗ ) ⊂ 𝐻𝑡2 (Ω), then
of 𝑀∗ (𝜆∗ ) and 𝑀𝑁,ℎ
∗
(𝜆∗ ) are analogous to 𝑀(𝜆) and 𝑀𝑁,ℎ (𝜆). there holds the following error estimates:
̂
Let 𝑀(𝜆) = {V ∈ 𝑀(𝜆) : ‖V‖1,Ω = 1}, and let 𝑀̂∗ (𝜆∗ ) = {V ∈
󵄨󵄨 𝑞 󵄨󵄨
𝑀∗ (𝜆∗ ) : ‖V‖1,Ω = 1}. 󵄨󵄨 1
󵄨󵄨 ∑𝜆
󵄨󵄨
󵄨
Note that when b = 0, both (24) and (26) are symmetric. 󵄨󵄨 𝑗,𝑁,ℎ − 𝜆󵄨󵄨󵄨
󵄨󵄨 𝑞 𝑗=1 󵄨󵄨
Thus, the ascent 𝜇 = 1 of 𝜆, and the ascent 𝑙 = 1 of 𝜆 𝑁,ℎ . 󵄨 󵄨
(40)
ℎ𝜏1 +𝜏2 −2
3.2. A Priori Error Estimates. We will analyze a prior error ≤ 𝐶 ( 𝑡 +𝑡 −2 ) sup ‖𝑢‖𝑡1 ,Ω sup ‖V‖𝑡2 ,Ω ,
𝑁1 2 ̂
𝑢∈𝑀(𝜆) ̂
V∈𝑀∗ ∗
(𝜆 )
estimates for spectral element methods which are suitable for
spectral methods with mesh fineness ℎ not considered. 󵄨󵄨 󵄨
󵄨󵄨𝜆 𝑗,𝑁,ℎ − 𝜆󵄨󵄨󵄨
Assume that 𝑅 and 𝑈 are two closed subspace in 𝐻01 (Ω). 󵄨 󵄨
Denote 1/𝜇
ℎ𝜏1 +𝜏2 −2
𝛿 (𝑅, 𝑈) = sup dist (V, 𝑈) , ≤ 𝐶(( 𝑡 +𝑡 −2 ) sup ‖𝑢‖𝑡1 ,Ω sup ‖V‖𝑡2 ,Ω ) (41)
V∈𝑅 𝑁1 2 ̂
𝑢∈𝑀(𝜆) ̂
V∈𝑀∗ ∗
(𝜆 )
‖V‖1,Ω =1 (34)
(𝑗 = 1, 2, . . . , 𝑞) ,
𝜃 (𝑅, 𝑈) = max (𝛿 (𝑅, 𝑈) , 𝛿 (𝑈, 𝑅)) .
We say that 𝜃(𝑅, 𝑈) is the gap between 𝑅 and 𝑈. ℎ𝜏1 −1
𝜃 (𝑀 (𝜆) , 𝑀𝑁,ℎ (𝜆)) ≤ 𝐶 sup ‖𝑢‖𝑡1 ,Ω . (42)
Denote 𝑁𝑡1 −1 𝑢∈𝑀(𝜆)
̂
𝜀𝑁,ℎ = 𝜀𝑁,ℎ (𝜆) = sup inf ‖𝑢 − V‖1,Ω ,

V∈𝑆𝑁,ℎ (Ω) 𝑙1
̂
𝑢∈𝑀(𝜆) Let ‖𝑢𝑁,ℎ ‖1,Ω = 1, and let (𝜆−1
𝑁,ℎ − 𝑇𝑁,ℎ ) 𝑢𝑁,ℎ = 0, for some
(35)
∗
𝜀𝑁,ℎ ∗
= 𝜀𝑁,ℎ (𝜆∗ ) = sup inf ‖𝑢 − V‖1,Ω . 𝑙1 ≤ 𝜇. Then, for every integer 𝑙2 (𝑙1 ≤ 𝑙2 ≤ 𝜇), there exists a
̂
𝑢∈𝑀∗ (𝜆∗ )V∈𝑆𝑁,ℎ (Ω) function 𝑢󸀠 , such that (𝜆−1 − 𝑇)𝑙2 𝑢󸀠 = 0 and
We give the following four lemmas from Theorem 8.1–8.4 (𝑙2 −𝑙1 +1)/𝜇
in [17], which are applications to spectral element methods. 󵄩󵄩 󵄩 ℎ𝜏1 −1
󵄩󵄩𝑢𝑁,ℎ − 𝑢󸀠 󵄩󵄩󵄩 ≤ 𝐶(( 𝑡 −1 ) sup ‖𝑢‖𝑡1 ,Ω ) ,
󵄩 󵄩1,Ω 𝑁1 ̂ 𝑢∈𝑀(𝜆)
Lemma 1. Assume ‖𝑇𝑁,ℎ − 𝑇‖1,Ω → 0 (𝑁 → ∞, ℎ → 0).
(43)
For small enough ℎ and big enough 𝑁, there holds
𝜃 (𝑀 (𝜆) , M𝑁,ℎ (𝜆)) ≤ 𝐶𝜀𝑁,ℎ . (36) where 𝜏1 = min(𝑁 + 1, 𝑡1 ), 𝜏2 = min(𝑁 + 1, 𝑡2 ).
Lemma 2. Assume ‖𝑇𝑁,ℎ − 𝑇‖1,Ω → 0 (𝑁 → ∞, ℎ → 0); Proof. We derive from the error estimate (20) that
then
󵄨󵄨 󵄨 󵄩󵄩 󵄩
󵄨󵄨 −1 1 𝑞 −1 󵄨󵄨󵄨 󵄩󵄩𝑇𝑁,ℎ − 𝑇󵄩󵄩󵄩1,Ω
󵄨󵄨𝜆 − ∑ 𝜆 󵄨󵄨 ∗
󵄨󵄨󵄨 𝑗,𝑁,ℎ 󵄨󵄨 ≤ 𝐶𝜀𝑁,ℎ 𝜀𝑁,ℎ . (37)
󵄩󵄩 󵄩
󵄨󵄨 𝑞 𝑗=1 󵄨
󵄨󵄨 󵄩󵄩(𝑇 − 𝑇𝑁,ℎ ) 𝑓󵄩󵄩󵄩1,Ω
= sup 󵄩󵄩 󵄩󵄩 (44)
Lemma 3. Assume that ‖𝑇𝑁,ℎ − 𝑇‖1,Ω → 0 (𝑁 → ∞, ℎ → 𝑓∈𝐻01 (Ω) 󵄩󵄩𝑓󵄩󵄩1,Ω
0); then there holds
≤ 𝐶 (1 + 𝑟1 ) ℎ𝑟1 𝑁−𝑟1 󳨀→ 0 (𝑁 󳨀→ ∞, ℎ 󳨀→ 0) .
󵄨󵄨 󵄨 1/𝜇
󵄨󵄨𝜆 − 𝜆 𝑗,𝑁,ℎ 󵄨󵄨󵄨 ≤ 𝐶(𝜀𝑁,ℎ 𝜀𝑁,ℎ
∗
) (𝑗 = 1, 2, . . . , 𝑞) . (38)
󵄨 󵄨
By (14),
Since ker((𝜆−1 − 𝑇)𝑙 ) (𝑙 ≥ 1) is a finite dimensional space, there
exists a direct-sum decomposition 𝐻01 (Ω) = ker((𝜆−1 − 𝑇)𝑙 ) ⊕ 𝜀𝑁,ℎ = 𝜀𝑁,ℎ (𝜆) = sup inf ‖𝑢 − V‖1,Ω
𝑀𝑙 . We define the operator 𝐸𝑙 as a projection along 𝑀𝑙 from ̂
𝑢∈𝑀(𝜆)
V∈𝑆𝑁,ℎ (Ω)
𝐻01 (Ω) to ker((𝜆−1 − 𝑇)𝑙 ). (45)

ℎ𝜏1 −1
≤ 𝐶 ( 𝑡 −1 ) sup ‖𝑢‖𝑡1 ,Ω .
Lemma 4. Assume ‖𝑇𝑁,ℎ − 𝑇‖1,Ω → 0 (𝑁 → ∞, ℎ → 0). 𝑁1 ̂
𝑢∈𝑀(𝜆)
Let 𝜆 𝑁,ℎ be an eigenvalue of 𝑇𝑁,ℎ and lim𝑁 → ∞,ℎ → 0 𝜆 𝑁,ℎ = 𝜆.
𝑘
𝑢𝑁,ℎ satisfies (𝜆−1 𝑁,ℎ − 𝑇𝑁,ℎ ) 𝑢𝑁,ℎ = 0 and ‖𝑢𝑁,ℎ ‖1,Ω = 1, where Analogically,
𝑘 ≤ 𝜇 is a positive integer. Then, for every integer 𝑙 ∈ [𝑘, 𝜇],
there holds ℎ𝜏2 −1
∗
󵄩󵄩 󵄩 (𝑙−𝑘+1)/𝜇 𝜀𝑁,ℎ ≤ 𝐶( ) sup ‖𝑢‖𝑡2 ,Ω . (46)
󵄩󵄩𝑢𝑁,ℎ − 𝐸𝑙 𝑢𝑁,ℎ 󵄩󵄩󵄩1,Ω ≤ 𝐶𝜀𝑁,ℎ . (39) 𝑁𝑡2 −1 𝑢∈𝑀
̂ ∗ (𝜆∗ )
Plugging the two inequalities above into (36), (38), and (39) Let (𝜆, 𝑢) be an eigenpair of (24). If 𝜆 𝑁,ℎ is an eigenvalue of
yields (42), (41), and (43), respectively. We find from (37) that (26) convergence to 𝜆, then there exists 𝑢𝑁,ℎ ∈ ker(𝜆−1 𝑁,ℎ −𝑇𝑁,ℎ ),
󵄨󵄨 𝑞 󵄨󵄨 󵄨󵄨 𝑞 −1 󵄨 such that (51)–(53) hold.
󵄨󵄨 1 󵄨󵄨 󵄨󵄨 1 𝜆 𝑗,𝑁,ℎ − 𝜆−1 󵄨󵄨󵄨
󵄨󵄨 ∑ 𝜆 󵄨 󵄨
− 𝜆󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∑ −1 −1 󵄨󵄨
󵄨󵄨 󵄨
󵄨󵄨 𝑞 𝑗=1 𝑗,𝑁,ℎ 󵄨󵄨 󵄨󵄨 𝑞 𝑗=1 𝜆 𝜆 𝑗,𝑁,ℎ 󵄨󵄨󵄨 Proof. We deduce (53) immediately from (41). We derive
󵄨 󵄨 󵄨 󵄨 from (22) and (7) that
󵄨󵄨 𝑞 󵄨󵄨 (47)
󵄨󵄨 1 󵄨
󵄨 󵄩󵄩󵄩𝑇𝑓 − 𝑇𝑁,ℎ 𝑓󵄩󵄩󵄩 ≤ 𝐶𝑁−𝑟1 −𝑟2 ℎ𝑟1 +𝑟2 󵄩󵄩󵄩𝑓󵄩󵄩󵄩 ;
≤ 𝐶 󵄨󵄨󵄨󵄨 ∑ 𝜆−1 − 𝜆−1 󵄨󵄨󵄨󵄨 ≤ 𝐶𝜀𝑁,ℎ 𝜀𝑁,ℎ
∗
, 󵄩 󵄩0,Ω 󵄩 󵄩0,Ω (54)
󵄨󵄨 𝑞 𝑗=1 𝑗,𝑁,ℎ 󵄨󵄨
󵄨 󵄨 thus,‖𝑇 − 𝑇𝑁,ℎ ‖0,Ω → 0 (𝑁 → ∞, ℎ → 0). Taking 𝑢 =
−1
combining with (45) and (46) yields (40). 𝐸𝑢𝑁,ℎ and by virtue of 𝑅(𝑇𝑁,ℎ , 𝑧)𝑢𝑁,ℎ = (𝜆−1 𝑁,ℎ − 𝑧) 𝑢𝑁,ℎ ,
Supposing that ‖𝑇𝑁,ℎ − 𝑇‖0,Ω → 0 (𝑁 → ∞, ℎ → 0), Lemma 6 and (22), we have
󵄩󵄩 󵄩 󵄩 󵄩
𝜌(𝑇) is a regular set of 𝑇, and Γ ⊂ 𝜌(𝑇) is a closed Jordan 󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩0,Ω = 󵄩󵄩󵄩𝐸𝑢𝑁,ℎ − 𝐸𝑁,ℎ 𝑢𝑁,ℎ 󵄩󵄩󵄩0,Ω
curve enclosing 𝜆−1 .
󵄩 󵄩
Denote ≤ 𝐶󵄩󵄩󵄩(𝑇 − 𝑇𝑁,ℎ ) 𝑢𝑁,ℎ 󵄩󵄩󵄩0,Ω
𝑅 (𝑧) = (𝑇 − 𝑧)−1 , 󵄩 󵄩 (55)
≤ 𝐶 (󵄩󵄩󵄩(𝑇 − 𝑇𝑁,ℎ ) 𝑢󵄩󵄩󵄩0,Ω
(48)
−1
𝑅 (𝑇𝑁,ℎ , 𝑧) = (𝑇𝑁,ℎ − 𝑧) . 󵄩 󵄩
+󵄩󵄩󵄩(𝑇 − 𝑇𝑁,ℎ ) (𝑢𝑁,ℎ − 𝑢)󵄩󵄩󵄩0,Ω ) ,
Define the spectral projection operators
from which follows
−1 󵄩󵄩 󵄩 󵄩 󵄩
𝐸= ∫ 𝑅 (𝑇, 𝑧) dz : 𝐻01 (Ω) 󳨀→ 𝑀 (𝜆) , 󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩0,Ω ≤ 𝐶󵄩󵄩󵄩(𝑇 − 𝑇𝑁,ℎ ) 𝑢󵄩󵄩󵄩0,Ω
2𝑖𝜋 Γ
(49) (56)
−1 𝐶ℎ𝑟2 +𝜏1 −1 ‖𝑢‖𝑡1 ,Ω
𝐸𝑁,ℎ = ∫ 𝑅 (𝑇𝑁,ℎ , 𝑧) dz : 𝐻01 (Ω) 󳨀→ 𝑀𝑁,ℎ (𝜆) . ≤ ,
2𝑖𝜋 Γ 𝑁𝑟2 +𝑡1 −1
We give the following lemma by referring to [18, 19] (see which is (52). By direct calculation, we have
proposition 5.3 in [18] and theorem 1.3.2 in [19]).
󵄩󵄩 󵄩
󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,Ω
Lemma 6. If ‖𝑇𝑁,ℎ − 𝑇‖0,Ω → 0 (𝑁 → ∞, ℎ → 0), then
󵄩 󵄩
there holds that 𝐸𝑁,ℎ → 𝐸(𝑝), 𝑅(𝑇𝑁,ℎ , 𝑧) is uniformly bound- = 󵄩󵄩󵄩𝜆𝑇𝑢 − 𝜆 ℎ 𝑇𝑁,ℎ 𝑢𝑁,ℎ 󵄩󵄩󵄩1,Ω
ed with 𝑁 and ℎ, and 󵄩 󵄩 󵄩 󵄩 (57)
≤ 󵄩󵄩󵄩𝜆𝑇𝑢 − 𝜆𝑇𝑁,ℎ 𝑢󵄩󵄩󵄩1,Ω + 󵄩󵄩󵄩𝜆𝑇𝑁,ℎ 𝑢 − 𝜆 ℎ 𝑇𝑁,ℎ 𝑢𝑁,ℎ 󵄩󵄩󵄩1,Ω
󵄩󵄩󵄩(𝐸𝑁,ℎ − 𝐸) V󵄩󵄩󵄩 ≤ 𝐶max󵄩󵄩󵄩(𝑇 − 𝑇𝑁,ℎ ) 𝑅 (𝑧) V󵄩󵄩󵄩 ,
󵄩 󵄩0,Ω 𝑧∈Γ 󵄩 󵄩0,Ω
󵄩 󵄩 󵄩 󵄩
≤ 󵄩󵄩󵄩(𝑇 − 𝑇𝑁,ℎ ) (𝜆𝑢)󵄩󵄩󵄩1,Ω + 𝐶󵄩󵄩󵄩𝜆𝑢 − 𝜆 𝑁,ℎ 𝑢𝑁,ℎ 󵄩󵄩󵄩0,Ω .
∀V ∈ 𝐻01 (Ω) ,
Plugging (20), (52), and (53) into (57) yields (51).
󵄩󵄩 󵄩 󵄩 󵄩
󵄩󵄩(𝐸𝑁,ℎ − 𝐸) V󵄩󵄩󵄩0,Ω ≤ 𝐶max󵄩󵄩󵄩(𝑇 − 𝑇𝑁,ℎ ) 𝑅 (𝑇𝑁,ℎ , 𝑧) V󵄩󵄩󵄩0,Ω , If (𝜆, 𝑢) is an eigenpair of (24), let 𝑢𝑁,ℎ = 𝐸𝑁,ℎ 𝑢; by the
𝑧∈Γ
same argument we can prove (51) and (52).
∀V ∈ 𝐻01 (Ω) .
(50) 4. A Posteriori Error Estimates
Theorem 7. Under the assumptions of Theorem 5, further Based on [20], we will discuss a posteriori error estimates.
assume that the ascent of 𝜆 is 𝜇 = 1. Let (𝜆 𝑁,ℎ , 𝑢𝑁,ℎ ) be an We further assume that Ω ⊂ 𝑅2 , the partition 𝐾ℎ is 𝛾-shape
eigenpair of (26) with ‖𝑢𝑁,ℎ ‖0,Ω = 1; then there exists an regular, and the polynomial degree of neighboring elements
eigenpair (𝜆, 𝑢) of (24), such that are comparable; that is, there exists 𝛾 > 0, such that for all
𝜅, 𝜅󸀠 ∈ 𝐾ℎ , 𝜅 ∩ 𝜅󸀠 ≠
0,
󵄩󵄩 󵄩 𝐶ℎ𝜏1 −1 ‖𝑢‖𝑡1 ,Ω
󵄩󵄩𝑢𝑁,ℎ − 𝑢󵄩󵄩󵄩1,Ω ≤ , (51)
𝑁𝑡1 −1 𝛾−1 ℎ𝜅 ≤ ℎ𝜅󸀠 ≤ 𝛾ℎ𝜅 ,
(58)
󵄩󵄩 󵄩 𝐶ℎ𝑟2 +𝜏1 −1 ‖𝑢‖𝑡1 ,Ω 𝛾−1 (𝑁𝜅 + 1) ≤ 𝑁𝜅󸀠 + 1 ≤ 𝛾 (𝑁𝜅 + 1) .
󵄩󵄩𝑢𝑁,ℎ − 𝑢󵄩󵄩󵄩0,Ω ≤ , (52)
𝑁𝑟2 +𝑡1 −1
󵄨󵄨 󵄨 We refer to the ℎ𝑝-clément interpolation estimates given
󵄨󵄨𝜆 𝑁,ℎ − 𝜆󵄨󵄨󵄨 by [20, 21] (see theorems 2.2 and 2.3, respectively), which gen-
eralize the well-known clément type interpolation operators
ℎ𝜏1 +𝜏2 −2 (53)
studied in [22] and [23] to the hp context.
≤ 𝐶 (( 𝑡 +𝑡 −2
) sup ‖𝑢‖𝑡1 ,Ω sup ‖V‖𝑡2 ,Ω ) ,
𝑁1 2 ̂
𝑢∈𝑀(𝜆) ̂
V∈𝑀∗ ∗
(𝜆 )
Lemma 8. Assume that the partition 𝐾ℎ is 𝛾-shape regular
where 𝜏1 = min(𝑁 + 1, 𝑡1 ) and 𝜏2 = min(𝑁 + 1, 𝑡2 ). and the polynomial distribution N is comparable. Then there
exists a positive constant 𝐶 = 𝐶(𝛾) and the clément operator Let 𝐷𝜅 , b𝜅 , and 𝑐𝜅 be the interpolations of 𝐷, b, and 𝑐 in
𝐼 : 𝐻01 (Ω) → 𝑆𝑁,ℎ (Ω), such that 𝜅 with the polynomial degree 𝑁𝜅 (resp. degree 𝑁𝜅 in every
direction), respectively, or the 𝐿2 (𝜅)-projection on the space
ℎ𝜅 of polynomials with degree 𝑁𝜅 . For convenient argument,
‖V − 𝐼V‖0,𝜅 ≤ 𝐶 ‖∇V‖0,𝜔𝜅 , (59)
𝑁𝜅 here and hereafter we assume that (𝜆, 𝑢) and (𝜆∗ = 𝜆, 𝑢∗ )
are the eigenpairs of the eigenvalue problem (24) and its
ℎ𝑒 adjoint problem (31), respectively. (𝜆 𝑁,ℎ , 𝑢𝑁,ℎ ) and (𝜆∗𝑁,ℎ =
‖V − 𝐼V‖0,𝑒 ≤ 𝐶√ ‖∇V‖0,𝜔𝑒 , (60)
𝑁𝑒 ∗
𝜆 𝑁,ℎ , 𝑢𝑁,ℎ ) are the solutions of the corresponding spectral
element approximations (26) and (32), respectively.
where ℎ𝑒 is the length of the edge 𝑒 and 𝑁𝑒 = max(𝑁𝜅1 , 𝑁𝜅2 ), Denote
where 𝜅1 , 𝜅2 are elements sharing the edge 𝑒 and 𝜔𝜅 , 𝜔𝑒 are
patches covering 𝜅 and 𝑒 with a few layers, respectively. 𝐿 𝜅 𝑢𝑁,ℎ : = −∇ ⋅ (𝐷𝜅 ∇𝑢𝑁,ℎ )
Define interval 𝐼̂ = (0, 1) and weight function Φ𝐼̂(𝑥) := + b𝜅 ⋅ ∇𝑢𝑁,ℎ + 𝑐𝜅 𝑢𝑁,ℎ ,
𝑥(1 − 𝑥). Denote the reference square and triangle element by (67)
𝜅̂ = (0, 1)2 and 𝜅̂ = {(𝑥, 𝑦)|0 < 𝑥 < 1, 0 < 𝑦 < √3(1/2 − |1/2 − 𝐿∗𝜅 𝑢𝑁,ℎ
∗ ∗
: = −∇ ⋅ (𝐷𝜅 ∇𝑢𝑁,ℎ )
𝑥|)}, respectively. Define weight function Φ𝜅̂ (𝑥) := dist(𝑥, 𝜕̂
𝜅).
∗ ∗
− b𝜅 ⋅ ∇𝑢𝑁,ℎ + (𝑐𝜅 − ∇ ⋅ b𝜅 ) 𝑢𝑁,ℎ .
The following three lemmas are given by [20]. Lemmas
9–10 provide the polynomial inverse estimates in standard Define the local error indicators
2 2 2
interval and element, while Lemma 11 provides a result for the 𝜂𝛼;𝜅 := 𝜂𝛼;𝐵𝜅
+ 𝜂𝛼;𝐸𝜅
,
extension from an edge to the element. (68)
∗2 ∗2 ∗2
𝜂𝛼;𝜅 := 𝜂𝛼;𝐵𝜅
+ 𝜂𝛼;𝐸𝜅
.
Lemma 9. Let −1 < 𝛼 < 𝛽, 𝜎 ∈ [0, 1]. Then there exists 𝐶 =
2 ∗2
𝐶(𝛼, 𝛽), such that for all 𝑁 ∈ N and all univariate polynomials Their first terms 𝜂𝛼;𝐵𝜅
, 𝜂𝛼;𝐵𝜅
are the weighted element internal
𝜋𝑁 of degree 𝑁, residuals given by
󵄨 󵄨2 ℎ𝜅2 󵄩󵄩 󵄩󵄩2
∫ Φ𝛼𝐼̂ (𝑥) 󵄨󵄨󵄨𝜋𝑁 (𝑥)󵄨󵄨󵄨 dx 2
𝜂𝛼;𝐵 := 2
󵄩󵄩(−𝐿 𝜅 𝑢𝑁,ℎ + 𝜆 𝑁,ℎ 𝑢𝑁,ℎ ) Φ𝛼/2
󵄩 𝜅 󵄩
󵄩 ,
󵄩0,𝜅
𝐼̂
𝜅
𝑁𝜅
(61) (69)
𝛽󵄨 󵄨2 ℎ2 󵄩
≤ 𝐶𝑁2(𝛽−𝛼) ∫ Φ𝐼̂ 󵄨󵄨󵄨𝜋𝑁 (𝑥)󵄨󵄨󵄨 dx. 󵄩󵄩2
𝐼̂
∗2
𝜂𝛼;𝐵 := 𝜅2 󵄩󵄩󵄩󵄩(−𝐿∗𝜅 𝑢𝑁,ℎ
∗
+ 𝜆∗𝑁,ℎ 𝑢𝑁,ℎ
∗
) Φ𝛼/2
𝜅 󵄩
󵄩 .
󵄩0,𝜅
𝜅
𝑁𝜅
Lemma 10. Let −1 < 𝛼 < 𝛽, 𝜎 ∈ [0, 1]. Then there exist 2 ∗2
Their second terms 𝜂𝛼;𝐸 , 𝜂𝛼;𝐸 are the weighted element
𝐶1 = 𝐶(𝛼, 𝛽), 𝐶2 = 𝐶𝜎 > 0, such that for all 𝑁 ∈ N and 𝜅 𝜅
all polynomials 𝜋𝑁 of degree bi-𝑁, boundary residuals given by
2 ℎ𝑒 󵄩󵄩󵄩󵄩 𝜕𝑢𝑁,ℎ 󵄩󵄩2

󵄩󵄩
∫
󵄨 󵄨2
Φ𝛼𝜅̂ 󵄨󵄨󵄨𝜋𝑁󵄨󵄨󵄨 dxdy 𝜂𝛼;𝐸 := ∑ 󵄩󵄩𝐷𝜅 [ ] Φ𝛼/2
𝑒 󵄩󵄩 ,
𝜅
𝑒⊂𝜕𝜅∩Ω
2𝑁 𝑒󵄩
󵄩 𝜕𝑛 󵄩󵄩0,𝑒
𝜅̂
(62) (70)
𝛽󵄨 󵄨2 󵄩 𝜕𝑢∗ 󵄩󵄩2
≤ 𝐶1 𝑁 2(𝛽−𝛼)
∫ Φ𝜅̂ 󵄨󵄨󵄨𝜋𝑁󵄨󵄨󵄨 dxdy, ℎ𝑒 󵄩󵄩󵄩 󵄩󵄩
∗2
𝜂𝛼;𝐸 := ∑ 󵄩󵄩𝐷𝜅 [ 𝑁,ℎ ] Φ𝛼/2
𝑒 󵄩
󵄩 ,
𝜅̂ 𝜅
2𝑁 𝑒󵄩
󵄩
󵄩 𝜕𝑛 󵄩󵄩
𝑒⊂𝜕𝜅∩Ω 󵄩0,𝑒
󵄨󵄨 󵄨󵄨2
∫ Φ2𝜎
𝜅̂ 󵄨󵄨∇𝜋𝑁 󵄨󵄨 dxdy where we denote the jump of the normal derivatives of 𝑢𝑁,ℎ
𝜅̂ ∗ ∗
(63) and 𝑢𝑁,ℎ across the edges by [𝜕𝑢𝑁,ℎ /𝜕𝑛] and [𝜕𝑢𝑁,ℎ /𝜕𝑛],
󵄨 󵄨2 respectively. ℎ𝑒 is the length of edge 𝑒. The weight functions
≤ 𝐶2 𝑁2(2−𝜎) ∫ Φ𝜎𝜅̂ 󵄨󵄨󵄨𝜋𝑁󵄨󵄨󵄨 dxdy. Φ𝜅 and Φ𝑒 are scaled transformations of the weight functions
𝜅̂
Φ𝜅̂ and Φ𝑒̂; that is, if 𝐹𝜅 is the element map for element 𝜅 and
Lemma 11. Let 𝛼 ∈ (1/2, 1]. 𝑒̂ := (0, 1) × {0}, Φ𝑒̂ := 𝑥(1 − 𝑥); 𝑒 is the image of the edge 𝑒̂ under 𝐹𝜅 , then
then there exists 𝐶𝛼 > 0 such that for all 𝑁 ∈ N, 𝜀 ∈ (0, 1],
and all univariate polynomials 𝜋 of degree 𝑁, there exists an Φ𝜅 = 𝐶𝜅 Φ𝜅̂ ∘ 𝐹𝜅−1 , Φ𝑒 = 𝐶𝑒 Φ𝑒̂ ∘ 𝐹𝜅−1 , (71)
extension V𝑒̂ ∈ 𝐻1 (̂
𝜅) and holds where we choose 𝐶𝜅 , 𝐶𝑒 > 0, such that
V𝑒̂|𝑒̂ = 𝜋 ⋅ Φ𝛼𝑒̂ , V𝑒̂|𝜕̂𝜅\̂𝑒 = 0, (64) ∫ Φ𝜅 dxdy = ∫ dxdy, ∫ Φ𝑒 ds = ∫ ds. (72)
𝜅 𝜅 𝑒 𝑒
󵄩󵄩 󵄩󵄩2 󵄩 𝛼/2 󵄩2
󵄩󵄩V𝑒̂󵄩󵄩0,̂𝜅 ≤ 𝐶𝛼 𝜀󵄩󵄩󵄩󵄩𝜋Φ𝑒̂ 󵄩󵄩󵄩󵄩0,̂𝑒, (65) We define the global error indicators as follows:
󵄩󵄩 󵄩󵄩2 󵄩 󵄩󵄩2 𝜂𝛼2 := ∑ 𝜂𝛼;𝜅
2
,
󵄩󵄩∇V𝑒̂󵄩󵄩0,̂𝜅 ≤ 𝐶𝛼 (𝜀𝑁
2(2−𝛼)
+ 𝜀−1 ) 󵄩󵄩󵄩󵄩𝜋Φ𝛼/2
𝑒̂ 󵄩
󵄩 .
󵄩0,̂𝑒
(66) 𝜅∈𝐾ℎ
(73)
It is easy to know that the three lemmas above hold for 𝜂𝛼∗2 := ∑ 𝜂𝛼;𝜅
∗2
.
complex-valued polynomials. 𝜅∈𝐾ℎ
Theorem 12. Let 𝛼 ∈ [0, 1]. Then there exists a constant 𝐶 > 0 1 𝜕𝑢𝑁,ℎ
+ ∑ ∑ ∫ 𝐷[ ]𝑤
independent of ℎ, N, and 𝜅, such that 2 𝜅∈𝐾 𝑒⊂𝜕𝜅∩Ω 𝑒
𝜕𝑛
ℎ
󵄩󵄩 󵄩2
󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,Ω ≤ 𝐶 ∑ 𝑁𝜅 𝜂𝛼;𝜅
2𝛼 2
+ ∫ (𝜆𝑢 − 𝜆 𝑁,ℎ 𝑢𝑁,ℎ ) (𝑢 − 𝑢𝑁,ℎ ), (76)
𝜅∈𝐾ℎ Ω
ℎ𝜅2 󵄩󵄩 󵄩2 which together with

+𝐶 ∑ { 󵄩𝐿 𝑢 − 𝐿𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝜅
𝜅∈𝐾ℎ 𝑁𝜅2 󵄩 𝜅 𝑁,ℎ 𝜕𝑢𝑁,ℎ 𝜕𝑢𝑁,ℎ
∫ 𝐷[ ] 𝑤 = ∫ (𝐷 − 𝐷𝜅 ) [ ]𝑤
ℎ𝑒 󵄩󵄩 𝑒 𝜕𝑛 𝑒 𝜕𝑛
󵄩2
+ ∑ 󵄩󵄩𝐷 − 𝐷𝜅 󵄩󵄩󵄩0,𝑒 (74)
𝑁
𝑒⊂𝜕𝜅∩Ω 𝑒 𝜕𝑢𝑁,ℎ
+ ∫ 𝐷𝜅 [ ] 𝑤,
󵄩󵄩 𝜕𝑢 󵄩󵄩2 𝑒 𝜕𝑛
󵄩 𝑁,ℎ 󵄩󵄩
× 󵄩󵄩󵄩 󵄩 }
󵄩󵄩 𝜕𝑛 󵄩󵄩󵄩0,∞,𝑒 ∫ (−𝐿𝑢𝑁,ℎ + 𝜆 𝑁,ℎ 𝑢𝑁,ℎ ) 𝑤 = ∫ (−𝐿 𝜅 𝑢𝑁,ℎ + 𝜆 𝑁,ℎ 𝑢𝑁,ℎ ) 𝑤
𝜅 𝜅
󵄩 󵄩2
+ 𝐶󵄩󵄩󵄩𝜆𝑢 − 𝜆 𝑁,ℎ 𝑢𝑁,ℎ 󵄩󵄩󵄩0,Ω .
+ ∫ (𝐿 𝜅 𝑢𝑁,ℎ − 𝐿𝑢𝑁,ℎ ) 𝑤
𝜅
Proof. We denote 𝑤 := 𝑢 − 𝑢𝑁,ℎ − 𝐼(𝑢 − 𝑢𝑁,ℎ ), where 𝐼 is ℎ𝑝- (77)
clément operator given by Lemma 8. We derive from 𝐻01 (Ω)-
elliptic of 𝑎(⋅, ⋅) that and using Cauchy-Schwartz inequality, the ℎ𝑝-clément inter-
polation estimates in Lemma 8 then yield
󵄩 󵄩2
𝐶󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,Ω ≤ 𝑎 (𝑢 − 𝑢𝑁,ℎ , 𝑤) 󵄩󵄩 󵄩2
󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,Ω
+ 𝑎 (𝑢 − 𝑢𝑁,ℎ , 𝐼 (𝑢 − 𝑢𝑁,ℎ ))
ℎ𝜅2 󵄩󵄩 󵄩2
󵄩𝐿 𝑢 − 𝐿𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝜅
2 2
≤ 𝐶 { ∑ [𝜂0;𝐵 + 𝜂0;𝐸 +
= 𝜆 ∫ 𝑢𝑤 − 𝑎 (𝑢𝑁,ℎ , 𝑤) 𝜅∈𝐾ℎ
𝜅 𝜅
𝑁𝜅2 󵄩 𝜅 𝑁,ℎ
Ω
1/2
+ ∫ (𝜆𝑢 − 𝜆 𝑁,ℎ 𝑢𝑁,ℎ ) ℎ𝑒 󵄩󵄩󵄩󵄩 𝜕𝑢𝑁,ℎ 󵄩󵄩󵄩2
󵄩
+ ∑ 󵄩(𝐷 − 𝐷𝜅 ) [ ]󵄩 ]}
Ω 𝑁 󵄩󵄩
𝑒⊂𝜕𝜅∩Ω 𝑒 󵄩
𝜕𝑛 󵄩󵄩󵄩0,𝑒
× 𝐼 (𝑢 − 𝑢𝑁,ℎ ) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩
× 󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,Ω +𝐶󵄩󵄩󵄩𝜆𝑢 − 𝜆 𝑁,ℎ 𝑢𝑁,ℎ 󵄩󵄩󵄩0,Ω 󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩0,Ω .
(78)
= ∫ (𝜆 𝑁,ℎ 𝑢𝑁,ℎ ) 𝑤
Ω
Using scaled transformation and setting 𝛼 = 0, 𝛽 = 𝛼 in
+ ∫ (𝜆𝑢 − 𝜆 𝑁,ℎ 𝑢𝑁,ℎ ) 𝑢 − 𝑢𝑁,ℎ (75) (61) and (62), we get 𝜂0;𝐸𝜅 ≤ 𝐶𝑁𝜅𝛼 𝜂𝛼;𝐸𝜅 and 𝜂0;𝐵𝜅 ≤ 𝐶𝑁𝜅𝛼 𝜂𝛼;𝐵𝜅 ;
Ω then this proof concludes.
− 𝑎 (𝑢𝑁,ℎ , 𝑤) , For the adjoint eigenvalue problem, we still have the
following.
𝑎 (𝑢𝑁,ℎ , 𝑤) = ∑ ∫ 𝐿𝑢𝑁,ℎ 𝑤
𝜅∈𝜅ℎ 𝜅 Theorem 13. Let 𝛼 ∈ [0, 1]. Then there exists a constant 𝐶 > 0
independent of ℎ, N, and 𝜅, such that
𝜕𝑢𝑁,ℎ
− ∑∫ 𝐷 𝑤
𝜕𝑛 󵄩󵄩 ∗ ∗ 󵄩 󵄩2
𝜅∈𝜅ℎ 𝜕𝜅 󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩
󵄩 󵄩1,Ω
= ∑ ∫ 𝐿𝑢𝑁,ℎ 𝑤 ≤ 𝐶 ∑ 𝑁𝜅2𝛼 𝜂𝛼;𝜅
∗2
𝜅∈𝜅ℎ 𝜅 𝜅∈𝐾ℎ
1 𝜕𝑢𝑁,ℎ ℎ𝜅2 󵄩󵄩 ∗ ∗
− ∑ ∑ ∫ 𝐷[ ] 𝑤. ∗ 󵄩 󵄩󵄩2
2 𝜅∈𝜅ℎ 𝑒⊂𝜕𝜅∩Ω 𝑒 𝜕𝑛 +𝐶 ∑ { 󵄩󵄩𝐿 𝜅 𝑢𝑁,ℎ − 𝐿∗ 𝑢𝑁,ℎ 󵄩󵄩
𝑁𝜅 2 󵄩
𝜅∈𝐾ℎ
Therefore, 󵄩 ∗ 󵄩2
ℎ𝑒 󵄩󵄩 󵄩2 󵄩󵄩 𝜕𝑢𝑁,ℎ 󵄩󵄩󵄩󵄩
+ ∑ 󵄩󵄩𝐷 − 𝐷𝜅 󵄩󵄩󵄩0,𝑒 󵄩󵄩󵄩󵄩 󵄩 }
󵄩 󵄩2
𝐶󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,Ω 𝑁
𝑒⊂𝜕𝜅∩Ω 𝑒
󵄩󵄩 𝜕𝑛 󵄩󵄩󵄩0,∞,𝑒
󵄩 ∗ 󵄩 󵄩󵄩2 .
≤ ∑ ∫ (−𝐿𝑢𝑁,ℎ + 𝜆 𝑁,ℎ 𝑢𝑁,ℎ ) 𝑤 + 𝐶󵄩󵄩󵄩󵄩𝜆∗ 𝑢∗ − 𝜆∗𝑁,ℎ 𝑢𝑁,ℎ 󵄩󵄩0,Ω
𝜅∈𝐾ℎ 𝜅 (79)
Lemma 14. Let 𝛼 ∈ [0, 1], 𝜀 > 0. Then there exists a constant Since 𝜂𝛼;𝐵𝜅 = ℎ𝜅 /𝑁𝜅 ‖V𝜅 Φ−𝛼/2
𝜅 ‖0,𝜅 , we obtain
𝐶(𝜀) > 0 independent of ℎ, N, and 𝜅, such that
󵄩 󵄩
2
𝜂𝛼;𝐵
󵄩 󵄩2
≤ 𝐶 (𝜀) {𝑁𝜅2(1−𝛼) 󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,𝜅 𝜂𝛼;𝐵𝜅 ≤ 𝐶 (𝑁𝜅1−𝛼 󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,𝜅
𝜅
ℎ𝜅 󵄩󵄩
ℎ𝜅2 + 󵄩𝐿𝑢 − 𝐿 𝜅 𝑢𝑁,ℎ (83)
+ 𝑁𝜅max{1+2𝜀−2𝛼,0} 𝑁𝜅 󵄩 𝑁,ℎ
𝑁𝜅2
󵄩
󵄩 󵄩2 + 𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢󵄩󵄩󵄩0,𝜅 ) .
× (󵄩󵄩󵄩𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢 + 𝐿𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝑘 )} .
(80) To obtain an upper bound in the case of 0 ≤ 𝛼 ≤ 1/2, we use
Proof. We denote V𝜅 := (−𝐿 𝜅 𝑢𝑁,ℎ + ∈ 𝜆 𝑁,ℎ 𝑢𝑁,ℎ )Φ𝛼𝜅 𝐻01 (𝜅) the polynomial inverse estimate (62) in Lemma 10; for 𝛽 >
with 𝛼 ∈ (0, 1] and extend V𝜅 to Ω by V𝜅 = 0 on Ω \ 𝜅; then 1/2, we derive from (62) that
󵄩󵄩 󵄩󵄩2 𝑁𝜅 󵄩󵄩 󵄩󵄩
󵄩󵄩V𝜅 Φ−𝛼/2 󵄩󵄩 = ∫ (−𝐿 𝜅 𝑢𝑁,ℎ + 𝜆 𝑁,ℎ 𝑢𝑁,ℎ ) V𝜅 𝜂 = Φ𝛼/2
󵄩 𝜅 󵄩0,𝜅 𝜅 ℎ𝜅 𝛼;𝐵𝜅 𝜅 󵄩 󵄩(𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ )󵄩󵄩0,𝜅
󵄩 󵄩󵄩
= − ∫ (𝐿 𝜅 𝑢𝑁,ℎ ) V𝜅 + 𝑎 (𝑢, V𝜅 ) ≤ 𝐶𝑁𝜅𝛽−𝛼 󵄩󵄩󵄩󵄩(𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ ) Φ𝛽/2
𝜅 󵄩
󵄩
󵄩0,𝜅
𝜅
𝑁𝜅
+ ∫ (𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢) V𝜅 = 𝐶𝑁𝜅𝛽−𝛼 𝜂
𝜅 ℎ𝜅 𝛽;𝐵𝜅
= 𝑎 (𝑢 − 𝑢𝑁,ℎ , V𝜅 ) 𝑁𝜅 󵄩󵄩 󵄩
≤ 𝐶𝑁𝜅𝛽−𝛼 (𝑁𝜅1−𝛽 󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,𝜅
ℎ𝜅 󵄩
+ ∫ (𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢 + 𝐿𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ ) V𝜅
󵄩 󵄩
𝜅
+󵄩󵄩󵄩𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢+𝐿𝑢𝑁,ℎ −𝐿 𝜅 𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝜅 ) .
󵄩 󵄩 󵄨 󵄨
≤ 𝐶󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,𝜅 󵄨󵄨󵄨V𝜅 󵄨󵄨󵄨1,𝜅
(84)
󵄩
+ 󵄩󵄩󵄩󵄩 (𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢 Setting 𝛽 = 1/2 + 𝜀, 𝜀 > 0,
󵄩󵄩
+ 𝐿𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ ) Φ𝛼/2
𝜅 󵄩
󵄩 󵄩 󵄩
󵄩0,𝜅 𝜂𝛼;𝐵𝜅 ≤ 𝐶 (𝜀) {𝑁𝜅1−𝛼 󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,𝜅
󵄩󵄩 󵄩
󵄩󵄩 .
× 󵄩󵄩󵄩V𝜅 Φ−𝛼/2
𝜅 󵄩󵄩0,𝜅 ℎ𝜅
+ 𝑁𝜅1/2+𝜀−𝛼
(81) 𝑁𝜅
We consider the 𝐻1 semi norm for V𝜅 . Using the polynomial 󵄩 󵄩
× 󵄩󵄩󵄩𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢 + 𝐿𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝜅 } .
inverse estimates (62)-(63) in Lemma 10, by transformation
(85)
between the reference element 𝜅̂ and 𝜅, we find for 𝛼 > 1/2
that
We obtain the desired result immediately from (83) and (85).
󵄨󵄨 󵄨󵄨2 2𝛼 󵄨 󵄨2
󵄨󵄨V𝜅 󵄨󵄨1,𝜅 ≤ 2 ∫ Φ𝜅 󵄨󵄨󵄨∇ (𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ )󵄨󵄨󵄨
𝜅
󵄨 󵄨2 󵄨 󵄨2 In order to obtain a local upper bound for the error

+ 2 ∫ 󵄨󵄨󵄨∇Φ𝛼𝜅 󵄨󵄨󵄨 󵄨󵄨󵄨𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ 󵄨󵄨󵄨 indicator 𝜂𝛼;𝜅 , we consider the edge residual term 𝜂𝛼;𝐸𝜅 . we
𝜅
introduce the set
𝑁𝜅2(2−𝛼) 󵄨 󵄨2
≤𝐶 ∫ Φ𝛼𝜅 󵄨󵄨󵄨𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ 󵄨󵄨󵄨 󵄨
ℎ𝜅2 𝜅 𝜔𝜅 := ∪ {𝜅󸀠 󵄨󵄨󵄨󵄨𝜅󸀠 and 𝜅 share at least one edge} . (86)
1 (82)
󵄨 󵄨2 Lemma 15. Let 𝛼 ∈ [0, 1], 𝜀 > 0. Then there exists a constant
+𝐶 ∫ Φ2(𝛼−1) 󵄨󵄨󵄨𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ 󵄨󵄨󵄨
ℎ𝜅2 𝜅 𝜅 𝐶(𝜀) > 0 independent of ℎ, N, and 𝜅, such that
𝑁𝜅2(2−𝛼) 󵄨 󵄨2 2
≤𝐶 ∫ Φ𝛼𝜅 󵄨󵄨󵄨𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ 󵄨󵄨󵄨 𝜂𝛼;𝐸 ≤ 𝐶 (𝜀) 𝑁𝜅max(1−2𝛼+2𝜀,0)
ℎ𝜅2 𝜅
𝜅
󵄩 󵄩2 ℎ2
𝑁2 × {𝑁𝜅 󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,𝜔 + 𝑁𝜅2𝜀 𝜅2
= 𝐶𝑁𝜅2(1−𝛼) 2𝜅 ‖ V𝜅 Φ−𝛼/2
𝜅 ‖20,𝜅 . 𝜅 𝑁𝜅
ℎ𝜅
Note that (62) is applicable since 𝛼 > 1/2 implies 2(𝛼 − 1) > 󵄩 󵄩2
× 󵄩󵄩󵄩𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢 + 𝐿𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝜔 } .
−1; thus, we set 𝛽 = 𝛼, 𝛼 = 2(𝛼 − 1) in (62); then the third 𝜅
inequality above holds. (87)

Proof. We will use weight functions on edge and a suitable It follows from (89)-(90) that
extension operator. For a given element 𝜅 with edge 𝑒, we 󵄩󵄩 󵄩󵄩
󵄩󵄩 𝜕𝑢𝑁,ℎ 󵄩󵄩
choose the element 𝜅1 so that 𝜕𝜅1 ∩ 𝜕𝜅 = 𝑒. Denote 𝜅𝑒 := 󵄩󵄩𝐷𝜅 [ ] Φ𝛼/2
𝑒 󵄩
𝜅1 ∪ 𝜅; we construct a function 𝑤𝑒 ∈ 𝐻01 (𝜅𝑒 ) with 𝑤𝑒 |𝑒 = 󵄩󵄩 𝜕𝑛 󵄩󵄩󵄩0,𝑒
𝐷𝜅 [𝜕𝑢𝑁,ℎ /𝜕𝑛]Φ𝛼𝑒 as follows. 1/2
Let V𝑒̂ = 𝐶𝑒 𝐷𝜅 [𝜕𝑢𝑁,ℎ /𝜕𝑛]Φ𝛼𝑒̂ (𝐶𝑒 is defined by (71)). 1
≤ 𝐶 {( (𝜀𝑁𝜅2(2−𝛼) + 𝜀−1 ))
Using Lemma 11, we extend V𝑒̂ to 𝜅̂, where the polynomial 𝜋 ℎ𝜅
corresponds to 𝐶𝑒 𝐷𝜅 [𝜕𝑢𝑁,ℎ /𝜕𝑛]. Define 𝑤𝑒 |𝜅 and 𝑤𝑒 |𝜅1 as the
󵄩 󵄩 1/2 󵄩 󵄩
affine transformation of V𝑒̂ in 𝜅̂; Thus, 𝑤𝑒 is a piecewise 𝐻1 - × 󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,𝜅 + (ℎ𝜅 𝜀) ‖ 󵄩󵄩󵄩𝐿𝑢𝑁,ℎ − 𝜆𝑢󵄩󵄩󵄩0,𝜅 } .
𝑒 𝑒
function. From (64), we know 𝑤𝑒 vanishes on 𝜕𝜅𝑒 ; Therefore,
𝑤𝑒 ∈ 𝐻01 (𝜅𝑒 ). It is trivial to extend 𝑤𝑒 to Ω, such that 𝑤𝑒 = 0 (91)
in Ω \ 𝜅𝑒 . We find 2
By the definition of 𝜂𝛼;𝐸 and setting 𝛼 = 0 in Lemma 14, we
𝜅
󵄩󵄩 󵄩󵄩2 get
󵄩󵄩 𝜕𝑢𝑁,ℎ 󵄩󵄩
󵄩󵄩𝐷𝜅 [ ] Φ𝛼/2
𝑒 󵄩 󵄩󵄩 󵄩󵄩 󵄩2
󵄩󵄩 𝜕𝑛 󵄩0,𝑒 󵄩󵄩𝐿 𝜅 𝑢𝑁,ℎ − 𝜆 𝑁,ℎ 𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝑘
𝜕𝑢𝑁,ℎ 𝜕𝑢𝑁,ℎ 𝑁𝜅4 󵄩󵄩 󵄩2
= ∫ 𝐷[ ] 𝑤𝑒 + ∫ (𝐷𝜅 − 𝐷) [ ] 𝑤𝑒 󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,𝑘 + 𝑁𝜅
1+2𝜀
≤ 𝐶 (𝜀) {
𝑒 𝜕𝑛 𝑒 𝜕𝑛 ℎ𝜅2 󵄩 (92)
= ∫ 𝐿𝑢𝑁,ℎ 𝑤𝑒 − 𝑎 (𝑢𝑁,ℎ , 𝑤𝑒 ) 󵄩 󵄩2
𝜅𝑒 ×󵄩󵄩󵄩𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢 + 𝐿𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝑘 } ,
𝜕𝑢𝑁,ℎ
+ ∫ (𝐷𝜅 − 𝐷) [ ] 𝑤𝑒 by the triangle inequality
𝑒 𝜕𝑛
󵄩󵄩󵄩𝐿𝑢𝑁,ℎ − 𝜆𝑢󵄩󵄩󵄩
(88) 󵄩 󵄩0,𝜅𝑒
= ∫ (𝐿𝑢𝑁,ℎ − 𝜆𝑢) 𝑤𝑒 + 𝑎 (𝑢 − 𝑢𝑁,ℎ , 𝑤𝑒 ) 󵄩󵄩 󵄩
𝜅𝑒 ≤ 󵄩󵄩𝐿 𝜅 𝑢𝑁,ℎ − 𝜆 𝑁,ℎ 𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝜅 (93)
𝑒
𝜕𝑢𝑁,ℎ 󵄩 󵄩
+ ∫ (𝐷𝜅 − 𝐷) [ ] 𝑤𝑒 + 󵄩󵄩𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢 + 𝐿𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝜅 .
󵄩
𝑒 𝜕𝑛 𝑒
󵄩 󵄩 󵄩 󵄩 Combining the three inequalities above and summing, we

≤ 󵄩󵄩󵄩𝐿𝑢𝑁,ℎ − 𝜆𝑢󵄩󵄩󵄩0,𝜅 󵄩󵄩󵄩𝑤𝑒 󵄩󵄩󵄩0,𝜅𝑒 have
𝑒
󵄩󵄩 󵄩󵄩 󵄨󵄨 󵄨󵄨 1
+ 𝐶󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩1,𝜅 󵄨󵄨𝑤𝑒 󵄨󵄨1,𝜅𝑒 2
𝜂𝛼;𝐸 ≤ 𝐶{ (𝜀𝑁𝜅2(2−𝛼) + 𝜀−1 ) + 𝑁𝜅3 𝜀}
𝑒 𝜅
𝑁𝜅
󵄩󵄩 (𝐷 − 𝐷) 󵄩󵄩 󵄩󵄩 󵄩2
󵄩 󵄩󵄩 󵄩󵄩 𝜕𝑢𝑁,ℎ 𝛼/2 󵄩
󵄩󵄩 ℎ2
+ 󵄩󵄩󵄩󵄩 𝜅 󵄩󵄩 󵄩 𝐷 [ ] Φ 𝑒 󵄩 󵄩󵄩 . 󵄩 󵄩2
× 󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,𝜔 + 𝐶𝜀𝑁𝜅2(1+𝜀) 𝜅2
󵄩󵄩 𝐷𝜅 󵄩󵄩󵄩0,∞,𝑒 󵄩󵄩󵄩 𝜅 𝜕𝑛
(94)
󵄩0,𝑒 𝜅 𝑁𝜅
󵄩 󵄩2
Therefore, × 󵄩󵄩󵄩𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢 + 𝐿𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝜔 .
𝜅
󵄩󵄩 𝜕𝑢𝑁,ℎ 󵄩󵄩2
󵄩󵄩 󵄩󵄩 Setting 𝜀 = 1/𝑁𝜅2
󵄩󵄩𝐷𝜅 [ ] Φ𝛼/2
𝑒 󵄩󵄩 in the above inequality yields the
󵄩󵄩 𝜕𝑛 󵄩󵄩0,𝑒 assertion for 𝛼 > 1/2. For the case of 𝛼 ∈ [0, 1/2], we set
󵄩 󵄩 󵄩 󵄩 (89) 𝛽 = 1/2 + 𝜀, use (62) in Lemma 10 to get 𝜂𝛼;𝐸𝜅 ≤ 𝑁𝜅𝛽−𝛼 𝜂𝛽;𝐸𝜅 ,
≤ 𝐶󵄩󵄩󵄩𝐿𝑢𝑁,ℎ − 𝜆𝑢󵄩󵄩󵄩0,𝜅 󵄩󵄩󵄩𝑤𝑒 󵄩󵄩󵄩0,𝜅𝑒 and find the desired result.
𝑒
󵄩
󵄩 󵄩 󵄨󵄨 󵄨󵄨
󵄩
+ 𝐶󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩1,𝜅 󵄨󵄨𝑤𝑒 󵄨󵄨1,𝜅𝑒 . Combining Lemmas 14 and 15, we obtain the following
𝑒
theorem.
We consider the case of 𝛼 ∈ (1/2, 1] first. Using the affine
equivalence and (65)-(66) in Lemma 11, we obtain the upper Theorem 16. Let 𝛼 ∈ [0, 1], 𝜀 > 0. Then there exists a constant
bounds for ‖𝑤𝑒 ‖0,𝜅𝑒 and |𝑤𝑒 |1,𝜅𝑒 as follows: 𝐶 > 0 independent of ℎ, N, and 𝜅, such that
2
󵄨󵄨 󵄨󵄨2 1 2(2−𝛼)
𝜂𝛼;𝜅 ≤ 𝐶 (𝜀) 𝑁𝜅max(1−2𝛼+2𝜀,0)
󵄨󵄨𝑤𝑒 󵄨󵄨1,𝜅𝑒 ≤ 𝐶 (𝜀𝑁𝜅 + 𝜀−1 )
ℎ𝜅
󵄩 󵄩2 ℎ2
󵄩󵄩 󵄩2 × {𝑁𝜅 󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,𝜔 + 𝑁𝜅2𝜀 𝜅2
󵄩 𝜕𝑢𝑁,ℎ 󵄩󵄩 𝜅 𝑁𝜅 (95)
× 󵄩󵄩󵄩𝐷𝜅 [ ] Φ𝛼/2
𝑒
󵄩󵄩 ,
󵄩󵄩 (90)
󵄩󵄩 𝜕𝑛 󵄩0,𝑒
󵄩 󵄩2
󵄩󵄩 󵄩󵄩2 ×󵄩󵄩󵄩𝜆 𝑁,ℎ 𝑢𝑁,ℎ − 𝜆𝑢 + 𝐿𝑢𝑁,ℎ − 𝐿 𝜅 𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝜔 } .
󵄩󵄩 󵄩󵄩 󵄩 𝜕𝑢𝑁,ℎ 󵄩󵄩 𝜅
󵄩󵄩𝑤𝑒 󵄩󵄩0,𝜅𝑒 ≤ 𝐶ℎ𝜅 𝜀󵄩󵄩󵄩𝐷𝜅 [ ] Φ𝛼/2

𝑒 󵄩 󵄩󵄩 .
󵄩󵄩 𝜕𝑛 󵄩0,𝑒 Similarly, we have Theorem 17.
Theorem 17. Let 𝛼 ∈ [0, 1], 𝜀 > 0. Then there exists a constant Note that the formula (51) gives the optimal orders of
𝐶 > 0 independent of ℎ, N, and 𝜅, such that convergence; thus, we deduce that the second and third terms
on the right side of (74) are higher order infinitesimals. We
∗2
𝜂𝛼;𝜅 ≤ 𝐶 (𝜀) 𝑁𝜅max(1−2𝛼+2𝜀,0) derive from (52) and (53), and 𝑁 = 𝑁𝜅 , that
2 󵄩󵄩󵄩𝜆𝑢 − 𝜆 𝑁,ℎ 𝑢𝑁,ℎ 󵄩󵄩󵄩
󵄩 󵄩2 2𝜀 ℎ𝜅 󵄩 󵄩0,Ω
× {𝑁𝜅 󵄩󵄩󵄩󵄩𝑢∗ − 𝑢𝑁,ℎ
∗ 󵄩
󵄩󵄩
󵄩1,𝜔𝜅 + 𝑁𝜅
𝑁𝜅2 󵄨󵄨 󵄨󵄨
≤ 󵄨󵄨𝜆 − 𝜆 𝑁,ℎ 󵄨󵄨 ‖𝑢‖0,Ω
󵄩 󵄩2 󵄨 󵄨󵄩 󵄩
×󵄩󵄩󵄩󵄩𝜆∗𝑁,ℎ 𝑢𝑁,ℎ
∗
− 𝜆∗ 𝑢∗ + 𝐿∗ 𝑢𝑁,ℎ
∗ ∗ 󵄩
− 𝐿∗𝜅 𝑢𝑁,ℎ 󵄩󵄩 } . + 󵄨󵄨󵄨𝜆 𝑁,ℎ 󵄨󵄨󵄨 󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩0,Ω (104)
󵄩0,𝜔 𝜅
(96) 𝐶ℎ𝜏1 +𝜏2 −2 𝐶ℎ𝑟2 +𝜏1 −1 𝐶ℎ𝑟2 +𝜏1 −1

≤ + 𝑟 +𝑡 −1 ≤ 𝑟 +𝑡 −1 .
𝑁𝑡1 +𝑡2 −2 𝑁2 1 𝑁2 1
In order to estimate bounds of |𝜆 − 𝜆 𝑁,ℎ |, we also need
Lemma 18 (see [8, 10]). Therefore, the fourth term on the right side of (74) is also a
higher order infinitesimal. Up to higher order terms, we get
Lemma 18. Let (𝜆, 𝑢) be an eigenpair of (24), and let (𝜆∗ = (98). We ignore higher order infinitesimals in (95) and get
∗
𝜆, 𝑢∗ ) be the associated eigenpair of the adjoint problem (31). (99). From Lemma 4 in [10], we know that (𝑢𝑁,ℎ , 𝑢𝑁,ℎ ) = 1
∗
Then for all 𝑤, 𝑤∗ ∈ 𝐻01 (Ω), (𝑤, 𝑤∗ ) ≠0, and 𝑢𝑁,ℎ is uniformly bounded with ℎ and 𝑁. By the same
argument of (98), we can deduce that
𝑎 (𝑤, 𝑤∗ ) 󵄩󵄩 ∗
−𝜆 ∗ 󵄩 󵄩󵄩2 ≤ 𝐶 ∑ 𝑁2𝛼 𝜂∗2 .
󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩1,Ω
(𝑤, 𝑤∗ ) 󵄩 𝜅 𝛼;𝜅 (105)
(97) 𝜅∈𝐾ℎ
𝑎 (𝑤 − 𝑢, 𝑤∗ − 𝑢∗ ) (𝑤 − 𝑢, 𝑤∗ − 𝑢∗ )
= − 𝜆 . From (97), we have
(𝑤, 𝑤∗ ) (𝑤, 𝑤∗ )
∗
𝑎 (𝑢𝑁,ℎ , 𝑢𝑁,ℎ )
Theorem 19. Under the assumptions of Theorem 7, we assume −𝜆
∗ )
(𝑢𝑁,ℎ , 𝑢𝑁,ℎ
that 𝐷, b, and 𝑐 are smooth enough, and let 𝛼 ∈ [0, 1]. Then
there exists an eigenpair (𝜆, 𝑢) of (24), such that
∗
𝑎 (𝑢𝑁,ℎ − 𝑢, 𝑢𝑁,ℎ − 𝑢∗ )
1/2 = (106)
󵄩󵄩 󵄩 ∗ )
󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,Ω ≤ 𝐶( ∑ 𝑁𝜅 𝜂𝛼;𝜅 )
2𝛼 2
, (98)
𝜅∈𝐾ℎ ∗
(𝑢𝑁,ℎ − 𝑢, 𝑢𝑁,ℎ − 𝑢∗ )
󵄩 󵄩2 −𝜆 ;
2
𝜂𝛼;𝜅 ≤ 𝐶 (𝜀) 𝑁𝜅max(2−2𝛼+2𝜀,1) 󵄩󵄩󵄩𝑢 − 𝑢𝑁,ℎ 󵄩󵄩󵄩1,𝜔 . (99) ∗ )
𝜅
Further let the ascent of 𝜆 𝑁,ℎ be 𝑙 = 1, and let (𝜆∗𝑁,ℎ , 𝑢𝑁,ℎ

∗
) be that is,
the corresponding adjoint eigenpair of (32), then there exists an ∗
adjoint eigenpair (𝜆∗ , 𝑢∗ ) of (31), such that 𝜆 𝑁,ℎ − 𝜆 = 𝑎 (𝑢𝑁,ℎ − 𝑢, 𝑢𝑁,ℎ − 𝑢∗ )
(107)
∗
1/2 1/2 − 𝜆 (𝑢𝑁,ℎ − 𝑢, 𝑢𝑁,ℎ − 𝑢∗ ) .
󵄨󵄨 󵄨
󵄨󵄨𝜆 𝑁,ℎ − 𝜆󵄨󵄨󵄨 ≤ 𝐶( ∑ 𝑁𝜅 𝜂𝛼;𝜅 )
2𝛼 2
( ∑ 𝑁𝜅2𝛼 𝜂𝛼;𝜅
∗2
) . (100)
𝜅∈𝐾ℎ 𝜅∈𝐾ℎ Substituting (98) and (105) into the above equality, we obtain
(100).
Particularly, if the eigenvalue problem (23) is symmetric (i.e., If the eigenvalue problem (23) is symmetric (i.e., b = 0),
b = 0), then then
󵄨 󵄨 𝜆 𝑁,ℎ − 𝜆 = 𝑎 (𝑢𝑁,ℎ − 𝑢, 𝑢𝑁,ℎ − 𝑢)
𝐶𝐶(𝜀)−1 ∑ 𝑁𝜅min(2𝛼−2−2𝜀,−1) 𝜂𝛼;𝜅
2
≤ 󵄨󵄨󵄨𝜆 𝑁,ℎ − 𝜆󵄨󵄨󵄨 . (101)
𝜅∈𝐾ℎ
(108)
− 𝜆 (𝑢𝑁,ℎ − 𝑢, 𝑢𝑁,ℎ − 𝑢) .
Proof. We know from the assumption 𝐷, 𝑐 ∈ 𝐻𝑡1 (𝜅), b ∈ Up to higher order term 𝜆(𝑢𝑁,ℎ − 𝑢, 𝑢𝑁,ℎ − 𝑢), by (99) we get
(𝐻𝑡1 (𝜅))2 . By the interpolation error estimates (14) and (15), (101).
we have
󵄩󵄩 󵄩 Remark 20. Babuska ̌ and Osborn [17] have discussed hp finite
󵄩󵄩𝐿 𝜅 𝑢𝑁,ℎ − 𝐿𝑢𝑁,ℎ 󵄩󵄩󵄩0,𝜅 ≤ 𝐶ℎ𝜅
min(𝑁𝜅 +1,𝑡1 )−1 −𝑡1 +1
𝑁𝜅 . (102) element approximation with simplex partition for eigenvalue
problems. Obviously, the Interpolation estimates (14) and (15)
From 𝐷 ∈ 𝐻𝑡1 (𝜅), we know that 𝐷 ∈ 𝐻𝑡1 −1/2 (𝑒). By the hold for hp finite element with simplex partition (see [24]).
interpolation error estimate on edge of element (see formula Therefore, our theoretical results of spectral methods and
(5.4.42) in [2]), we get spectral methods for eigenvalue problems, which have been
󵄩󵄩 󵄩 discussed in Sections 3 and 4, hold for hp finite element with
󵄩󵄩𝐷 − 𝐷𝜅 󵄩󵄩󵄩0,𝑒 ≤ 𝐶ℎ𝑒
min(𝑁𝜅 +1,𝑡1 −1/2) −𝑡1 +1/2
𝑁𝜅 . (103) simplex partition.
Table 1: Errors of LGL-SM, modal, and Eq-SM for 1st eigenvalue.

14
LGL-SM Modal-SM Eq-SM
Condition number of first eigenvalue

𝑁 DOF
𝜆1 𝜆1 𝜆1
12
4 9 5.19𝐸 + 00 5.19𝐸 + 00 5.19𝐸 + 00
5 16 4.51𝐸 − 01 4.51𝐸 − 01 4.51𝐸 − 01
10
6 25 7.68𝐸 − 03 7.68𝐸 − 03 7.68𝐸 − 03
7 36 1.07𝐸 − 05 1.07𝐸 − 05 1.07𝐸 − 05
8
8 49 1.21𝐸 − 05 1.21𝐸 − 05 1.21𝐸 − 05
9 64 9.16𝐸 − 07 9.16𝐸 − 07 9.16𝐸 − 07
6
10 81 2.46𝐸 − 08 2.46𝐸 − 08 2.48𝐸 − 08
11 100 2.91𝐸 − 10 2.91𝐸 − 10 4.35𝐸 − 09
4
12 121 9.31𝐸 − 13 1.06𝐸 − 12 2.79𝐸 − 08
13 144 5.68𝐸 − 14 1.28𝐸 − 13 1.41𝐸 − 07
14 169 2.84𝐸 − 14 1.28𝐸 − 13 2.28𝐸 − 06 5 10 15 20 25 30
Degree of polynomial space
15 196 7.82𝐸 − 14 2.13𝐸 − 14 3.60𝐸 − 05
Eq-SM
LGL-SM
5. Numerical Experiments Modal-SM
In this section, we simply denote spectral methods, spectral Figure 1: Condition number of first eigenvalue for SM.
element methods, and finite element methods with SM,
SEM, and FEM, respectively. And spectral methods with
equidistant nodal basis, modal basis, and LGL nodal basis are
replaced by Eq-SM, Modal-SM, and LGL-SM, respectively. decreasing the mesh fineness h can decrease the errors of the
Note that all these methods employ the tensorial basis. first eigenvalue. But it is expensive to increase polynomial
∗
In our experiment, we compute 1/|(𝑢𝑁,ℎ , 𝑢𝑁,ℎ )| as condi- degree and decrease mesh fineness h at the same time.
tion number for simple eigenvalue (see Remark 2.1 in [25]), For ℎ = 1/4 and ℎ = 1/16, we obtain from Table 2
∗
where 𝑢𝑁,ℎ and 𝑢𝑁,ℎ are eigenfunctions of eigenvalue problem
the first eigenvalue errors 2.8𝐸 − 14 and 1.3𝐸 − 13 and
(25) and its adjoint problem (32) normalized with ‖ ⋅ ‖0,Ω ,
the corresponding degree of freedom 1225 and 6241 for
respectively.
hp-SEM, respectively, Whereas from Table 1, to reach this
accuracy, LGL-SM and Modal-SM should merely perform the
5.1. Example 1. Consider the nonsymmetric eigenvalue prob- interpolation approximations with polynomial degree bi-14
lem and bi-13 or so, and the corresponding degrees of freedom
are merely 169 and 144, respectively. Therefore, we conclude
−Δ𝑢 + 10𝑢𝑥 + 𝑢𝑦 = 𝜆𝑢, in Ω = (0, 1)2 , that LGL-SM and Modal-SM are highly accurate and efficient
(109) for this kind of nonsymmetric eigenvalue problems.
𝑢 = 0, on 𝜕Ω. In Figure 2 from [9], when the degree of freedom is up to
1000, the error of linear FEM is about 1E-2; the function value
The first eigenvalue of (109) 𝜆 1 = 101/4 + 2𝜋2 is a recovery techniques in [9] obviously improves the accuracy
simple eigenvalue. And the corresponding eigenfunctions are up to 1E-5. Comparing Tables 1 and 2 in this paper with Figure
sufficiently smooth. 2 in [9], we can also find the advantages of LGL-SM, Modal-
SM, and hp-SEM over the function value recovery techniques
5.1.1. Comparisons between LGL-SM, Modal, and Eq-SM. for FEM given by [9] from accuracy and degree of freedom.
Figure 1 shows that the condition numbers of the first eigen-
value for LGL-SM, Modal-SM, and Eq-SM coincide with each 5.1.3. hp-SEM versus hp-FEM. From Table 4, we find that
other at the beginning but perform abnormally with 𝑁 > 19 the condition number of the first eigenvalue for hp-version
for Eq-SM. Table 1 tells us that when 𝑁 > 11, the accuracy of methods (hp-SEM and hp-FEM) stays at 4.27. It is indicated
first eigenvalue obtained by Eq-SM is not as good as obtained from Tables 2 and 3 that, when 𝑁 is greater than 7, compared
that by LGL-SM and Modal-SM. When 𝑁 = 15, the error of with hp-SEM, the errors of hp-FEM tend to become large,
the first eigenvalues obtained by Eq-SM is greater than 1E-5; whereas the errors of hp-SEM still keep stable or even stay
however, the order of the magnitude of errors for LGL-SM a decreasing tendency; however, this phenomenon is not
and Modal-SM still keeps below 1E-13. The best result of first apparent for ℎ = 1/2.
eigenvalue error for Eq-SM is merely 1E-9 or so.
Remark 21. Condition numbers of 1st eigenvalue for hp-FEM
5.1.2. LGL-SM and Modal-SM versus hp-SEM. Tables 1 and (not listed in Table 4) are almost the same to those for hp-
2 indicate that increasing the polynomial degree 𝑁 or SEM.
Table 2: Errors and DOF of hp-SEM for the first eigenvalue.
ℎ = 1/2 ℎ = 1/4 ℎ = 1/8 ℎ = 1/16

𝑁
Error DOF Error DOF Error DOF Error DOF
2 5.18𝐸 + 00 9 2.54𝐸 − 01 49 1.50𝐸 − 02 225 9.00𝐸 − 04 961
3 7.00𝐸 − 03 25 6.10𝐸 − 04 121 1.20𝐸 − 05 529 1.90𝐸 − 07 2209
4 8.40𝐸 − 03 49 2.60𝐸 − 05 225 9.70𝐸 − 08 961 3.70𝐸 − 10 3969
5 1.64𝐸 − 04 81 1.60𝐸 − 07 361 1.50𝐸 − 10 1521 1.30𝐸 − 13 6241
6 4.10𝐸 − 07 121 2.30𝐸 − 11 529 9.90𝐸 − 13 2209 3.60𝐸 − 12 9025
7 3.10𝐸 − 08 169 1.70𝐸 − 12 729 3.10𝐸 − 13 3025 1.60𝐸 − 12 12321
8 1.90𝐸 − 10 225 1.90𝐸 − 13 961 2.10𝐸 − 12 3969 4.80𝐸 − 12 16129
9 5.50𝐸 − 13 289 2.80𝐸 − 14 1225 6.00𝐸 − 13 5041 1.10𝐸 − 12 20449
10 3.80𝐸 − 13 361 1.10𝐸 − 12 1521 4.40𝐸 − 12 6241 1.50𝐸 − 11 25281
Table 3: Errors of hp-FEM for the first eigenvalue. Table 5: The Approximate eigenvalues and indicator 𝜓0 of P-SEM.
𝑁 ℎ = 1/2 ℎ = 1/4 ℎ = 1/8 ℎ = 1/16 𝑁 𝜆 𝑁,ℎ 𝜓0

3 7.00𝐸 − 03 6.10𝐸 − 04 1.20𝐸 − 05 1.90𝐸 − 07 3 28.56900 2.72𝐸 + 01
4 8.40𝐸 − 03 2.60𝐸 − 05 9.70𝐸 − 08 3.70𝐸 − 10 4 31.99175 3.49𝐸 + 00
5 1.60𝐸 − 04 1.60𝐸 − 07 1.50𝐸 − 10 1.30𝐸 − 12 5 34.82082 2.25𝐸 − 01
6 4.10𝐸 − 07 2.40𝐸 − 11 3.60𝐸 − 13 8.60𝐸 − 12 6 34.65087 1.31𝐸 − 02
7 3.10𝐸 − 08 6.10𝐸 − 12 1.30𝐸 − 11 3.00𝐸 − 11 7 34.65057 3.32𝐸 − 03
8 1.80𝐸 − 10 3.10𝐸 − 11 2.30𝐸 − 10 2.10𝐸 − 10 8 34.64765 1.92𝐸 − 03
9 7.50𝐸 − 11 3.40𝐸 − 11 6.80𝐸 − 10 7.40𝐸 − 10 9 34.64567 1.22𝐸 − 03
10 2.50𝐸 − 11 9.90𝐸 − 10 8.70𝐸 − 09 6.60𝐸 − 09 10 34.64432 8.11𝐸 − 04
11 2.00𝐸 − 09 9.60𝐸 − 09 8.90𝐸 − 09 5.40𝐸 − 07 11 34.64335 5.62𝐸 − 04
12 34.64265 4.02𝐸 − 04
Table 4: Condition number of first eigenvalue for hp-SEM. 13 34.64212 2.95𝐸 − 04
14 34.64171 2.22𝐸 − 04
𝑁 ℎ = 1/2 ℎ = 1/4 ℎ = 1/8 ℎ = 1/16
15 34.64139 1.71𝐸 − 04
3 4.284381324 4.270132842 4.269625046 4.269615821
16 34.64114 1.33𝐸 − 04
4 4.267343095 4.269607452 4.269615638 4.26961567
17 34.64094 1.06𝐸 − 04
5 4.269636446 4.269615725 4.26961567 4.26961567
18 34.64078 8.49𝐸 − 05
6 4.269619135 4.26961567 4.26961567 4.26961567
7 4.269615617 4.26961567 4.26961567 4.26961567
8 4.26961567 4.26961567 4.26961567 4.26961567 is large enough, which is caused by round-off errors derived
9 4.26961567 4.26961567 4.26961567 4.26961567 from the bad condition number of eigenvalue. In Figure 3, we
give the comparison between the error indicator 𝜓0 and the
true errors for hp-SEM.
5.1.4. Validity of the Error Indicator. Denote
1/2 1/2
5.2. Example 2. Consider the nonsymmetric eigenvalue
problem
𝜓𝛼 = ( ∑ 𝑁𝜅2𝛼 𝜂𝛼;𝜅
2
) ( ∑ 𝑁𝜅2𝛼 𝜂𝛼;𝜅
∗2
) . (110)
𝜅∈𝐾ℎ 𝜅∈𝐾ℎ (−1, 1)2
−Δ𝑢 + 10𝑢𝑥 = 𝜆𝑢, in Ω = ,
From Theorem 19, we know that 𝜓𝛼 is a reliable error (0, 1)2 (111)
indicator for 𝜆 𝑁,ℎ . We choose 𝜓0 (setting 𝛼 = 0 in (110)) as 𝑢 = 0, on 𝜕Ω.
a posteriorii error indicator.
In Figures 2 and 3, we denote the true error and est. error A reference value for the first eigenvalue (simple eigen-
with |𝜆 𝑁,ℎ − 𝜆| and 𝜓0 , respectively. value) of (111) is 34.6397 given by [5]. And the corresponding
As is depicted in Figure 2, when the polynomial degree eigenfunctions have the singularity at the origin. Next, we
𝑁 ≤ 12, the error indicator 𝜓0 can properly estimate the shall compare the relevant numerical results between P-SEM
true errors of LGL-SM for the first eigenvalue, however, also and the other methods adopted in this paper. Note that
slightly underestimate the true errors. It is easy to see that 𝜓0 here and hereafter P-version methods are for the fixed mesh
shows almost the same algebraic decay as the true error with fineness ℎ = 1. Table 5 lists part data of the approximate
the polynomial degree 𝑁 (≤12) increasing. Nevertheless, the eigenvalues computed by P-SEM and the corresponding error
error indicator 𝜓0 cannot approximate the true errors if 𝑁 indicator 𝜓0 for reference.
40
0
34.6404
30
−5
20
First eigenvalue
log10 (error)
−10
10
−15 0
−20 −10
−20
−25
−30
5 10 15 20 25 2 4 6 8 10 12 14 16 18 20 22
Degree of polynomial space Degree of polynomial space
Est. error P-FEM

True error P-SEM
Figure 2: The Error indicator 𝜓0 of LGL-SM. Figure 4: The Approximate 1st eigenvalue of P-SEM and P-FEM.
2 100
0 90
Condition number of first eigenvalue
80
−2
70
−4
log10 (error)
60
−6
50
−8
40
−10 30
−12 20
−14 10
2 3 4 5 6 7 8 9 10 0
Degree of polynomial space 0 5 10 15 20 25 30
Degree of polynomial space
Est. error
True error P-FEM
P-SEM
Figure 3: The Error indicator 𝜓0 of hp-SEM (ℎ = 1/2).
Figure 5: Condition number of first eigenvalue for P-SEM and P-
FEM.
5.2.1. Stability of P-Version Methods. Figure 4 indicates that
the eigenvalues computed by P-FEM will not seriously deviate
from the results computed by P-SEM until the interpolation degree of freedom is merely 1365. Compared with the linear
polynomial degree 𝑁 is up to 19. This phenomenon coincides FEM, hp-SEM can obtain a higher accuracy with less degrees
with the abnormity of condition number of first eigenvalue of freedom as follows: for fixed ℎ = 1/16 and 𝑁 = 10, the
for P-FEM (see Figure 5). The reason is that the singularities approximate eigenvalue is 34.63984 with degree of freedom
of the eigenfunctions limit the accuracy of both kinds of 76161 but P-SEM with polynomial degree bi-44 can reach
methods; this is slightly different from the case of the eigen- this accuracy. Therefore, P-SEM is more efficient for the
value problem with the sufficiently smooth eigenfunctions. eigenvalue problems with the singular solutions than the
other methods.
5.2.2. P-SEM versus Other Methods. By calculations, we find
that, in the case of the linear FEM, for fixed mesh fineness Acknowledgments
ℎ = 1/256, the approximate eigenvalue is 34.6403 with degree
of freedom up to 195585. But P-SEM with the polynomial This work was supported by the National Natural Science
degree bi-22 can reach this accuracy, and the corresponding Foundation of China (Grant no. 11161012) and the Educational
Administration and Innovation Foundation of Graduate for the Stokes eigenvalue problem,” Science China Mathematics,
Students of Guizhou Normal University (no. 2012(11)). vol. 56, no. 6, pp. 1313–1330, 2013.
[17] I. Babuška and J. Osborn, “Eigenvalue problems,” in Handbook
of Numerical Analysis: Finite Element Methods (Part 1), P. G.
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http://dx.doi.org/10.1155/2013/108026
Research Article
Nonlinear Hydroelastic Waves beneath a Floating Ice
Sheet in a Fluid of Finite Depth
Ping Wang1,2 and Zunshui Cheng1,3

1
School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266061, China
2
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
3
Research Center for Complex Systems and Network Sciences, Department of Mathematics, Southeast University,
Nanjing 210096, China
Correspondence should be addressed to Ping Wang; pingwang2003@126.com
Received 21 May 2013; Revised 29 August 2013; Accepted 29 August 2013
Copyright © 2013 P. Wang and Z. Cheng. This is an open access article distributed under the Creative Commons Attribution
cited.
The nonlinear hydroelastic waves propagating beneath an infinite ice sheet floating on an inviscid fluid of finite depth are
investigated analytically. The approximate series solutions for the velocity potential and the wave surface elevation are derived,
respectively, by an analytic approximation technique named homotopy analysis method (HAM) and are presented for the second-
order components. Also, homotopy squared residual technique is employed to guarantee the convergence of the series solutions. The
present formulas, different from the perturbation solutions, are highly accurate and uniformly valid without assuming that these
nonlinear partial differential equations (PDEs) have small parameters necessarily. It is noted that the effects of water depth, the ice
sheet thickness, and Young’s modulus are analytically expressed in detail. We find that, in different water depths, the hydroelastic
waves traveling beneath the thickest ice sheet always contain the largest wave energy. While with an increasing thickness of the
sheet, the wave elevation tends to be smoothened at the crest and be sharpened at the trough. The larger Young’s modulus of the
sheet also causes analogous effects. The results obtained show that the thickness and Young’s modulus of the floating ice sheet all
greatly affect the wave energy and wave profile in different water depths.
1. Introduction proposed by Greenhill [1] in 1887. A comprehensive summary

on mathematical method and modeling for the problem can
In recent decades, the ice cover in the polar region has be found in some review articles such as Squire et al. [2, 3].
attracted more and more attention in the field of ocean In addition to ice sheets, this work can apply to very large
engineering and polar engineering in view of their practical floating structures (VLFSs) such as floating airports, mobile
importance and theoretical investigations. The motivations offshore bases, offshore port facilities, offshore storage and
for the research work are to study damage to offshore con- waste disposal provisions, energy islands including some
structions by floating ice sheets, the transportation systems in wave power configurations, and ultralarge ships, where there
the cold region where the ice cover can be considered as roads is an extensive complementary literature [4–6].
and aircraft runways and air-cushioned vehicles are used to Most theoretical works on the problem are still in the
break the ice, for example. One of the important problems scope of linear theory based on the assumption that the
in this field would appear to be the accurate measurement of wave amplitudes generated are very small in comparison
the characteristics of waves traveling beneath a floating ice with the wave lengths. So such models are not appropriate
sheet. And such wave may have been generated in the ice to describe waves of arbitrary amplitude considered here.
cover itself by the wind, or it may have originated by a moving According to hydrodynamics and elasticity, we can construct
load on the ice sheets. Considerable work has been done since the nonlinear partial differential equations (PDEs) (1)–(5) to
the first theoretical model of wave propagation in sea ice was describe nonlinear hydroelastic waves of arbitrary amplitude
traveling through water covered by an ice sheet in finite water The objective of the present work is to analytically study
depth. Unfortunately, it is very difficult to solve analytically the nonlinear hydroelastic waves under an ice sheet lying
the coupled nonlinear PDEs mathematically. Further, most over an incompressible inviscid fluid of finite uniform depth
of the most works literature on the nonlinear theory of sea by means of the HAM. According to the potential theory in
waves ice sheet interaction are necessarily in the context of hydrodynamics and elasticity, the nonlinear partial differen-
weakly nonlinear analysis due to the limitation of present tial equations (PDEs) (1)–(5) are composed of the Laplace
mathematical tools. Now the main analytical study on such equation taken as the governing equation for inviscid flows,
complex nonlinear PDEs still follows the well-known pertur- the kinematic and dynamic boundary conditions on the
bation technique. For example, Forbes [7] derived nonlinear unknown ice sheet-water interface with a zero draft, a simple
PDEs to describe two-dimensional periodic waves beneath an linear model for the thin sheet that includes the effects of
elastic sheet floating on the surface of an infinitely deep fluid. flexural rigidity and vertical inertia, and a bottom boundary
The periodic solutions are sought using the Fourier series and condition. The convergent homotopy-series solutions for the
perturbation expansions for the Fourier coefficients. And it velocity potential and the wave surface elevation are formally
is found that the solutions have certain features in common derived by applying the HAM with the consideration of the
with capillary-gravity waves. Following the framework in minimum of the squared residual, respectively. It should
[7], Forbes continued his study of finite-amplitude surface be mentioned that we study the effects of the water depth
waves beneath a floating elastic sheet in infinitely deep water and two important physical parameters including Young’s
[8], and optimized their previous perturbation technology modulus and the thickness of the ice sheet on the wave energy
directly by developing the Fourier coefficients as expansions and its elevation in detail. Discussion and conclusions are
in the wave height. Waves of extremely large amplitude are made in Sections 4 and 5, respectively. All of results obtained
found to exist, and results are presented for waves belonging will help enrich our understanding of nonlinear hydroelastic
to several different nonlinear solution branches. Recently, waves propagating under a floating ice sheet on a fluid of finite
Vanden-Broeck and Părău [9] further extended the results of depth.
Forbes for periodic waves to the arbitrary-amplitude waves.
It is noted that perturbation and asymptotic approximations 2. Mathematical Description
of nonlinear PDEs often break down as nonlinearity becomes
strong. So the weakly nonlinear solutions of small-amplitude The problem under consideration is a train of nonlinear
waves are derived by the perturbation approach, while fully hydroelastic waves propagating beneath a two-dimensional
nonlinear solutions of large-amplitude waves have to be infinite elastic plate floating on a fluid of finite depth ℎ
calculated numerically by means of the numerical series and a zero draft. A Cartesian coordinate 𝑜𝑥𝑧 is used in
truncation method in Vanden-Broeck’s study. which the 𝑧-axis points vertically upward, while 𝑧 = 0
Furthermore, perturbation and asymptotic techniques represents the undisturbed surface. We follow Greenhill in
depend extremely on the small/large parameters in general, [1] assuming that this problem is capable of modeling ocean
while our nonlinear PDEs have no any small/large parame- waves in the presence of sea ice when the fluid is inviscid and
ters. Thus the perturbation techniques are not applicable to incompressible and the flow is irrotational, and the ice sheet
the nonlinear problem under consideration. In this paper, is mathematically idealized as a thin elastic plate. Then the
we apply a new analytic approximation method known as governing equations for a velocity potential 𝜙(𝑥, 𝑧, 𝑡) can be
the homotopy analysis method (HAM) to effectively solve written as
the nonlinear PDEs presented here. Based on the concept
of homotopy in algebraic topology, the HAM was proposed 𝜕2 𝜙 𝜕2 𝜙
+ = 0, (−ℎ ≤ 𝑧 ≤ 𝜁 (𝑥, 𝑡)) , (1)
by Liao [10] in 1992. Unlike the perturbation method, the 𝜕𝑥2 𝜕𝑧2
HAM is entirely independent of any small/large parameters.
Moreover, it provides us with extremely large freedom to where 𝜁(𝑥, 𝑡) is the wave surface elevation. The bottom
choose base functions and initial approximations (16) and boundary condition reads
(17) of solutions and auxiliary linear operators (21)–(23) 𝜕𝜙
only under some basic rules [11, 12]. More importantly, in = 0, (𝑧 = −ℎ) . (2)
contrast to all other previous analytic techniques, the HAM 𝜕𝑧
provides us a convenient way to control and adjust the The motion of the fluid and the plate is coupled through
convergence of the approximate series solutions by means of the dynamic free-surface condition. We also assume that
introducing an auxiliary parameter 𝑐0 . The method has been any particle which is once between the elastic plate and the
systematically described by Liao [11, 12]. Recently the HAM water surface remains on it. So the kinematic and dynamic
has been successfully applied to the study of a number of boundary conditions on the unknown surface 𝑧 = 𝜁(𝑥, 𝑡) are,
classical nonlinear differential equations including nonlinear respectively, modeled as
equations arising in fluid mechanics [13–18], heat transfer
[19, 20], solitons and integrable models [21–24], and finance 𝜕𝜁 𝜕𝜙 𝜕𝜁 𝜕𝜙
+ − = 0, (3)
[25, 26]. These aforementioned studies show the validity 𝜕𝑡 𝜕𝑥 𝜕𝑥 𝜕𝑧
and generality of the HAM for some highly nonlinear PDEs
𝜕𝜙 1 󵄨󵄨 󵄨󵄨2 𝑝𝑒
with multiple solutions, singularity, and unknown boundary + 󵄨∇𝜙󵄨 + + 𝑔𝜁 = 0, (4)
conditions. 𝜕𝑡 2 󵄨 󵄨 𝜌
where 𝑝𝑒 is the water-plate interface pressure, 𝜌 is the fluid We combine partially (10) and (11) to gain the boundary
density, and 𝑔 is the gravitational acceleration, for a thin conditions on 𝑧 = 𝜁(𝑋) as follows:
homogeneous elastic plate with uniform mass density 𝜌𝑒 and
constant thickness 𝑑. 𝜕2 𝜙 𝜕𝜙 𝜕𝑓 𝜔 5
4 d 𝜁
3
2 d 𝜁
𝜔2 + 𝑔 − 𝜔 − (𝐷𝑘 + 𝑚𝑒 𝜔 )
Since we are considering long waves here, the linear 𝜕𝑋2 𝜕𝑧 𝜕𝑋 𝜌 d𝑋5 d𝑋3
Kirchhoff (Euler-Bernoulli) beam theory is applied to the (13)
𝜕𝜙 d𝜁
floating elastic plate as follows: − 𝑘2 𝑔 = 0.
𝜕𝑋 d𝑋
𝜕4 𝜁 𝜕2 𝜁
𝑝𝑒 = 𝐷 + 𝑚𝑒 ( + 𝑔) , (5) Now the corresponding unknown potential function 𝜙(𝑋, 𝑧)
𝜕𝑥4 𝜕𝑡2 and the wave surface elevation 𝜁(𝑋) are governed by (8), (9),
(11), and (13).
where 𝑚𝑒 = 𝜌𝑒 𝑑, 𝐷 = 𝐸𝑑3 /[12(1 − ]2 )] is the flexural rigidity
of the plate, 𝐸 is the effective Young’s modulus of the plate,
and ] Poisson’s ratio. We substitute (5) into (4) to derive a new 3. Analytic Approach Based on
form of the dynamic boundary condition as follows: the Homotopy Analysis Method
𝜕𝜙 1 󵄨󵄨 󵄨󵄨2 1 𝜕4 𝜁 𝜕2 𝜁 3.1. Solution Expression and Initial Approximation. Using
+ 󵄨󵄨∇𝜙󵄨󵄨 + 𝑔𝜁 + [𝐷 4 + 𝑚𝑒 ( 2 + 𝑔)] = 0. (6) the homotopy analysis method, we should first of all start
𝜕𝑡 2 𝜌 𝜕𝑥 𝜕𝑡
from a set of base functions and solution expression which
Here, we consider a train of nonlinear waves traveling are very important to approximate the unknown solutions
beneath an elastic plate with constant wave number 𝑘 and of the nonlinear boundary problem under consideration.
constant angular frequency 𝜔 of the incident wave. For a Mathematically, it seems impossible to guess the expression
general case it should be emphasized that, by means of the forms of the unknown potential function and the wave
traveling-wave method directly, the progressive waves are vertical displacement. Fortunately, considering the physical
transferred from the temporal differentiation into the spatial background of our problem, we may gain proper solution
one, which is very different from the mathematical model expressions of it. From viewpoints of the physical considera-
obtained by simply eliminating the time-dependent terms tions here, our problem is composed of a train of progressive
from the kinematic and dynamic boundary conditions on waves cause by a load moving on the ice sheet, an infinite
the unknown free surface [7–9]. Namely, we introduce an elastic plate acting as an ice sheet floating on an fluid of finite
independent variable transformation depth. As is well known, in case of the pure water waves, the
progressive wave elevation can be expressed as
𝑋 = 𝑘𝑥 − 𝜔𝑡, (7)
+∞
where the angular frequency 𝜔 and the wave number 𝑘 are 𝜁 (𝑋) = ∑ 𝛽𝑛 cos (𝑛𝑋) (14)
given. Thus, we can express the potential function 𝜙(𝑥, 𝑧, 𝑡) = 𝑛=0
𝜙(𝑋, 𝑧) and the traveling wave profile 𝜁(𝑥, 𝑡) = 𝜁(𝑋). by a set of base functions {cos(𝑛𝑋), 𝑛 ≥ 0}, where 𝛽𝑛 are
Then the governing equation and the bottom boundary unknown coefficients. In the case of plate-covered surface,
condition for the velocity potential are transformed, respec- since we assume that there is no gap between the bottom
tively, by surface of the thin elastic plate and the top surface of the fluid
layer and a zero draft, the vertical displacement of the thin
𝜕2 𝜙 𝜕2 𝜙
𝑘2 + = 0, (−ℎ ≤ 𝑧 ≤ 𝜁 (𝑋)) , (8) plate is still periodic in the 𝑋 direction. Therefore, we clearly
𝜕𝑋2 𝜕𝑧2 know that 𝜁(𝑋) can be expressed in the above form (14) too.
𝜕𝜙 Besides, according to the linear wave theory, we can find
= 0, (𝑧 = −ℎ) . (9) the solutions of the Laplace equation (8) by the separation
𝜕𝑧
of variables method. To acquire those solutions, we have to
With the transformation (7), (3), and (6) on 𝑧 = 𝜁(𝑋) are use kinematic and dynamic boundary conditions of the free
given by surface and the boundary condition in finite water depth,
and we consider the solution derived here as the solution
d𝜁 𝜕𝜙 d𝜁 𝜕𝜙
−𝜔 + 𝑘2 − = 0, (10) expression of potential function
d𝑋 𝜕𝑋 d𝑋 𝜕𝑧
+∞
cosh [𝑛𝑘 (𝑧 + ℎ)]
𝜕𝜙 1 d4 𝜁 d2 𝜁 𝜙 (𝑋, 𝑧) = ∑ 𝛼𝑛 sin (𝑛𝑋) (15)
−𝜔 + 𝑓 + 𝑔𝜁 + [𝐷𝑘4 4 + 𝑚𝑒 (𝜔2 2 + 𝑔)] = 0, 𝑛=1 cosh (𝑛𝑘ℎ)
𝜕𝑋 𝜌 d𝑋 d𝑋
(11) by a set of base functions {cosh[𝑛𝑘(𝑧+ℎ)]/cosh(𝑛𝑘ℎ) sin(𝑛𝑋),
𝑛 ≥ 0}, where 𝛼𝑛 are unknown coefficients. Note that the
respectively, where potential function 𝜙(𝑋, 𝑧) defined by (15) automatically
satisfies the governing equation (8) and the bottom boundary
1 2 𝜕𝜙 2 𝜕𝜙 2
𝑓= [𝑘 ( ) + ( ) ] . (12) condition (9). The above expressions (14) and (15) are called
2 𝜕𝑋 𝜕𝑧 the solution expressions of 𝜙(𝑋, 𝑧) and 𝜁(𝑋), respectively,
which play important roles in the method of homotopy the HAM, we can completely forget the linear terms in (13)
analysis. and choose proper auxiliary linear operator of Φ(𝑋, 𝑧; 𝑞) by
According to the solution expression (15) and the bound- means of the solution expression (15) which is obtained under
ary condition (9), we construct the initial approximation of the physical considerations as
the potential function:
𝜕2 Φ (𝑋, 𝑧; 𝑞) 𝜕Φ (𝑋, 𝑧; 𝑞)
cosh [𝑘 (𝑧 + ℎ)] L1 [Φ (𝑋, 𝑧; 𝑞)] = 𝜔2 2
+𝑔 . (21)
𝜙0 (𝑋, 𝑧) = 𝛼0,1 sin (𝑋) , (16) 𝜕𝑋 𝜕𝑧
cosh (𝑘ℎ)
In particular, if the angular frequency 𝜔 is given, we
where 𝛼0,1 is an unknown coefficient. We choose can choose such an approximation based on the linear wave
theory to simplify the subsequent resolution of the nonlinear
𝜁0 (𝑋) = 0. (17) PDEs as follows:
as the initial approximation of wave profile 𝜁(𝑋) to simplify 𝜔 ≈ √𝑔𝑘 tanh (𝑘ℎ). (22)
the subsequent solution procedure [18, 20]. It should be
emphasized that higher order terms can hold the corrections So we simplify the auxiliary linear operator in (21) as follows:
of the analytic series solutions due to the nonlinearity
inherent in (11) and (13) although the initial guess 𝜁0 (𝑋) is 𝜕2 Φ (𝑋, 𝑧; 𝑞)
zero. L1 [Φ (𝑋, 𝑧; 𝑞)] = 𝑔𝑘 tanh (𝑘ℎ)
𝜕𝑋2
(23)
3.2. Continuous Variation. The HAM is based on a kind 𝜕Φ (𝑋, 𝑧; 𝑞)
+𝑔 ,
of continuous mapping of an initial approximation to the 𝜕𝑧
exact solution through a series of deformation equations. For
simplicity, based on the nonlinear boundary condition (13) where L1 [0] = 0. Note that, due to the weakly nonlinear
and (11), we define the two following nonlinear operators N1 effects, the actual frequency 𝜔 is often slightly different
and N2 as follows from the linear dispersion relation 𝜔0 = √𝑔𝑘 tanh(𝑘ℎ). In
Section 4, 𝜔/𝜔0 = 1.01 is chosen so that the perturbation
N1 [Φ (𝑋, 𝑧; 𝑞) , 𝜂 (𝑋; 𝑞)] theory is valid and corresponding results are highly accurate,
and then we can compare our results with those obtained by
𝜕2 Φ (𝑋, 𝑧; 𝑞) 𝜕Φ (𝑋, 𝑧; 𝑞) 𝜕𝐹 the perturbation method.
= 𝜔2 2
+𝑔 −𝜔 Based on the linear operator of the wave profile function
𝜕𝑋 𝜕𝑧 𝜕𝑋
𝜂(𝑋; 𝑞) in the nonlinear operator N2 , for simplicity, we may
𝜔 𝜕5 𝜂 (𝑋; 𝑞) 𝜕3 𝜂 (𝑋; 𝑞) (18)
choose another auxiliary linear operator:
− (𝐷𝑘4 + 𝜔2
𝑚𝑒 )
𝜌 𝜕𝑋5 𝜕𝑋3
𝜕4 𝜂 (𝑋; 𝑞) 𝜕2 𝜂 (𝑋; 𝑞)
𝜕Φ (𝑋, 𝑧; 𝑞) 𝜕𝜂 (𝑋; 𝑞) L2 [𝜂 (𝑋; 𝑞)] = + + 𝜂 (𝑋; 𝑞) , (24)
− 𝑘2 𝑔 , 𝜕𝑋4 𝜕𝑋2
𝜕𝑋 𝜕𝑋
where L2 [0] = 0.
N2 [𝜂 (𝑋; 𝑞) , Φ (𝑋, 𝑧; 𝑞)] We let 𝑐0 be an nonzero convergence-control parameter.
It is noted that both 𝑐0 and 𝑞 in the HAM are auxiliary
𝜕Φ (𝑋, 𝑧; 𝑞)
= −𝜔 + 𝐹 + 𝑔𝜂 (𝑋; 𝑞) parameters without any physical meaning. Instead of the
𝜕𝑋 nonlinear PDEs (8), (9), (11), and (13), we reconstruct the so-
𝜕4 𝜂 (𝑋; 𝑞) 2 called zeroth-order deformation equations as follows:
1 2 𝜕 𝜂 (𝑋; 𝑞)
+ [𝐷𝑘4 + 𝑚𝑒 (𝜔 + 𝑔)] ,
𝜌 𝜕𝑋4 𝜕𝑋2 𝜕2 Φ (𝑋, 𝑧; 𝑞) 𝜕2 Φ (𝑋, 𝑧; 𝑞)
(19) 𝑘2 + = 0, (−ℎ ≤ 𝑧 ≤ 𝜂 (𝑋; 𝑞)) ,
𝜕𝑋2 𝜕𝑧2
(25)
where
𝜕Φ (𝑋, 𝑧; 𝑞)
1 𝜕Φ 2 𝜕Φ 2 = 0, (𝑧 = −ℎ) , (26)
𝐹 = [𝑘2 ( ) + ( ) ] (20) 𝜕𝑧
2 𝜕𝑋 𝜕𝑧
(1 − 𝑞) L1 [Φ (𝑋, 𝑧; 𝑞) − 𝜙0 (𝑋, 𝑧)]
and 𝑞 ∈ [0, 1] is the embedding parameter of the HAM.
Here, it should be emphasized that, as mentioned by Liao = 𝑞𝑐0 N1 [Φ (𝑋, 𝑧; 𝑞) , 𝜂 (𝑋; 𝑞)] , (𝑧 = 𝜂 (𝑋; 𝑞)) ,
and Cheung and Tao et al. [14, 15], the HAM provides us (27)
with extremely large freedom to choose the auxiliary linear
operators and the initial guess. Note that both linear terms (1 − 𝑞) L2 [𝜂 (𝑋; 𝑞) − 𝜁0 (𝑋)]
of Φ(𝑋, 𝑧; 𝑞) and linear terms of 𝜂(𝑋; 𝑞) are all contained in
(18). If we choose all linear terms, the subsequent iterative = 𝑞𝑐0 N2 [𝜂 (𝑋; 𝑞) , Φ (𝑋, 𝑧; 𝑞)] , (𝑧 = 𝜂 (𝑋; 𝑞)) .
procedure will become very complex. Fortunately, based on (28)
Then, from (27) and (28), two mapping functions Φ(𝑋, 𝑧; 𝑞) expansion of Φ(𝑋, 𝑧; 𝑞) at the unknown surface is given in
and 𝜂(𝑋; 𝑞) vary respectively continuously from their initial Appendices (A.1)–(A.5). Upon the substitution of appropriate
approximation 𝜙0 (𝑋, 𝑧) and 𝜁0 (𝑋) to the exact solutions series (A.5) and (30) into the boundary conditions (27) and
𝜙(𝑋, 𝑧) and 𝜁(𝑋) of the original problem. The Taylor series (28), we have two linear boundary conditions on 𝑧 = 0 as
of Φ(𝑋, 𝑧; 𝑞) and 𝜂(𝑋; 𝑞) at 𝑞 = 0 are follows:
+∞
󵄨 𝜙
Φ (𝑋, 𝑧; 𝑞) = 𝜙0 (𝑋, 𝑧) + ∑ 𝜙𝑚 (𝑋, 𝑧) 𝑞𝑚 , (29) L1 (𝜙𝑚 )󵄨󵄨󵄨𝑧=0 = 𝑐0 Δ 𝑚−1 + 𝜒𝑚 𝑆𝑚−1 − 𝑆𝑚 , (35)
𝑚=1
d4 𝜁𝑚−1 d2 𝜁𝑚−1
+∞ L2 (𝜁𝑚 ) = 𝑐0 Δ𝜁𝑚−1 + 𝜒𝑚 ( + + 𝜁𝑚−1 ) ,
𝜂 (𝑋; 𝑞) = 𝜁0 (𝑋) + ∑ 𝜁𝑚 (𝑋) 𝑞𝑚 , (30) d𝑋4 d𝑋2
𝑚=1 (36)
where
where
1 𝜕𝑚 󵄨
{𝜙𝑚 (𝑋, 𝑧) , 𝜁𝑚 (𝑋)} = {Φ (𝑋, 𝑧; 𝑞) , 𝜂 (𝑋; 𝑞)}󵄨󵄨󵄨𝑞=0 .
𝑚! 𝜕𝑞𝑚 0, 𝑚⩽1
𝜒𝑚 = { (37)
(31) 1, 𝑚 > 1.
Assume that 𝑐0 is so properly chosen that the series in
(29) and (30) converge at 𝑞 = 1; then we have the so-called The detailed derivation of the above equations and the
homotopy-series solutions as follows: expression for 𝜙𝑚 and 𝜁𝑚 are given in Appendix A. It should
be noted that (27) and (28) holds on the unknown boundary
+∞ 𝑧 = 𝜂(𝑋; 𝑞), while (35) and (36) hold on 𝑧 = 0. Furthermore,
𝜙 (𝑋, 𝑧) = Φ (𝑋, 𝑧; 1) = 𝜙0 (𝑋, 𝑧) + ∑ 𝜙𝑚 (𝑋, 𝑧) , the original nonlinear DPEs (1)–(5) are transferred into an
𝑚=1 infinite number of linear decoupled high-order deformation
(32) equations (34)–(36). Namely, given 𝜙𝑚−1 and 𝜁𝑚−1 , 𝜙𝑚 and
+∞
𝜁 (𝑋) = 𝜂 (𝑋; 1) = 𝜁0 (𝑋) + ∑ 𝜁𝑚 (𝑋) . 𝜁𝑚 can be obtained easily by means of the inverse operators
𝑚=1 of the right-hand sides of (35) and (36), respectively, and a
computer algebra system such as Mathematica. The resulting
At the 𝑛th-order of approximations, we have expressions for 𝜙𝑚 and 𝜁𝑚 are presented to the second order
+𝑛 in the coming subsection.
𝜙 (𝑋, 𝑧) ≈ 𝜙0 (𝑋, 𝑧) + ∑ 𝜙𝑚 (𝑋, 𝑧) ,
𝑚=1 3.4. First-Order and Second-Order Approximations. Substi-
(33)
+𝑛 tuting initial approximations (16) and (17) into (36), we can
𝜁 (𝑋) ≈ 𝜁0 (𝑋) + ∑ 𝜁𝑚 (𝑋) . get 𝜁1 (𝑋) using the inverse linear operator L2 in (36) as
𝑚=1 follows:
As shown later in the following section, the unknown 1 2
terms 𝜙𝑚 (𝑋, 𝑧) and 𝜁𝑚 (𝑋) are governed by the linear PDEs 𝜁1 (𝑋) = [4𝑑𝑔𝑐0 + 𝑐0 𝑎0,1 + 𝑘2 𝑐0 𝑎0,1
2
tanh2 (ℎ𝑘)]
4
(34)–(36).
− 𝜔𝑐0 𝛼0,1 cos (𝑋) (38)
3.3. High-Order Deformation Equations. High-order defor- 1
mation equations for the unknown 𝜙𝑚 (𝑋, 𝑧), 𝜁𝑚 (𝑋) can be + [𝑐 𝑎2 − 𝑘2 𝑐0 𝑎0,1
2
tanh2 (ℎ𝑘)] cos (2𝑋) .
derived directly from the zeroth-order deformation equa- 52 0 0,1
tions. Firstly, substituting the homotopy-Maclaurin series
(29) and (30) into the governing equation (25) and the But now the coefficient 𝛼0,1 in the initial approximation
boundary condition in finite water depth (26) and then of 𝜙0 (𝑋, z) in (16) is still unknown. So we introduce an
equating the like-power of the embedding parameter 𝑞, we additional equation to relate the solutions with the wave
have height:
𝜕2 𝜙𝑚 (𝑋, 𝑧) 𝜕2 𝜙𝑚 (𝑋, 𝑧) 𝜁1 (𝑚𝜋) − 𝜁1 (𝑛𝜋) = 𝐻, (39)

𝑘2 + = 0, (−ℎ ≤ 𝑧 ≤ 0) ,
𝜕X2 𝜕𝑧2
(34)
𝜕𝜙𝑚 (𝑋, 𝑧) in which 𝑚 is an even integer, 𝑛 is an odd integer, and 𝐻 is
= 0, (𝑧 = −ℎ) , the wave height to the first order based on the HAM. The
𝜕𝑧
relation (39) for the wave height and its vertical displacement
where 𝑚 ≥ 1. can determine the solution of 𝛼0,1 .
Note that, Φ(𝑋, 𝑧; 𝑞) at the unknown surface 𝑧 = 𝜂(𝑋; 𝑞) Further, in the analogous manner as for the first-order
may be expressed in terms of the Taylor expansion at 𝑧 = approximation, by using the inverse linear operator L1 in
0 instead of 𝑧 = 𝜂(𝑋; 𝑞). The detailed derivation of the (35), it is easy to get the solution of 𝜙1 (𝑋, 𝑧), especially
by means of the symbolic computation software such as

Mathematica: 8
𝐻
𝛼0,1 = − , 6
2𝜔𝑐0
cosh [𝑘 (ℎ + 𝑧)] 4
𝜙1 (𝑋, 𝑧) = 𝛼1,1 sin (𝑋)
cosh (𝑘ℎ)
(40) 2
log10 𝜀Tm
−𝐻2 + 𝐻2 𝑘2 tanh2 (ℎ𝑘)
+
16𝑔𝑘𝜔𝑐0 [2 tanh (ℎ𝑘) − tanh (2ℎ𝑘)] 0
cosh [2𝑘 (ℎ + 𝑧)]

× sin (2𝑋) . −2
cosh (2𝑘ℎ)
We find the common solution 𝜙1 (𝑋, 𝑧) has one unknown −4
coefficient 𝛼1,1 which can be determined by avoiding the
“secular” term sin(𝑋) in 𝜙2 (𝑋, 𝑧). We note that all subsequent −6
functions occur recursively. Utilizing the linear equations −0.8 −0.6 −0.4 −0.2 0
(35) and (36) to continue with the first-order approximations C0
we have 𝑇
Figure 1: Residual squares of log10 𝜀𝑚 versus 𝑐0 . Solid line: first-
𝜁2 (𝑋) = 𝛽2,0 + 𝛽2,1 cos (𝑋) + 𝛽2,2 cos (2𝑋) order approximation; dashed line: third-order approximation;
dash-dotted line: fifth-order approximation; dash-dot-dotted line:
+ 𝛽2,3 cos (3𝑋) + 𝛽2,4 cos (4𝑋) , seventh-order approximation.
cosh [𝑘 (ℎ + 𝑧)]
𝜙2 (𝑋, 𝑧) = 𝛼2,1 sin (𝑋)
cosh (𝑘ℎ)
0.06
+ 𝛼2,2 sin (2𝑋)
cosh (2𝑘ℎ) (41)
0.04

+ 𝛼2,3 sin (3𝑋) 0.02
cosh (3𝑘ℎ)
cosh [4𝑘 (ℎ + 𝑧)] 𝜁 0
+ 𝛼2,4 sin (4𝑋)
cosh (4𝑘ℎ)
−0.02
+ 𝛼2,5 sin (5𝑋) ,
cosh (5𝑘ℎ) −0.04
where 𝛼𝑖,𝑗 is the 𝑗th unknown coefficient of 𝜙𝑖 (𝑋, 𝑧) and
𝛽𝑖,𝑗 is the 𝑗th unknown coefficient of 𝜁𝑖 (𝑋). The detailed −0.06
expressions of these coefficients for 𝜙2 and 𝜁2 are given in
−2 −1 0 1 2 3 4 5 6
Appendix B.
X
In order to obtain higher-order functions 𝜙𝑚 (𝑋, 𝑧) and
𝜁𝑚 (𝑋), we need only to continue this approach. In principle, Figure 2: Comparison of our present 3rd-order surface elevation
we can acquire infinite-order solutions for our physical 𝜁 with those obtained by the perturbation method. Solid line:
model. It is also worthwhile to mention that these solutions perturbation-series solution; dashed line: homotopy-series solution.
will retain model parameters and the convergence control
parameter 𝑐0 .
where
3.5. Optimal Convergence-Control Parameter. If we fix all
1 𝑀 󵄨 2
model parameters in our approximate series solutions, there 𝜙
𝜀𝑚 = ∑( N [𝜙 (𝑋, 𝑧) , 𝜁 (𝑋)]󵄨󵄨󵄨𝑋=𝑖Δ𝑋 ) ,
is still an unknown convergence control parameter 𝑐0 in 1 + 𝑀 𝑖=0 1
them, which is used to guarantee the convergence of our (43)
approximation solutions. According to Liao [12], it is the 1 𝑀 󵄨 2
𝜁
𝜀𝑚 = ∑( N [𝜙 (𝑋, 𝑧) , 𝜁 (𝑋)]󵄨󵄨󵄨𝑋=𝑖Δ𝑋 ) ,
convergence control parameter 𝑐0 that essentially differs the 1 + 𝑀 𝑖=0 2
HAM from all other analytic methods. And the optimal value
of 𝑐0 is determined by the minimum of the total squared- 𝜙 𝜁
where 𝜀𝑚 and 𝜀𝑚 are two residual square errors of boundary
𝑇
residual 𝜀𝑚 of our nonlinear problem, defined by conditions (27) and (28), respectively. 𝑀 is the number of the
𝑇 𝜙 𝜁 discrete points, and Δ𝑋 = 𝜋/𝑀. In this paper, we choose 𝑀 =
𝜀𝑚 = 𝜀𝑚 + 𝜀𝑚 , (42) 10.
0.001 𝑇
Table 1: The total residual square error 𝜀𝑚 for different approxima-
tion order 𝑚 with 𝑐0 = −0.18.
𝑇
𝑚 𝜀𝑚
0.0008
1 3.497 × 10−3
3 3.404 × 10−4
0.0006 5 3.700 × 10−5
7 7.910 × 10−6
P.E.
10 4.803 × 10−8
0.0004 15 5.382 × 10−11
0.0002 take these data hereinafter for computation unless otherwise

𝑇
stated. The total residual square error 𝜀𝑚 at several orders
of approximation versus the convergence-control parameter
𝑇
0 𝑐0 is shown in Figure 1. It is found that 𝜀𝑚 at every order
5 10 15 20 25 30 has the smallest values which corresponds to the optimal 𝑐0 .
h For example, as 𝑚 = 7, the optimal 𝑐0 = −0.18, and the
Figure 3: P.E. for (44) versus the water depth ℎ for different plate smallest value of 𝜀7𝑇 = 7.910 × 10−6 . So, let the optimal
thicknesses 𝑑. Solid line: 𝑑 = 0.001; dashed line: 𝑑 = 0.005; dash- convergence-control parameter 𝑐0 = −0.18, the total residual
𝑇
dot-dotted line: 𝑑 = 0.01. square error 𝜀𝑚 decreases quickly as the order 𝑚 increases, as
𝑇
shown in Table 1. It is also found that 𝜀15 is down to 5.382 ×
−11
0.001 10 at the 15th-order of approximation, which indicates the
convergence of our series solutions. In this way, we ensure
that all our solutions are highly accurate.
Also, we compare our HAM solutions of waves propagat-
0.0008
ing beneath an elastic plate floating on a fluid of finite depth
with those results obtained by perturbation techniques, as
shown in Figure 2. It should be noted that the perturbation-
0.0006
series solution is derived by substituting the series expansions
(4.5) and (4.6) in [9] into the nonlinear PDEs (8)–(12), and
P.E.
equating power of small parameter 𝜖 leads to a succession

0.0004 of linear PDEs, and then the linear PDEs can be solved by
the separation of variables. In Figure 2. It is seen that our
homotopy-series approximation of the surface elevation 𝜁
0.0002 agrees well with the perturbation-series approximation, and
only slight derivations occur at the trough of the wave profile
as in Figure 2, which further indicates the validity of our
0 present theory about nonlinear hydroelastic waves beneath
5 10 15 20 25 30
a floating ice sheet.
h
We define quantities which measure how much energy
Figure 4: P.E. for (44) versus the water depth ℎ for different Young’s there is in the wave propagating beneath an infinite elastic
moduli of the plate 𝐸. Solid line: 𝐸 = 108 ; dashed line: 𝐸 = 109 ; plate. Let P.E. be the mean potential density per unit length
dash-dot-dotted line: 𝐸 = 1010 . in the 𝑋-axis [27]. In terms of the wave surface elevation
function, the energy density can be written as
1 2𝜋 2
Theorem 2.1 given by Liao in [12] can guarantee the P.E. = ∫ 𝜁 (𝑋) d𝑋. (44)
rationality of (42). So we obtain the optimal convergence 4𝜋2 0
control parameter 𝑐0 by the minimum of the squared-residual
𝑇 𝑇 Different from all research objectives in [7–9], we firstly
𝜀𝑚 , generally corresponding to 𝑑𝜀𝑚 /𝑑𝑐0 = 0.
consider in this paper the effect of water depth on nonlinear
hydroelastic waves beneath a floating elastic plate in detail.
4. Results and Analysis The energy of hydroelastic waves for different Young’s moduli
of the plate 𝐸 and different plate thicknesses ℎ in various water
In order to show the convergence of the analytical series solu- depths are as shown in Figures 3 and 4 and Tables 2 and 3,
tion to our problems by means of the HAM, we consider the respectively. We find that, when water depth ℎ is about more
cases of 𝑘 = 𝜋/5 m−1 , 𝑑 = 0.01 m, 𝜌𝑒 = 900 kgm−3 , ] = 0.33, than 2, the hydroelastic waves traveling beneath the thickest
𝐸 = 1010 Nm−2 , ℎ = 5 m, 𝐻 = 0.1 m, and 𝜔/𝜔0 = 1.01 and plate always contain the largest wave energy in different water
0.04 0.04
0.03 0.03
𝜁 𝜁
0.02 0.02
0.01 0.01
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
X X
Figure 5: Variation of the plate deflection 𝜁(𝑋) near the crest versus Figure 7: Variation of the plate deflection 𝜁(𝑋) near the crest versus
𝑋 for different Young’s moduli of the plate 𝐸. Solid line: 𝐸 = 108 ; 𝑋 for different plate thicknesses 𝑑. Solid line: 𝑑 = 0.001; dashed line:
dashed line: 𝐸 = 109 ; dash-dot-dotted line: 𝐸 = 1010 . 𝑑 = 0.005; dash-dot-dotted line: 𝑑 = 0.01.
−0.02 −0.02
−0.04 −0.04
𝜁 𝜁
−0.06 −0.06
−0.08 −0.08
2 2.4 2.8 3.2 3.6 4 2 2.4 2.8 3.2 3.6 4
X X
Figure 6: Variation of the plate deflection 𝜁(𝑋) near the trough Figure 8: Variation of the plate deflection 𝜁(𝑋) near the trough
versus 𝑋 for different Young’s moduli of the plate 𝐸. Solid line: versus 𝑋 for different plate thicknesses 𝑑. Solid line: 𝑑 = 0.001;
𝐸 = 108 ; dashed line: 𝐸 = 109 ; dash-dot-dotted line: 𝐸 = 1010 . dashed line: 𝑑 = 0.005; dash-dot-dotted line: 𝑑 = 0.01.
depths. And with an increasing Young’s modulus of the plate, indicates that the results are very similar to the effects due to
the wave energy becomes large too. different Young’s moduli 𝐸 of the plate.
The effect of Young’s modulus 𝐸 of the plate on the wave
elevation 𝜁(𝑋) under a floating elastic plate is studied. Figures 5. Conclusions
5 and 6 show the differences in 𝜁(𝑋) for 𝐸 = 108 , 109 ,
and 1010 . According to Figures 5 and 6, respectively, we can In this paper, the nonlinear hydroelastic waves propagating
see that the nonlinear hydroelastic response of the waves beneath a two-dimensional infinite elastic plate floating on
becomes flatter at the crest and steeper at the trough due to a fluid of finite depth are investigated analytically by the
the larger value of Young’s modulus 𝐸. Finally, we consider HAM. Mathematically, for a train of nonlinear hydroelastic
the impact the plate thickness 𝑑 by increasing 𝑑 from 0.001 waves traveling at a constant velocity in a fluid of finite or
to 0.01. In Figures 7 and 8, we show several displacements infinite depth, the PDEs in [7–9] were obtained by simply
𝜁(𝑋) with 𝑑 = 0.001, 𝑑 = 0.005, and 𝑑 = 0.01, respectively. It eliminating the time-dependent terms from the kinematic
Table 2: P.E. for (44) with different plate thicknesses and various Appendices
water depths ℎ.
A. The Detailed Derivation of (35) and (36)
P.E. P.E. P.E.
ℎ
(𝑑 = 0.001) (𝑑 = 0.005) (𝑑 = 0.01)
and the Expressions for 𝜙𝑚 and 𝜁𝑚
1 0.00067245 0.00100093 0.00040836 Let
3 0.00031356 0.00055509 0.00069567
5 0.00030650 0.00053375 0.00074396 𝑛
+∞ +∞
10 0.00030590 0.00053158 0.00074694 𝜂 = ( ∑ 𝜁𝑖 𝑞 ) = ∑ 𝜇𝑛,𝑖 𝑞𝑖 .
𝑛 𝑖
(A.1)
15 0.00030592 0.00053159 0.00074694 𝑖=1 𝑖=𝑛
20 0.00030595 0.00053159 0.00074696
30 0.00030600 0.00053159 0.00074696
∞ 0.00030600 0.00053159 0.00074696
For any 𝑧, we have a Maclaurin series as follows:
+∞
1 𝜕𝑛 𝜙𝑚 󵄨󵄨󵄨󵄨
𝜙𝑚 (𝑋, 𝑧) = ∑ 𝑛 󵄨󵄨󵄨
𝑧𝑛 . (A.2)
𝑛=0 𝑛! 𝜕𝑧 󵄨𝑧=0
Table 3: P.E. for (44) with different values of Young’s modulus of the
plate 𝐸 and various water depths ℎ.
For 𝑧 = 𝜂(𝑋; 𝑞), it follows from (A.1) and (A.2) that
P.E. P.E. P.E.
ℎ
(𝐸 = 108 ) (𝐸 = 109 ) (𝐸 = 1010 )
1 0.00076484 0.0009946 0.00040836 +∞
1 𝜕𝑛 𝜙𝑚 󵄨󵄨󵄨󵄨 +∞
3 0.00038884 0.00054698 0.00069567 𝜙𝑚 (𝑋, 𝜂) = ∑ ( 󵄨
󵄨 ) ( ∑ 𝜇𝑛,𝑖 𝑞𝑖 )
𝑛=0 𝑛! 𝜕𝑧𝑛 󵄨󵄨𝑧=0 𝑖=𝑛
5 0.00030138 0.00047932 0.00074396
(A.3)
10 0.00028886 0.00046970 0.00074694 +∞
15 0.00028884 0.00046969 0.00074694 = ∑ 𝜓𝑚,𝑖 𝑞𝑖 ,
20 0.00028884 0.00046969 0.00074696 𝑖=0
30 0.00028884 0.00046969 0.00074696
∞ 0.00028884 0.00046969 0.00074696
where
𝑖
1 𝜕𝑛 𝜙𝑚 󵄨󵄨󵄨󵄨
𝜓𝑚,𝑖 = ∑ ( 󵄨 )𝜇 . (A.4)
𝑛=0 𝑛! 𝜕𝑧𝑛 󵄨󵄨󵄨𝑧=0 𝑛,𝑖
and dynamic boundary conditions on the unknown free
surface in the frame of reference moving with the wave.
Here, for a general case it should be noted that we construct Thus we have, for 𝑧 = 𝜂(𝑋; 𝑞),
the PDEs by directly applying the traveling-wave method to
transfer the temporal differentiation into the spatial one in a
fixed Cartesian coordinate 𝑜𝑥𝑧. Furthermore, the convergent +∞ +∞ +∞
homotopy-series solutions for the PDES are derived by the Φ (𝑋, 𝜂; 𝑞) = ∑ 𝜙𝑚 (𝑋, 𝜂) 𝑞𝑚 = ∑ ( ∑ 𝜓𝑚,𝑖 𝑞𝑖 ) 𝑞𝑚
HAM with the optimal convergence control parameter. 𝑚=0 𝑚=0 𝑛=0
Physically, we study the effect of the water depth on the (A.5)

+∞
nonlinear hydroelastic waves under an elastic plate in detail.
= ∑ 𝜑𝑚 𝑞𝑚 ,
It is found that, in different water depths, the wave energy 𝑚=0
density (P.E.) tends to become larger with an increasing
thickness of the sheet. The same conclusions are obtained in
various water depths by means of different values of Young’s where
modulus of the plate. Additionally, the influences of Young’s
modulus and the thickness of the plate on the wave elevation 𝑚
𝜁(𝑋) are investigated, respectively. As Young’s modulus of 𝜑𝑚 = ∑𝜓𝑚−𝑖,𝑖 . (A.6)
the plate increases, the wave elevation becomes lower. And 𝑖=0
the increasing thickness of the plate flattens the crest and
sharpens the trough of the wave profile. The results obtained
here demonstrate that Young’s modulus and the thickness of Substituting the series expansions (A.1) and (A.5) into
the sheet have important effects on the energy and the profile the boundary conditions (27) and (28) and then equating
of nonlinear hydroelastic waves under an ice sheet floating on the like-power of the embedding parameter 𝑞, we have two
a fluid of finite depth. linear boundary conditions (35) and (36), respectively. And
𝜙
the explicit expressions for Δ 𝑚−1 , 𝑆𝑚−1 , 𝑆𝑚 , and Δ𝜁𝑚−1 in these 𝛽2,1 = ([𝐻3 𝜌 + 32𝑔𝐻𝑘𝜌𝜔2 𝑐0 tanh (ℎ𝑘)
two conditions are given by
+ 32𝐷𝑔𝐻𝑘5 𝜔2 𝑐02 tanh (ℎ𝑘)
𝜙 d2 𝜑𝑚
Δ 𝑚−1 = 𝜔2 + 𝑔𝜑𝑚
d𝑋2 + 32𝑔2 𝐻𝑘𝜌𝜔2 𝑐02 tanh (ℎ𝑘)
𝑚
d𝜑𝑛 d2 𝜑𝑚−𝑛 d𝜑 − 32d𝑔𝐻𝑘𝜌𝜔4 𝑐02 tanh (ℎ𝑘)
− 𝜔∑ ( 2
+ 𝜑𝑛 𝑚−𝑛 )
𝑛=0 d𝑋 d𝑋 d𝑋
− 64𝑔𝑘𝜌𝜔3 𝑐02 𝛼1,1 tanh (ℎ𝑘)
𝜔 4 d5 𝜁𝑚 𝜔3 𝑚𝑒 d3 𝜁𝑚 𝑚
d𝜑 d𝜁 − 16𝑔𝐻𝑘𝜌𝜔2 𝑐0 tanh (2ℎ𝑘)
− 𝐷𝑘 5
− 3
− 𝑘2 𝑔 ∑ 𝑛 𝑚−𝑛 ,
𝜌 d𝑋 𝜌 d𝑋 𝑛=0 d𝑋 d𝑋
− 16𝐷𝑔𝐻𝑘5 𝜔2 𝑐02 tanh (2ℎ𝑘)
𝑚−2
d2 𝜓𝑚−1−𝑖,𝑖
𝑆𝑚−1 = ∑ ( + 𝛾𝑚−1−𝑖,𝑖 ) , − 16𝑔2 𝐻𝑘𝜌𝜔2 𝑐02 tanh (2ℎ𝑘)
𝑖=0 d𝑋2
+ 16d𝑔𝐻𝑘𝜌𝜔4 𝑐02 tanh (2ℎ𝑘)
𝑚−1 2
d 𝜓𝑚−𝑖,𝑖
𝑆𝑚 = ∑ ( + 𝛾𝑚−𝑖,𝑖 ) , + 32𝑔𝑘𝜌𝜔3 𝑐02 𝛼1,1 tanh (2ℎ𝑘)
𝑖=1 d𝑋2
+ 𝐻3 𝑘2 𝜌 tanh (ℎ𝑘) tanh (2ℎ𝑘)
d𝜑𝑚−1
Δ𝜁𝑚−1 = − 𝜔 − 𝐻3 𝑘4 𝜌tanh3 (ℎ𝑘) tanh (2ℎ𝑘)
d𝑋
−𝐻3 𝑘2 𝜌tanh2 (ℎ𝑘)])
1 𝑚−1 d𝜑 d𝜑
+ ∑ ( 𝑛 𝑚−1−𝑛 + 𝜑𝑛 𝜑𝑚−1−𝑛 ) + 𝜁𝑚−1 −1
2 𝑛=0 d𝑋 d𝑋 × ([32𝑔𝑘𝜌𝜔2 𝑐0 (2 tanh (ℎ𝑘) − tanh (2ℎ𝑘))]) ,
𝐷𝑘4 d4 𝜁𝑚−1 𝑚𝑒 𝜔2 d2 𝜁𝑚−1 1 d𝐻2 𝑔𝐻2 𝐷𝐻2 𝑘4

+ + , (𝑚 ≥ 2) , 𝛽2,2 = − + +
𝜌 d𝑋4 𝜌 d𝑋2 13 52 208𝜔2 13𝜌𝜔2
𝐻2 𝐻𝛼1,1 1
1 d𝜑 2 d𝜑 𝑚𝑔 + − + d𝐻2 𝑘2 tanh2 (ℎ𝑘)
Δ𝜁0 = [( 0 ) + 𝜑20 ] − 𝜔 0 + 𝑒 , 2
16𝜔 𝑐0 4𝜔 52
2 d𝑋 d𝑋 𝜌
(A.7) 𝑔𝐻2 𝑘2 tanh2 (ℎ𝑘) 𝐷𝐻2 𝑘6 tanh2 (ℎ𝑘)
− −
where 208𝜔2 13𝜌𝜔2
𝑚
𝜑𝑚 = ∑𝛾𝑚−𝑖,𝑖 , 𝐻2 𝑘2 tanh2 (ℎ𝑘)
−
𝑖=0 16𝜔2 𝑐0
(A.8)
󵄨 𝐻𝑘2 𝛼1,1 tanh2 (ℎ𝑘)
𝑖
1 𝜕 𝜙𝑚−𝑖 󵄨󵄨󵄨 𝑛+1 +
𝛾𝑚−𝑖,𝑖 =∑ ( 󵄨󵄨 ) 𝜇 . 4𝜔
𝑛=0 𝑛! 𝜕𝑧𝑛+1 󵄨󵄨󵄨𝑧=0 𝑛,𝑖
𝐻2
+
8𝑔𝑘 (2 tanh (ℎ𝑘) − tanh (2ℎ𝑘))
B. Expressions of the Coefficients 𝐻2 𝑘tanh2 (ℎ𝑘)

− ,
8𝑔 (2 tanh (ℎ𝑘) − tanh (2ℎ𝑘))
𝛽2,3 = (𝐻3 − 𝐻3 𝑘2 tanh2 (ℎ𝑘)

1
𝛽2,0 = [𝐻2 + 𝑔𝐻2 𝑐0 + 16d𝑔𝜔2 𝑐02
16𝜔2 𝑐0 − 𝐻3 𝑘2 tanh (ℎ𝑘) tanh (2ℎ𝑘)
+ 16d𝑔2 𝜔2 𝑐03 − 4𝐻𝜔𝑐0 𝛼1,1 +𝐻3 𝑘4 tanh3 (ℎ𝑘) tanh (2ℎ𝑘))

−1
+ 𝐻2 𝑘2 tanh2 (ℎ𝑘) + 𝑔𝐻2 𝑘2 𝑐0 tanh2 (ℎ𝑘) × (2336𝑔𝑘𝜔2 𝑐0 (2 tanh (ℎ𝑘) − tanh (2ℎ𝑘))) ,
𝛽2,4 = 0,
2 2
−4𝐻𝑘 𝜔𝑐0 𝛼1,1 tanh (ℎ𝑘)] , (B.1)
𝛼1,1 = ([𝐻 ( − 13𝐻2 (1 + 2𝑔𝑘2 ) 𝜌𝜔2 𝐻2 𝜔

+
8𝑔𝑘 tanh (ℎ𝑘) − 4𝑔𝑘 tanh (2ℎ𝑘)
− 𝑔𝑘 (𝑔𝐻2 𝑘2 𝜌 − 208 (𝐷𝑘4 𝜔4 − 𝑑𝜌𝜔6 ) 𝑐02 )
𝐻2 tanh (2ℎ𝑘) (1 + 𝑘2 tanh2 (ℎ𝑘))
× tanh (2ℎ𝑘) −
16𝜔 tanh (ℎ𝑘) − 8𝜔 tanh (2ℎ𝑘)
+ 𝐻2 𝑘2 𝜌 tanh2 (ℎ𝑘) d𝑔𝐻2 𝑘 (1 − 𝑘2 tanh2 (ℎ𝑘))

+
8𝜔 tanh (ℎ𝑘) − 4𝜔 tanh (2ℎ𝑘)
× (13 (1 + 2𝑔𝑘2 ) 𝜔2 + 𝑔2 𝑘3 tanh (2ℎ𝑘))
2 4 3
+ (d𝑔𝐻2 𝑘 tanh (ℎ𝑘) tanh (2ℎ𝑘)
+ 𝐻 𝑘 𝜌 tanh (ℎ𝑘)
2 2
× (−1 + 𝑘2 tanh2 (ℎ𝑘)))
× (−2𝑔 𝑘 − 13 (−1 + 𝑔) 𝜔 tanh (2ℎ𝑘))
× (4𝜔 tanh (ℎ𝑘) − 2𝜔 tanh (2ℎ𝑘))−1
+ 𝑘 tanh (ℎ𝑘)
−𝐻2 tanh (2ℎ𝑘) (−1 + 𝑘2 tanh2 (ℎ𝑘))
× (416𝑔𝜔4 (−𝐷𝑘4 + d𝜌𝜔2 ) 𝑐02 + 𝐻2 𝑘𝜌 +
16𝜔𝑐0 tanh (ℎ𝑘) − 8𝜔𝑐0 tanh (2ℎ𝑘)
2 2
× (2𝑔 𝑘 + 13 (−1 + 𝑔) 𝜔 tanh (2ℎ𝑘))
𝐻2 tanh (ℎ𝑘) (1 − 𝑘2 tanh2 (ℎ𝑘) tanh (2ℎ𝑘))
+
+ 𝐻2 𝑘𝜌 (2𝑔2 𝑘 + 13 (−1 + 𝑔) 8𝜔𝑐0 tanh (ℎ𝑘) − 4𝜔𝑐0 tanh (2ℎ𝑘)
×𝜔2 tanh (2ℎ𝑘))))]) 𝐻4 𝑘 (1 − 𝑘4 tanh4 (ℎ𝑘))

+
128𝜔3 𝑐02 tanh (ℎ𝑘) − 64𝜔3 𝑐02 tanh (2ℎ𝑘)
× ([ − 416𝜔2 𝑐02
𝐻4 𝑘 tanh (ℎ𝑘) tanh (2ℎ𝑘) (−1 + 𝑘4 tanh4 (ℎ𝑘))
× (d𝑔2 𝑘2 + 𝜔2 𝑔𝑘𝜌𝜔 + ],
64𝜔3 𝑐02 tanh (ℎ𝑘) − 32𝜔3 𝑐02 tanh (2ℎ𝑘)
× (𝑔𝐻2 𝑘2 (−1 + tanh2 (ℎ𝑘)) 𝛼2,3
× (25 + 27𝑘2 tanh2 (ℎ𝑘)) 1

=
−9𝑔𝑘 tanh (ℎ𝑘) + 3𝑔𝑘 tanh (3ℎ𝑘)
−𝑔𝑘 tanh (ℎ𝑘) − d𝑔2 𝑘2 tanh2 (ℎ𝑘)))
× [(𝑔𝐻2 𝑘4 𝛼1,1 tanh2 (ℎ𝑘)
−1
× (2 tanh (ℎ𝑘) − tanh (2ℎ𝑘)) ]) , −1
−9𝑔𝐻2 𝑘4 𝛼1,1 tanh4 (ℎ𝑘)) × (416𝜔2 𝑐0 )
1
𝛼2,2 = + (−𝑔𝐻3 𝑘2 − 𝑔𝐻2 𝑘2 𝛼1,1 + 𝑔𝐻3 𝑘4 tanh2 (ℎ𝑘)
−4𝑔𝑘 tanh (ℎ𝑘) + 2𝑔𝑘 tanh (2ℎ𝑘)
𝑔𝐻2 𝑘2 2𝐷𝐻2 𝑘4 d𝐻2 𝜔 𝐻𝛼1,1 +9𝑔𝐻2 𝑘2 𝛼1,1 tanh2 (ℎ𝑘))

× [− + − −
8𝜔 13𝜌𝜔 26 2 −1
× (416𝜔2 𝑐0 )
𝑔𝐻𝑘2 𝛼1,1 2𝐷𝐻2 𝑘6 tanh2 (ℎ𝑘)
− − 5𝐻3 𝑘 tanh (ℎ𝑘)tanh (2ℎ𝑘)(1 − 𝑘2 tanh2 (ℎ𝑘))
4 13𝜌𝜔 +
2 (32𝜔𝑐0 tanh (ℎ𝑘) − 16𝜔𝑐0 tanh (2ℎ𝑘))
1
+ d𝐻2 𝑘2 𝜔 tanh2 (ℎ𝑘)
26 𝐻3 𝑘 (1 − 𝑘2 tanh2 (ℎ𝑘))
+
1 2 (16𝜔𝑐0 tanh (ℎ𝑘) − 8𝜔𝑐0 tanh (2ℎ𝑘))
+ 𝐻𝑘2 𝛼1,1 tanh2 (ℎ𝑘)
2
+ (𝐻3 𝑘 tanh (ℎ𝑘) (− tanh (ℎ𝑘) − tanh (2ℎ𝑘)
2 2
+ 𝑔𝐻𝑘 𝛼1,1 tanh (ℎ𝑘)
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2 2
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http://dx.doi.org/10.1155/2013/416757
Research Article
Numerical Solution of Nonlinear Fredholm
Integrodifferential Equations by Hybrid of Block-Pulse
Functions and Normalized Bernstein Polynomials
S. H. Behiry
General Required Courses Department, Jeddah Community College, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to S. H. Behiry; salah behiry@hotmail.com
Received 24 April 2013; Accepted 1 September 2013
Copyright © 2013 S. H. Behiry. This is an open access article distributed under the Creative Commons Attribution License, which
A numerical method for solving nonlinear Fredholm integrodifferential equations is proposed. The method is based on hybrid
functions approximate. The properties of hybrid of block pulse functions and orthonormal Bernstein polynomials are presented
and utilized to reduce the problem to the solution of nonlinear algebraic equations. Numerical examples are introduced to illustrate
the effectiveness and simplicity of the present method.
1. Introduction polynomials in the solution of integral equations, differential

equations, and approximation theory; see for instance [14–
Integrodifferential equations are often involved in mathemat- 17].
ical formulation of physical phenomena. Fredholm integrod- The purpose of this work is to utilize the hybrid functions
ifferential equations play an important role in many fields consisting of combination of block-pulse functions with
such as economics, biomechanics, control, elasticity, fluid normalized Bernstein polynomials for obtaining numerical
dynamics, heat and mass transfer, oscillation theory, and solution of nonlinear Fredholm integrodifferential equation:
airfoil theory; for examples see [1–3] and references cited
therein. Finding numerical solutions for Fredholm integrod- 𝑠 1
𝑞
ifferential equations is one of the oldest problems in applied ∑ 𝑝𝑖 (𝑥) 𝑦(𝑖) (𝑥) = 𝑔 (𝑥) + 𝜆 ∫ 𝑘 (𝑥, 𝑡) [𝑦 (𝑡)] 𝑑𝑡,
𝑖=0 0 (1)
mathematics. Numerous works have been focusing on the
development of more advanced and efficient methods for 0 ≤ 𝑥, 𝑡 < 1,
solving integrodifferential equations such as wavelets method
[4, 5], Walsh functions method [6], sinc-collocation method with the conditions
[7], homotopy analysis method [8], differential transform
method [9], the hybrid Legendre polynomials and block- 𝑦(𝑖) (0) = 𝛼𝑖 , 0 ≤ 𝑖 ≤ 𝑠 − 1, (2)
pulse functions [10], Chebyshev polynomials method [11],
and Bernoulli matrix method [12]. where 𝑦(𝑖) (𝑥) is the 𝑖th derivative of the unknown function
Block-pulse functions have been studied and applied that will be determined, 𝑘(𝑥, 𝑡) is the kernel of the integral,
extensively as a basic set of functions for signals and functions 𝑔(𝑥) and 𝑝𝑖 (𝑥) are known analytic functions, 𝑞 is a positive
approximations. All these studies and applications show that integer, and 𝜆 and 𝛼𝑖 are suitable constants. The proposed
block-pulse functions have definite advantages for solving approach for solving this problem uses few numbers of basis
problems involving integrals and derivatives due to their and benefits of the orthogonality of block-pulse functions and
clearness in expressions and their simplicity in formulations; the advantages of orthonormal Bernstein polynomials prop-
see [13]. Also, Bernstein polynomials play a prominent role in erties to reduce the nonlinear integrodifferential equation to
various areas of mathematics. Many authors have used these easily solvable nonlinear algebraic equations.
This paper is organized as follows. In the next section, where

we present Bernstein polynomials and hybrid of block-pulse 𝑇
functions. Also, their useful properties such as functions C = [C𝑇1 , C𝑇2 , . . . , C𝑇𝑗 , . . . , C𝑇𝑚 ] ,
approximation, convergence analysis, operational matrix of (8)
𝑇
product, and operational matrix of differentiation are given. C𝑗 = [𝑐𝑗0 , 𝑐𝑗1 , 𝑐𝑗2 , . . . , 𝑐𝑗𝑛 ] , 𝑗 = 1, 2, . . . , 𝑚,
In Section 3, the numerical scheme for the solution of (1) and
𝑇
(2) is described. In Section 4, the proposed method is applied H (𝑥) = [H𝑇1 (𝑥) , H𝑇2 (𝑥) , . . . , H𝑇𝑗 (𝑥) , . . . , H𝑇𝑚 (𝑥)] , (9)
to some nonlinear Fredholm integrodifferential equations,
and comparisons are mad with the existing analytic or and H𝑗 (𝑥) = [ℎ𝑗0 (𝑥), ℎ𝑗1 (𝑥), . . . , ℎ𝑗𝑛 (𝑥)]𝑇 , 𝑗 = 1, 2, . . . , 𝑚. The
numerical solutions that were reported in other published constant coefficients 𝑐𝑗𝑖 are (𝑦(𝑥), ℎ𝑗𝑖 (𝑥)), 𝑖 = 0, 1, 2, . . . , 𝑛,
works in the literature. Finally conclusions are given in 𝑗 = 1, 2, . . . , 𝑚, and (⋅, ⋅) is the standard inner product on
Section 5. 𝐿2 [0, 1).
We can also approximate the function 𝑘(𝑥, 𝑡) ∈ 𝐿2 ([0, 1)×
2. Properties of Hybrid Functions [0, 1)) by
𝑚 𝑚 𝑛 𝑛
𝑖𝑗
2.1. Hybrid of Block-Pulse Functions and Orthonormal Bern- 𝑘 (𝑥, 𝑡) ≈ ∑ ∑ ∑ ∑ 𝑘𝑙𝑟 ℎ𝑖𝑙 (𝑥) ℎ𝑗𝑟 (𝑡) = H𝑇 (𝑥) KH (𝑡) ,
stein Polynomials. The Bernstein polynomials of 𝑛th degree 𝑖=1 𝑗=1 𝑙=0 𝑟=0
are defined on the interval [0, 1] as [16] (10)
𝑛 where K = [K𝑖𝑗 ] is an 𝑚(𝑛 + 1) × 𝑚(𝑛 + 1) matrix, such that
𝐵𝑖,𝑛 (𝑥) = ( ) 𝑥𝑖 (1 − 𝑥)𝑛−𝑖 , for 𝑖 = 0, 1, 2, . . . , 𝑛, (3)
𝑖 the elements of the sub matrix k𝑖𝑗 are
𝑖/𝑚 𝑗/𝑚
𝑖𝑗
where 𝑘𝑙𝑟 = ∫ ∫ 𝑘 (𝑥, 𝑡) ℎ𝑖(𝑙−1) (𝑥) ℎ𝑗(𝑟−1) (𝑡) 𝑑𝑥𝑑𝑡,
𝑖−1/𝑚 𝑗−1/𝑚 (11)
𝑛 𝑛!
( )= . (4) 𝑙, 𝑟 = 1, 2, . . . , 𝑛 + 1, 𝑖, 𝑗 = 1, 2, . . . , 𝑚,
𝑖 𝑖! (𝑛 − 𝑖)!
utilizing properties of block-pulse function and orthonormal
There are (𝑛 + 1) 𝑛th degree Bernstein polynomials. Using
Bernstein polynomials.
Gram-Schmidt orthonormalization process on 𝐵𝑖,𝑛 (𝑥), we
obtain a class of orthonormal polynomials from the Bernstein
polynomials. We call them orthonormal Bernstein polynomi- 2.3. Convergence Analysis. In this section, the error bound
als of degree 𝑛 and denote them by 𝑏𝑖,𝑛 (𝑥), 0 ≤ 𝑖 ≤ 𝑛. For 𝑛 = 3, and convergence is established by the following lemma.
the four orthonormal Bernstein polynomials are given by
Lemma 1. Suppose that 𝑓 ∈ 𝐶(𝑛+1) [0, 1) is 𝑛 + 1 times
3 2
𝑏0,3 (𝑥) = −√7 [𝑥 − 3𝑥 + 3𝑥 − 1] , continuously differentiable function such that 𝑓 = ∑𝑚 𝑗=1 𝑓𝑗 , and
let 𝑌𝑗 = Span{ℎ𝑗0 (𝑥), ℎ𝑗1 (𝑥), . . . , ℎ𝑗𝑛 (𝑥)}, 𝑗 = 1, 2, . . . , 𝑚. If
𝑏1,3 (𝑥) = √5 [7𝑥3 − 15𝑥2 + 9𝑥 − 1] , C𝑇𝑗 H𝑗 (𝑥) is the best approximation to 𝑓𝑗 from 𝑌𝑗 , then C𝑇 H(𝑥)
(5) approximates 𝑓 with the following error bound:
𝑏2,3 (𝑥) = −√3 [21𝑥3 − 33𝑥2 + 13𝑥 − 1] , 󵄩󵄩 󵄩 𝛾
󵄩󵄩𝑓 − C𝑇 H (𝑥)󵄩󵄩󵄩 ≤
󵄩 󵄩2 𝑚𝑛+1 (𝑛 + 1)!√2𝑛 + 3 ,
𝑏3,3 (𝑥) = 35𝑥3 − 45𝑥2 + 15𝑥 − 1. (12)
󵄨 󵄨
Hybrid functions ℎ𝑗𝑖 (𝑥), 𝑗 = 1, 2, . . . , 𝑚 and 𝑖 = 0, 1, . . . , 𝑛 𝛾 = max 󵄨󵄨󵄨󵄨𝑓(𝑛+1) (𝑥)󵄨󵄨󵄨󵄨 .
𝑥∈[0.1)
are defined on the interval [0, 1) as
Proof. The Taylor expansion for the function 𝑓𝑗 (𝑥) is
{√𝑚 𝑏𝑖,𝑛 (𝑚𝑥 − 𝑗 + 1) , 𝑗−1 𝑗
≤𝑥< , 𝑗−1 𝑗−1 𝑗−1
ℎ𝑗𝑖 (𝑥) = { 𝑚 𝑚 (6) 𝑓̃𝑗 (𝑥) = 𝑓𝑗 ( ) + 𝑓𝑗󸀠 ( ) (𝑥 − )
𝑚 𝑚 𝑚
{0, otherwise,
𝑛
𝑗 − 1 (𝑥 − (𝑗 − 1/𝑚))
where 𝑗 and 𝑛 are the order of block-pulse functions and + ⋅ ⋅ ⋅ + 𝑓𝑗(𝑛) ( ) , (13)
degree of orthonormal Bernstein polynomials, respectively. 𝑚 𝑛!
It is clear that these sets of hybrid functions in (6) are 𝑗−1 𝑗
orthonormal and disjoint. ≤𝑥< ,
𝑚 𝑚
for which it is known that
2.2. Functions Approximation. A function 𝑦(𝑥) ∈ 𝐿2 [0, 1) 𝑛+1
󵄨󵄨 󵄨 󵄨 󵄨 (𝑥 − (𝑗 − 1/𝑚))
may be approximated as 󵄨󵄨𝑓𝑗 (𝑥) − 𝑓̃𝑗 (𝑥)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝑓(𝑛+1) (𝜂)󵄨󵄨󵄨 ,
󵄨 󵄨 󵄨 󵄨 (𝑛 + 1)!
𝑚 𝑛 (14)
𝑦 (𝑥) ≈ ∑ ∑ 𝑐𝑗𝑖 ℎ𝑗𝑖 (𝑥) = C𝑇 H (𝑥) , (7) 𝑗−1 𝑗
𝜂∈[ , ) , 𝑗 = 1, 2, . . . , 𝑚.
𝑗=1 𝑖=0 𝑚 𝑚
Since C𝑇𝑗 H𝑗 (𝑥) is the best approximation to 𝑓𝑗 form 𝑌𝑗 and ̃T

=A ̃ (𝑥) ,
𝑓̃ ∈ 𝑌 , using (14) we have
𝑗 𝑗 (22)
󵄩󵄩 󵄩2 󵄩 󵄩2
󵄩󵄩𝑓𝑗 − C𝑇𝑗 H𝑗 (𝑥)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑓𝑗 − 𝑓̃𝑗 󵄩󵄩󵄩 where Ã = diag[A1 , A2 , . . . , A𝑗 , . . . , A𝑚 ] is the 𝑚(𝑛 + 1) ×
󵄩 󵄩2 󵄩 󵄩
𝑚(𝑛 + 1) coefficient matrix of the (𝑛 + 1) × (𝑛 + 1) coefficient
𝑗/𝑚
󵄨󵄨 󵄨2 ̃
submatrix A𝑗 , and T(𝑥) = [t1 (𝑥), t2 (𝑥), . . . , t𝑗 (𝑥), . . . , t𝑚 (𝑥)]𝑇
=∫ 󵄨󵄨𝑓𝑗 (𝑥) − 𝑓̃𝑗 (𝑥)󵄨󵄨󵄨 𝑑𝑥
󵄨 󵄨
𝑗−1/𝑚 is the 𝑚(𝑛 + 1) vector with t𝑗 (𝑥) = [1, 𝑥, 𝑥2 , . . . , 𝑥𝑛 ]𝑇 , such
𝑛+1 2 that H𝑗 (𝑥) = A𝑗 t𝑗 (𝑥). Now
𝑗/𝑚 𝑓(𝑛+1) (𝜂) (𝑥− (𝑗−1/𝑚))
≤∫ [ ] 𝑑𝑥
𝑗−1/𝑚 (𝑛+1)! 𝑑 ̃Q̃T
̃ (𝑥) = A
̃Q̃A
̃−1 H (𝑥) ,
H (𝑥) = A (23)
2 𝑗/𝑚 𝑑𝑥
𝛾 𝑗 − 1 2𝑛+2
≤[ ] ∫ (𝑥 − ) 𝑑𝑥 ̃ = diag[Q, . . . , Q] is the 𝑚(𝑛 + 1) × 𝑚(𝑛 + 1) matrix
(𝑛 + 1)! 𝑗−1/𝑚 𝑚 where Q
2 of the (𝑛 + 1) × (𝑛 + 1) sub-matrix Q, such that
𝛾 1
=[ ] 2𝑛+3 .
(𝑛 + 1)! 𝑚 (2𝑛 + 3) 0 0 0
⋅⋅⋅ 0 0
(15) [1 0 ⋅ ⋅ ⋅ 0 0]
0
[ ]
[ ⋅ ⋅ ⋅ 0 0]
Now, Q = [0 2 0 ]. (24)
[ .. .. .. .. .. ]
𝑚 [. . ⋅ ⋅ ⋅ . .]
.
󵄩󵄩 󵄩2 󵄩 󵄩2
󵄩󵄩𝑓 − C𝑇 H (𝑥)󵄩󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩𝑓𝑗 − C𝑇𝑗 H𝑗 (𝑥)󵄩󵄩󵄩
󵄩 󵄩2 󵄩 󵄩2 [0 0 0 ⋅ ⋅ ⋅ 𝑛 0]
𝑗=1
(16) Hence,
𝛾2
≤ 2𝑛+2 . ̃Q̃A
̃−1 .
𝑚 [(𝑛 + 1)!]2 (2𝑛 + 3) D=A (25)
By taking the square roots we have the above bound. In general, we can have
2.4. The Operational Matrix of Product. In this section, we 𝑑𝑘

present a general formula for finding the 𝑚(𝑛 + 1) × 𝑚(𝑛 + 1) H (𝑥) = D𝑘 H (𝑥) , 𝑘 = 1, 2, 3, . . . . (26)
𝑑𝑥𝑘
operational matrix of product C̃ whenever
C𝑇 H (𝑥) H𝑇 (𝑥) ≈ H𝑇 (𝑥) C,̃ (17) 3. Outline of the Solution Method
where This section presents the derivation of the method for solving
̃ = diag [C
C ̃2 , . . . , C
̃1 , C ̃𝑗 , . . . , C
̃𝑚 ] . (18) 𝑠th-order nonlinear Fredholm integrodifferential equation
(1) with the initial conditions (2).
𝑗
̃ = [𝑐 ] are (𝑛 + 1) × (𝑛 + 1) symmetric matrices
In (18), C 𝑗 𝑙𝑟
Step 1. The functions 𝑦(𝑖) (𝑥), 𝑖 = 0, 1, 2, . . . , 𝑠 are being
depending on 𝑛, where
approximated by
𝑗/𝑚 𝑛
𝑗
𝑐𝑙𝑟 = ∫ (ℎ𝑗(𝑙−1) (𝑥) ℎ𝑗(𝑟−1) (𝑥) ∑ 𝑐𝑗𝑖 ℎ𝑗𝑖 (𝑥)) 𝑑𝑥, 𝑦(𝑖) (𝑥) = C𝑇 (H (𝑥))(𝑖) = C𝑇 D𝑖 H (𝑥) , 𝑖 = 0, 1, 2, . . . , 𝑠,
𝑗−1/𝑚 𝑖=0 (19)
(27)
𝑙, 𝑟 = 1, 2, . . . , 𝑛 + 1.
where D is given by (25).
Furthermore, the integration of cross-product of two
hybrid functions vectors is Step 2. The function 𝑘(𝑥, 𝑡) is being approximated by (10).
1
∫ H (𝑥) H𝑇 (𝑥) 𝑑𝑥 = I, (20) Step 3. In this step, we present a general formula for approxi-
0
mate 𝑦𝑞 (𝑥). By using (7) and (17), we can have
where I is the 𝑚(𝑛 + 1) identity matrix.
2
̃
𝑦2 (𝑥) = [C𝑇 H (𝑥)] = C𝑇 H (𝑥) H𝑇 (𝑥) C = H𝑇 (𝑥) CC,
2.5. The Operational Matrix of Differentiation. The opera- (28)
tional matrix of derivative of the hybrid functions vector H(𝑥)
2
is defined by ̃
𝑦3 (𝑥) = C𝑇 H (𝑥) [C𝑇 H (𝑥)] = C𝑇 H (𝑥) H𝑇 (𝑥) CC
𝑑 (29)
H (𝑥) = DH (𝑥) , (21) 𝑇 ̃CC ̃ 2 C,
̃ = H𝑇 (𝑥) (C)
𝑑𝑥 = H (𝑥) C
where D is the 𝑚(𝑛 + 1) × 𝑚(𝑛 + 1) operational matrix of
derivative given as and so by use of induction, 𝑦𝑞 (𝑥) will be approximated as
𝑇 ̃ 𝑞−1 C.
𝑦𝑞 (𝑥) = H𝑇 (𝑥) (C) (30)
H (𝑥) = [H𝑇1 (𝑥) , H𝑇2 (𝑥) , . . . , H𝑇𝑗 (𝑥) , . . . , H𝑇𝑚 (𝑥)]
Table 1: Numerical comparison of absolute difference errors for and hence we get
Example 3.
𝑠
𝑇 𝑞−1
Method of [17] The proposed method ̃ 𝑖 (D𝑖 ) C − 𝜆K(C)
∑P ̃ C = G. (35)
𝑥
𝑛=7 𝑛 = 2, 𝑚 = 30 𝑛 = 3, 𝑚 = 30 𝑖=0
0.0 3.2038𝐸 − 009 3.1309𝐸 − 007 4.0173𝐸 − 010
The matrix (35) gives a system of 𝑚(𝑛 + 1) nonlinear algebraic
0.2 7.1841𝐸 − 010 3.8241𝐸 − 007 4.9068𝐸 − 010
equations which can be solved utilizing the initial condition
0.4 1.4151𝐸 − 010 4.6707𝐸 − 007 5.9932𝐸 − 010
for the elements of C. Once C is known, 𝑦(𝑥) can be
0.6 4.0671𝐸 − 011 5.7048𝐸 − 007 7.3201𝐸 − 010 constructed by using (7).
0.8 9.1044𝐸 − 010 6.9679𝐸 − 007 8.9407𝐸 − 010
1.0 3.7002𝐸 − 009 8.2709𝐸 − 007 1.4907𝐸 − 010
4. Applications and Numerical Results
In this section, numerical results of some examples are
presented to validate accuracy, applicability, and convergence
Step 4. Approximate the functions 𝑔(𝑥) and 𝑝𝑖 (𝑥) by
of the proposed method. Absolute difference errors of this
𝑔 (𝑥) ≈ G𝑇 H (𝑥) , (31) method is compared with the existing methods reported in
the literature [5, 6, 17, 18]. The computations associated with
𝑝𝑖 (𝑥) ≈ P𝑇𝑖 𝐻 (𝑥) , 𝑖 = 0, 1, 2, . . . , 𝑠, (32) these examples were performed using MATLAB 9.0.
where G and P𝑖 are constant coefficient vectors which are Example 1. Consider the first-order nonlinear Fredholm
defined similarly to (7). integrodifferential equation [17, 18] as follows:
Now, using (27)–(32) and (10) to substitute into (1), we can
1
obtain 1
𝑦󸀠 (𝑥) = 1 − 𝑥 + ∫ 𝑥 𝑦2 (𝑡) 𝑑𝑡, 0 ≤ 𝑥 < 1, (36)
𝑠 3 0
𝑖 𝑇
∑ P𝑇𝑖 H (𝑥) H𝑇 (𝑥) (D ) C
𝑖=0 with the initial condition
1
̃ 𝑞−1 C𝑑𝑡.
= H𝑇 (𝑥) G + 𝜆 ∫ H𝑇 (𝑥) KH (𝑡) H𝑇 (𝑡) (C) 𝑦 (0) = 0. (37)
0
(33) In this example, we have 𝑝0 = 0, 𝑝1 = 1, 𝑔(𝑥) = 1 − (1/3)𝑥,
𝜆 = 1, 𝑘(𝑥, 𝑡) = 𝑥, and 𝑞 = 2.
Utilizing (17) and (20), we may have The matrix (35) for this example is
𝑠
̃ 𝑖 (D𝑖 )𝑇 C = H𝑇 (𝑥) G + 𝜆H𝑇 (𝑥) K(C)
∑H𝑇 (𝑥) P ̃ 𝑞−1 C, P ̃ C = G,
̃ 1 D𝑇 C − K (C) (38)
𝑖=0
(34) where for 𝑛 = 1 and 𝑚 = 2 we have
−3 3√3 0 0 𝑐10
[ − √3 3 0 0 ] [𝑐11 ]
̃ 1 = I,
P D𝑇 = [
[ 0
], C=[ ]
[ ,
0 −3 3 3]
√ 𝑐20 ]
[ 0 0 −√ 3 3 ] [𝑐21 ]
1 √3 1 √3
[ ]
[ 16 48 16 48 ]
[ ]
[ √3 1 √3 1 ]
[ ]
[ ]
[ 16 16 16 16 ]
K=[ ],
[ 1 √3 1 √3 ]
[ ]
[ ]
[ 4 12 4 12 ]
[ ]
[ √3 1 √3 1 ]
[ 8 8 8 8 ]
3√6𝑐10 − √2𝑐11 −√2𝑐10 + √6𝑐11 0 0
1 [−√2𝑐10 + √6𝑐11 √6𝑐10 + 5√2𝑐11 0 0 ]
̃= [
C ],
4 [ 0 0 3 6𝑐20 − 2𝑐21 − 2𝑐20 + 6𝑐21 ]
√ √ √ √
[ 0 0 −√2𝑐20 + √6𝑐21 √6𝑐20 + 5√2𝑐21 ]
17√6
[ ]
[ 72 ]
[ ]
[ 5√2 ]
[ ]
[ ]
[ 24 ]
G=[ ].
[ 7√6 ]
[ ]
[ ]
[ 36 ]
[ ]
[ √2 ]
[ 6 ]
(39)
Equation (38) gives a system of nonlinear algebraic equations Example 2. Consider the first-order nonlinear Fredholm
that can be solved utilizing the initial condition (37); that is, integrodifferential equation [6, 17] as follows:
√6𝑐10 − √2𝑐11 = 0, we obtain 1
1 4
𝑥𝑦󸀠 (𝑥) − 𝑦 (𝑥) = − + 𝑥2 + ∫ (𝑥2 + 𝑡) 𝑦2 (𝑡) 𝑑𝑡,
6 5 0 (41)
√6 √2 0 ≤ 𝑥 < 1.
𝑐10 = , 𝑐11 = ,
24 8 with the initial condition
(40)
√6 √2 𝑦 (0) = 0. (42)
𝑐20 = , 𝑐21 = .
6 4
In this example, we have 𝑝0 = −1, 𝑝1 = 𝑥, 𝑔(𝑥) = −(1/6) +
(4/5)𝑥2 , 𝜆 = 1, 𝑘(𝑥, 𝑡) = 𝑥2 + 𝑡, and 𝑞 = 2.
Substituting these values into (7), the result will be 𝑦(𝑥) = 𝑥, The matrix (35) for this example is
that is, the exact solution. It is noted that the result gives the (P̃0 + P̃ 1 D𝑇 ) C − K (C)
̃ C = G, (43)
exact solution as in [17], while in [18] using the sinc method
the maximum absolute error is 1.52165 × 10−3 . where for 𝑛 = 2 and 𝑚 = 2 we have
1 √15 −√5
[ 12 0 0 0 ]
[ 60 120 ]
[ √15 1 √3 ]
[ 0 0 0 ]
[ 60 4 24 ]
[ ]
[ −√5 √3 5 ]
[ 0 0 0 ]
[ 120 24 12 ]
̃ 0 = −I,
P ̃ [
P1 = [ ],
[ 0 7 √ 15 − 5 ]
√ ]
[ 0 0 ]
[ 12 60 120 ]
[ √ 15 3 √3 ]
[ 0 0 0 ]
[ ]
[ 60 4 24 ]
[ −√5 √3 11 ]
0 0 0
[ 120 24 12 ]
7√15 11√10
[ −5 −2√5 0 0 0 ] [− 450 ]
[ 3 ] [ ]
[ −√15 14√3 ] [ −√6 ]
[ −3 0 0 0 ] [ ]
[ 3 ] 𝑐10 [ 90 ]
[ 3 ] [ ]
[ √
−8 3 ] [𝑐11 ] [ √2 ]
[ 0 8 0 0 0 ] [ ] [ ]
[ ] [𝑐12 ] [ 180 ]
𝑇
D =[ 3 ], C=[ ] [ ],
7√15 [𝑐20 ] , G=[ ]
[
[ 0 0 0 −5
]
−2√5 ] [ ] [ 23√10 ]
[ 3 ] [𝑐21 ] [ ]
[ ] [ 900 ]
−√15 14√3 ] [ 13√6 ]
[
[ 0 0 0 −3 ] [𝑐22 ] [ ]
[ 3 3 ] [ ]
[ ] [ 180 ]
−8√3 [ 19√2 ]
0 0 0 0 8
[ 3 ] [ 180 ]
1 √15 7√5 13 41√5√15

[ ]
[ 24 45 240 72 20
720 ]
[ ]
[ √15 1 5√3 √15 1 √3 ]
[ ]
[ ]
[ 72 12 144 2416 16 ]
[ ]
[ √5 5√3 1 7√5√3 5 ]
[ ]
[ ]
[ 48
K=[
144 24 14416 72 ]
], ̃ = [̃c1 0 ] ,
C
[ 7 31√15 √15 41 17√15 7√5 ] 0 ̃c2
[ ]
[ ]
[ 48 720 20 144 240 90 ]
[ ]
[ 7√15 3 5√3 11√15 13 7√3 ]
[ ]
[ ]
[ 144 16 72 144 48 72 ]
[ ]
[ √5 11√3 1 13√5 5√3 1 ]
[ 16 144 12 144 48 9 ]
5√10 5√6 √2 5√6 11√10 8√30 √2 8√30 3√10
[ 𝑐𝑗0 − 𝑐𝑗1 + 𝑐𝑗2 − 𝑐𝑗0 + 𝑐𝑗1 − 𝑐𝑗2 𝑐𝑗0 − 𝑐𝑗1 + 𝑐 ]
[ 7 21 7 21 35 105 7 105 35 𝑗2 ]
[ ]
[ 5√6 11√10 8√30 11 √10 3√6 √2 8√30 √2 5√6 ]
[
̃c𝑗 = [− 𝑐 ]
𝑐𝑗0 + 𝑐 − 𝑐 𝑐 + 𝑐 + 𝑐 − 𝑐 + 𝑐 + ],
[ 21 35 𝑗1 105 𝑗2 35 𝑗0 7 𝑗1 7 𝑗2 105 𝑗0 7 𝑗1 21 𝑗2 ]
[ ]
[ √2 8√30 3√10 8√30 √2 5√6 3√10 5√6 13√2 ]
𝑐 − 𝑐 + 𝑐 − 𝑐 + 𝑐 + 𝑐 𝑐 + 𝑐 + 𝑐
[ 7 𝑗0 105 𝑗1 35 𝑗2 105 𝑗0 7 𝑗1 21 𝑗2 35 𝑗0 21 𝑗1 7 𝑗2 ]
𝑗 = 1, 2.
(44)
Equation (43) gives a system of nonlinear algebraic 𝑚 = 30. Comparison among the proposed method and
equations that can be solved utilizing the initial condition methods in [17] is shown in Table 1. It is clear from this table
(42); that is, √10𝑐10 − √6𝑐11 + √2𝑐12 = 0, we obtain that the results obtained by the proposed method, using few
numbers of basis, are very promising and superior to that of
√10 √6 [17].
𝑐10 = , 𝑐11 = ,
240 48
Example 4. Consider the following nonlinear Fredholm inte-
√2 √10
𝑐12 = , 𝑐20 = , (45) grodifferential equation [5, 17]:
24 15
1
√6 √2 1 −2
𝑐21 = , 𝑐22 = . 𝑦󸀠 (𝑥) + 𝑦 (𝑥) = (𝑒 − 1) + ∫ 𝑦2 (𝑡) 𝑑𝑡, 0 ≤ 𝑥 < 1,
8 6 2 0
(48)
Substituting these values into (7), the result will be 𝑦(𝑥) = 𝑥2 ,
that is, the exact solution. It is noted that the result gives the
exact solution as in [17], while in [6] approximate solution is with the initial conditions
obtained with maximum absolute error 1.0000 × 10−5 .
𝑦 (0) = 1. (49)
Example 3. Consider the second-order nonlinear Fredholm
integrodifferential equation [17] as follows:
The exact solution of this problem is 𝑦(𝑥) = 𝑒−𝑥 . In Table 2 we
1 have compared the absolute difference errors of the proposed
𝑦󸀠󸀠 (𝑥) + 𝑥𝑦󸀠 (𝑥) − 𝑥𝑦 (𝑥) = 𝑒𝑥 sin 𝑥 + ∫ sin 𝑥⋅𝑒−2𝑡 𝑦2 (𝑡) 𝑑𝑡, method with the collocation method based on Haar wavelets
0
in [5] and method in [17].
0 ≤ 𝑥 < 1, Maximum absolute errors of Example 4 for some different
values of 𝑛 and 𝑚 are shown in Table 3. As it is seen from
(46)
Table 3, for a certain value of 𝑛 as 𝑚 increases the accuracy
with the initial conditions increases, and for a certain value of 𝑚 as 𝑛 increases the
accuracy increases as well. In case of 𝑚 = 1, the numerical
𝑦 (0) = 𝑦󸀠 (0) = 1. (47) solution obtained is based on orthonormal Bernstein poly-
The exact solution is 𝑦(𝑥) = 𝑒𝑥 . We solve this example by nomials only, while in case of 𝑛 = 0, the numerical solution
using the proposed method with 𝑛 = 2, 𝑚 = 30 and 𝑛 = 3, obtained is based on block-pulse functions only.
Table 2: Numerical comparison of absolute difference errors for Example 4.
Method of [5] Method of [17] The proposed method

𝑥 Number of collocation points 𝑛=7 𝑛 = 3, 𝑚 = 35 𝑛 = 4, 𝑚 = 15
𝑁 = 128
0.125 3.7591𝐸 − 007 2.4509𝐸 − 010 5.5200𝐸 − 011 1.6710𝐸 − 011
0.250 6.6413𝐸 − 007 1.0202𝐸 − 010 8.9982𝐸 − 011 3.9705𝐸 − 012
0.375 8.6917𝐸 − 007 1.6139𝐸 − 010 9.4606𝐸 − 011 1.2126𝐸 − 011
0.500 1.0020𝐸 − 006 3.2362𝐸 − 010 9.2457𝐸 − 011 1.8312𝐸 − 012
0.625 1.0757𝐸 − 006 1.9197𝐸 − 010 7.4991𝐸 − 011 8.1299𝐸 − 012
0.750 1.1029𝐸 − 006 6.6120𝐸 − 011 4.9442𝐸 − 011 7.7237𝐸 − 012
0.875 1.0944𝐸 − 006 2.2417𝐸 − 010 2.6083𝐸 − 011 2.5547𝐸 − 012
Table 3: Maximum absolute errors for different values of 𝑛 and 𝑚 for Example 4.
𝑚
𝑛
1 5 10 15 20 25 30 35
0 5.7735𝐸 − 01 1.1547𝐸 − 01 5.7735𝐸 − 02 3.8490𝐸 − 02 2.8868𝐸 − 02 2.3094𝐸 − 02 1.9245𝐸 − 02 1.6496𝐸 − 02
1 2.2361𝐸 − 01 8.9443𝐸 − 03 2.2361𝐸 − 03 9.9381𝐸 − 04 5.5902𝐸 − 04 3.5777𝐸 − 04 2.4845𝐸 − 04 1.8254𝐸 − 04
2 6.2994𝐸 − 02 5.0395𝐸 − 04 6.2994𝐸 − 05 1.8665𝐸 − 05 7.8743𝐸 − 06 4.0316𝐸 − 06 2.3331𝐸 − 06 1.4693𝐸 − 06
3 1.3889𝐸 − 02 2.2222𝐸 − 05 1.3889𝐸 − 06 2.7435𝐸 − 07 8.6806𝐸 − 08 3.5556𝐸 − 08 1.7147𝐸 − 08 9.2554𝐸 − 09
4 2.5126𝐸 − 03 8.0403𝐸 − 07 2.5126𝐸 − 08 3.3088𝐸 − 09 7.8519𝐸 − 10 2.5729𝐸 − 10 1.0340𝐸 − 10 4.7839𝐸 − 11
5 3.8521𝐸 − 04 2.4653𝐸 − 08 3.8521𝐸 − 10 3.3818𝐸 − 11 6.0189𝐸 − 12 1.5778𝐸 − 12 5.2841𝐸 − 13 2.0955𝐸 − 13
6 5.1230𝐸 − 05 6.5574𝐸 − 10 5.1230𝐸 − 12 2.9984𝐸 − 13 4.0023𝐸 − 14 8.3935𝐸 − 15 2.3425𝐸 − 15 7.9625𝐸 − 16
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http://dx.doi.org/10.1155/2013/282593
Research Article
Semi-Idealized Study on Estimation of Partly and Fully Space
Varying Open Boundary Conditions for Tidal Models
Jicai Zhang1,2 and Haibo Chen3,4

1
Institute of Physical Oceanography, Ocean College, Zhejiang University, Hangzhou 310058, China
2
MOE Key Laboratory of Coast and Island Development, Nanjing University, Nanjing 210093, China
3
Laboratory of Physical Oceanography, Ocean University of China, Qingdao 266100, China
4
China Offshore Environmental Services Ltd., Qingdao 266061, China
Correspondence should be addressed to Jicai Zhang; jicai zhang@163.com
Received 5 June 2013; Revised 1 September 2013; Accepted 1 September 2013
Copyright © 2013 J. Zhang and H. Chen. This is an open access article distributed under the Creative Commons Attribution
cited.
Two strategies for estimating open boundary conditions (OBCs) with adjoint method are compared by carrying out semi-idealized
numerical experiments. In the first strategy, the OBC is assumed to be partly space varying and generated by linearly interpolating
the values at selected feature points. The advantage is that the values at feature points are taken as control variables so that the
variations of the curves can be reproduced by the minimum number of points. In the second strategy, the OBC is assumed to be fully
space varying and the values at every open boundary points are taken as control variables. A series of semi-idealized experiments
are carried out to compare the effectiveness of two inversion strategies. The results demonstrate that the inversion effect is in inverse
proportion to the number of feature points which characterize the spatial complexity of open boundary forcing. The effect of ill-
posedness of inverse problem will be amplified if the observations contain noises. The parameter estimation problems with more
control variables will be much more sensitive to data noises, and the negative effects of noises can be restricted by reducing the
number of control variables. This work provides a concrete evidence that ill-posedness of inverse problem can generate wrong
parameter inversion results and produce an unreal “good data fitting.”
1. Introduction ultimate purpose of applying data assimilation method is to

reduce the data misfit between model results and various
The tides and tidal currents are the basic motion forms of observations, by either improving the models or dynamically
ocean water and play an important role in the research on interpolating the observations. Among all the data assim-
other processes, such as the storm surge, the circulation ilation methods, the 4DVAR is one of the most effective
and the estuarine dynamics [1, 2]. For tidal models, open and powerful approaches. It is based on the optimal control
boundary conditions (OBCs) are one of the most important methods and perturbation theory [8, 9]. This technique
parameters, which are determined by the physics of tides allows us to retrieve an optimal data for a given model from
and tidal currents. Therefore, how to obtain reasonable and heterogeneous observation fields [9]. It is an advanced data
accurate OBCs for regional tidal models has been a subject assimilation method which involves the adjoint method and
of ongoing research. Data assimilation methods have been has the advantage of directly assimilating various observa-
commonly used to optimize the open boundary conditions tions distributed in time and space into numerical models
[3–7]. while maintaining dynamical and physical consistency with
Data assimilation methods, especially the complex ones the model. The adjoint method is a powerful tool for
like four-dimensional variational (4DVAR), are developed parameter estimation. Navon [10] presented an important
on the base of rigorous mathematical theories, such as overview on the state of the art of parameter estimation
inverse problem theory and optimal control theory. The in meteorology and oceanography in view of application of
4DVAR data assimilation techniques to inverse parameter height of equilibrium tides; 𝑅 is the radius of the earth,
estimation problems. Zhang and Lu [7] studied the parameter 𝑎 = 𝑅 cos 𝜙; 𝑓 = 2Ω sin 𝜙, where Ω represents the angular
estimation problems with a three-dimensional tidal model speed of earth rotation; 𝑔 is the acceleration due to gravity,
with 4DVAR and also summarized relative works. More ℎ is the undisturbed water depth and 𝐻 = ℎ + 𝜁 denotes
recently, Kazantsev [9] briefly revealed the history of data the total water depth; 𝐴 is the coefficient of horizontal
assimilation starting from Lorenz’s pioneering work and then eddy viscosity; Δ is the Laplace operator and Δ(𝑢, V) =
deeply studied the sensitivity of a shallow-water model to 𝑎−1 [𝑎−1 𝜕𝜆 (𝜕𝜆 (𝑢, V)) + 𝑅−1 𝜕𝜙 (cos 𝜙𝜕𝜙 (𝑢, V))]; 𝐹𝜆 and 𝐹𝜙 are east
parameters by applying adjoint based technique. and north components of bottom friction terms, respectively,
For parameter estimation problems, it is of great impor- and their expressions are given in quadratic form:
tance to reasonably reduce the number of spatially varying
control variables because of the ill-posedness of inverse √𝑢 2 + V 2 √𝑢2 + V2
problem. As noted by Yeh in the work of ground water 𝐹𝜆 = −𝐶𝑄 𝑢, 𝐹𝜙 = −𝐶𝑄 V. (2)
ℎ+𝜁 ℎ+𝜁
flow parameter estimation, the inverse or parameter esti-
mation problem is often ill-posed and beset by instability
and nonuniqueness, particularly if one seeks parameters dis- 2.2. The Adjoint. The general idea of the adjoint method
tributed in space and time domain [11]. The same viewpoint is described as follows. First, a model is defined by an
has been put forward by references [12–16]. Consequently, algorithm and its independent variables such as initial con-
how to reduce the number of parameters to be estimated ditions, boundary conditions, and empirical parameters. The
became an important aspect needing to draw attention to cost function which measures the data misfit between the
[13–17]. In this work two strategies for inverting the open modeling results and observations is then minimized through
boundary conditions with adjoint method are compared by optimizing the control variables. In detail, the cost function
carrying out semi-idealized numerical experiments. In the decreases along the opposite direction of the gradients with
first strategy, the OBC is assumed to be partly space varying respect to the control variables, and this gradient is calculated
and generated by linearly interpolating the values at selected by what has become known as the adjoint model. In order to
feature points. The feature points are selected by calculating construct the adjoint equations, the cost function is defined
the second-order derivatives of discrete curves and the values as
at selected feature points are taken as control variables to be 1 ̂ 2 𝑑𝑆 𝑑𝑇,
𝐽 (𝜁) = 𝐾𝜁 ∬ (𝜁 − 𝜁) (3)
estimated. The advantage is that most of the variations of the 2 Ω𝑇,𝑆
curves can be reproduced by the minimum number of points.
In the second strategy, the OBC is assumed to be fully space and the Lagrangian function is defined as
varying and the values at every open boundary points are
taken as control variables.
This paper is organized as follows. The 2D tidal model
𝐿=∬ [𝜇 ( 𝜕𝑢 + 𝑢 𝜕𝑢 + V 𝜕𝑢 − 𝑢V tan 𝜙
with adjoint is briefly described in Section 2. The two inver- Ω𝑇,𝑆 𝜕𝑡 𝑎 𝜕𝜆 𝑅 𝜕𝜙 𝑅
sion strategies are developed in Section 3. A series of semi- [
idealized numerical experiments are carried out and the
𝑔 𝜕 (𝜁 − 𝜁)
results are analyzed and discussed in Section 4. Conclusions −𝑓V − 𝐹𝜆 − 𝐴Δ𝑢 + )
in Section 5 complete the paper. 𝑎 𝜕𝜆
2. The Adjoint Tidal Model 𝜕V 𝑢 𝜕V V 𝜕V 𝑢2 tan 𝜙

+ ]( + + +
𝜕𝑡 𝑎 𝜕𝜆 𝑅 𝜕𝜙 𝑅
2.1. The 2D Tidal Model. The governing equations for the
tides used in the present study are the vertically integrated (4)
𝑔 𝜕 (𝜁 − 𝜁)
equations of continuity and momentum: +𝑓𝑢 − 𝐹𝜙 − 𝐴ΔV + )
𝑅 𝜕𝜙
𝜕𝜁 1 𝜕 [(ℎ + 𝜁) 𝑢] 1 𝜕 [(ℎ + 𝜁) V cos 𝜙]
+ + = 0,
𝜕𝑡 𝑎 𝜕𝜆 𝑎 𝜕𝜙 𝜕𝜁 1 𝜕 [(ℎ + 𝜁) 𝑢]
+ 𝜏( +
𝑔 𝜕 (𝜁 − 𝜁) 𝜕𝑡 𝑎 𝜕𝜆
𝜕𝑢 𝑢 𝜕𝑢 V 𝜕𝑢 𝑢V tan 𝜙
+ + − − 𝑓V − 𝐴Δ𝑢 + = 𝐹𝜆 ,
𝜕𝑡 𝑎 𝜕𝜆 𝑅 𝜕𝜙 𝑅 𝑎 𝜕𝜆
1 𝜕 [(ℎ + 𝜁) V cos 𝜙] ]
+ ) 𝑑𝑆 𝑑𝑇
𝜕V 𝑢 𝜕V V 𝜕V 𝑢 tan 𝜙2
𝑔 𝜕 (𝜁 − 𝜁) 𝑎 𝜕𝜙
+ + + + 𝑓𝑢 − 𝐴ΔV + = 𝐹𝜙 , ]
𝜕𝑡 𝑎 𝜕𝜆 𝑅 𝜕𝜙 𝑅 𝑅 𝜕𝜙
+ 𝐽 (𝜁) ,
(1)
where 𝑡 is time; 𝜆 and 𝜙 are the east longitude and north where 𝜁̂ is the observations of surface elevation; Ω𝑇,𝑆 stands
latitude, respectively; 𝜁 is the sea surface elevation above for the whole integration area of time and space; 𝜇, ], and 𝜏
the undisturbed sea level; 𝑢 and V are the east and north are the adjoint variables (namely, Lagrangian multipliers) of
components of fluid velocity, respectively, 𝜁 is the adjusted 𝑢, V, and 𝜁, respectively. Based on the theory of Lagrangian
0.8
FP(3) and GP[II(3)]
0.7
0.6
0.5
Value (m)
FP(1) and GP[II(1)] FP(M − 1) and GP[II(M − 1)]
0.4
FP(k) and GP[II(k)]
0.3
0.2
0.1 FP(k − 1) and GP[II(k − 1)] FP(k + 1) and GP[II(k + 1)]

FP(2) and GP[II(2)] FP(M) and GP[II(M)]
0
0 20 40 60 80 100 120 140 160 180
Open boundary grid points
General points
Feature points
Figure 1: Example of discrete curves and their feature points. GP stands for general points and FP indicates feature points.
multiplier method, we have the following first-order derivates where Ψ(𝑖, 𝑗) (1 ≤ 𝑖 ≤ 3, 1 ≤ 𝑗 ≤ 2) is a matrix whose
of Lagrangian function with respect to all the model variables: components denote the adjoint terms of bottom friction. The
components of Ψ for the quadratic parameterizations are
𝜕𝐿 𝜕𝐿 𝜕𝐿
= 0, = 0, = 0, (5a) given as
𝜕𝜁 𝜕𝑢 𝜕V
𝜕𝐿 𝜕𝐿 𝜕𝐿 𝐶𝑄𝑢√𝑢2 + V2 𝐶𝑄V√𝑢2 + V2 }
= 0, = 0, = 0, (5b) {
{ −𝜇 −] }
𝜕𝜏 𝜕𝜇 𝜕] {
{ 2 2 }
}
{
{ (ℎ + 𝜁) (ℎ + 𝜁) }
}
{
{ }
}
𝜕𝐿 𝜕𝐿 𝜕𝐿 {
{ }
}
= 0, = 0, = 0. {
{ 𝐶 (2𝑢 2
+ 2
) 𝐶𝑄𝑢V }
}
𝜕𝐶𝑄 𝜕𝑎 𝜕𝑏
(5c) { 𝑄 V }
Ψ = {𝜇 ] , (7)
{
{
{
√ 2
(ℎ + 𝜁) 𝑢 + V 2 (ℎ + 𝜁) 𝑢 + V }
√ 2 2 }
}
}
Equations (5b) give the original governing (1) and the adjoint {
{ }
}
{
{ }
equations can be developed from (5a). In (5c), 𝑎 and 𝑏 are the {
{ 𝐶𝑄𝑢V 𝐶𝑄 (𝑢2 + 2V2 ) }}
}
{ }
Fourier coefficients along the open boundary and 𝐶𝑄 denotes {𝜇
{
√ 2 2
]
√ 2 2
}
}
(ℎ + 𝜁) 𝑢 + V (ℎ + 𝜁) 𝑢 + V
the bottom friction coefficients. From (5c) we can obtain the { }
optimization formulae of model parameters.
Based on (5a) the adjoint equations can be obtained as The numerical schemes for the forward model and the adjoint
model in this section are both based on Lu and Zhang [17] and
𝜕𝜏 𝑢 𝜕𝜏 V 𝜕 (𝜏 cos 𝜙) 𝑔 𝜕𝜇 𝑔 𝜕] ̂ Zhang et al. [18].
+ + + + − 𝐾𝜁 (𝜁 − 𝜁)
𝜕𝑡 𝑎 𝜕𝜆 𝑎 𝜕𝜙 𝑎 𝜕𝜆 𝑎 𝜕𝜙
= Ψ (1, 1) + Ψ (1, 2) ,
3. Methodology
𝜕𝜇 𝜇 𝜕𝑢 ] 𝜕V 1 𝜕 1 𝜕
− 𝑓] − − + (𝜇𝑢) + (𝜇V) 3.1. Feature Points of a Curve. If the values of OBCs are
𝜕𝑡 𝑎 𝜕𝜆 𝑎 𝜕𝜆 𝑎 𝜕𝜆 𝑅 𝜕𝜙
plotted versus the location or index of grid points along open
ℎ + 𝜁 𝜕𝜏 ℎ + 𝜁 𝜕𝜏 𝜇V tan 𝜙 boundaries, they will form a discretized curve. Without loss
+ + + 𝐴Δ𝜇 + of generality, the curve can be presented by Figure 1. Assume
𝑎 𝜕𝜆 𝑎 𝜕𝜆 𝑅
(6) there are 𝑁 general (or, computational) points along open
2]𝑢 tan 𝜙 boundaries with index of GP(𝑘), 𝑘 = 1, 2, . . . , 𝑁. This type
− = Ψ (2, 1) + Ψ (2, 2) ,
𝑅 of curve can be approximately linearly expressed by a certain
𝜕] 𝜇 𝜕𝑢 ] 𝜕V 1 𝜕 1 𝜕 series of points which are defined as feature points in this
+ 𝑓𝜇 − − + (]𝑢) + (]V) paper. For the curve shown in Figure 1, one can easily obtain
𝜕𝑡 𝑅 𝜕𝜙 𝑅 𝜕𝜙 𝑎 𝜕𝜆 𝑅 𝜕𝜙
the feature points as indicated by symbol “+.” Assume the
ℎ + 𝜁 𝜕 (𝜏 cos 𝜙) ℎ + 𝜁 𝜕 (𝜏 cos 𝜙) number of feature points is 𝑀 with index of FP(𝑗), 𝑗 =
+ + + 𝐴Δ] 1, 2, . . . , 𝑀. Further assuming the feature point with index
𝑎 𝜕𝜙 𝑎 𝜕𝜙
of 𝑗 is coincident with the general point with index of II(𝑗),
𝜇𝑢 tan 𝜙 we can obtain the following relation: II(1) = 1, II(𝑀) = 𝑁,
+ = Ψ (3, 1) + Ψ (3, 2) , II(𝑗) = 𝑘, 2 < 𝑘 < 𝑁 − 1.
𝑅
It is easy to conclude that any general point can be linearly For the whole curve (or the whole boundary), the relation
expressed by two adjacent feature points. For example, as between general points and feature points can be similarly
shown in Figure 1, an arbitrary general point GP(𝑘) locates expressed in matrix form as
between two adjacent feature points FP(𝑗 − 1) and FP(𝑗),
where II(𝑗 − 1) ≤ 𝑘 ≤ II(𝑗). Through linear interpolation,
we can obtain the value of GP(𝑘) as
VGP = WFG × VFP , (9)
GP (𝑘)
II (𝑗) − 𝑘 𝑘 − II (𝑗 − 1)
= FP (𝑗 − 1) + FP (𝑗) .
II (𝑗) − II (𝑗 − 1) II (𝑗) − II (𝑗 − 1)
where VGP and VFP are both column vectors with dimensions
of 𝑁 and 𝑀, respectively, and WFG is the weighting matrix of
linear interpolation with dimensions of 𝑁 × 𝑀. The detailed
(8) forms of three matrixes are given as
VGP = [GP (1) , GP (2) , . . . , GP (𝑁)]𝑇 , (10)
VFP = [FP (1) , FP (2) , . . . , FP (𝑀)]𝑇 , (11)
𝑤1,1 0 0 0 0 0 0 0
𝑤2,1 𝑤22 0 0 0 0 0 0
.. ..
( . . 0 0 0 0 0 0 )
( )
(𝑤II(2)−1,1 𝑤II(2)−1,2 0 0 0 0 0 0 )
( )
( 0 𝑤II(2),2 0 0 0 0 0 0 )
( )
( 0 0 d 0 0 0 0 0 )
( )
( 0 0 0 𝑤II(𝑗−1),𝑗−1 0 0 0 0 )
( )
( 0 0 0 𝑤II(𝑗−1)+1,𝑗−1 𝑤II(𝑗−1)+1,𝑗 0 0 0 )
( )
( . . )
WFG =( 0 0 0 .
. .
. 0 0 0 ),
( ) (12)
( 0 0 0 𝑤II(𝑗)−1,𝑗−1 𝑤II(𝑗)−1,𝑗 0 0 0 )
( )
( 0 0 0 0 𝑤II(𝑗),𝑗 0 0 0 )
( )
( 0 0 0 0 0 0 0 )
( d )
( 0 0 0 0 0 0 𝑤 0 )
( II(𝑀−1),𝑀−1 )
( 0 0 0 0 0 0 𝑤 𝑤 )
( II(𝑀−1)+1,𝑀−1 II(𝑀−1)+1,𝑀 )
( .. .. )
0 0 0 0 0 0 . .
0 0 0 0 0 0 𝑤II(𝑀)−1,𝑀−1 𝑤II(𝑀)−1,𝑀
( 0 0 0 0 0 0 𝑤II(𝑀),𝑀 )
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
𝑀 columns, 𝑁 rows
where the nonzero components are the linear interpolation selected feature points. It indicates that the OBC identifica-
coefficients. Specifically, without loss of generality, tion problem can be transformed to seek the values of a few
selected feature points, which reduces the number of control
𝑤II(𝑗),𝑗 = 1.0, 𝑗 = 1, 2, . . . , 𝑀, (13a) variables.
II (𝑗) − II (𝑗 − 1) − 𝑚
𝑤II(𝑗−1)+𝑚,𝑗−1 = , 3.2. Selection of Feature Points for Periodic Tidal Open Bound-
II (𝑗) − II (𝑗 − 1) (13b) ary. Along a certain open boundary, we also assume that
1 ≤ 𝑚 < II (𝑗) − II (𝑗 − 1) , there are 𝑁 general grid points. The height of water level 𝜁
at the 𝑛th time step is given by
𝑚
𝑤II(𝑗−1)+𝑚,𝑗 = , 𝑛
𝜁GP(𝑘) = 𝑎0 + [𝑎GP(𝑘) cos (𝜔𝑛Δ𝑡) + 𝑏GP(𝑘) sin (𝜔𝑛Δ𝑡)] , (14)
II (𝑗) − II (𝑗 − 1)
(13c)
1 ≤ 𝑚 < II (𝑗) − II (𝑗 − 1) . where GP(𝑘) stands for the general points of open boundaries
and 1 ⩽ 𝑘 ⩽ 𝑁, 𝜔 is the frequency of 𝑀2 constituent, 𝑎GP(𝑘)
Using (9), any general points along open boundaries can and 𝑏GP(𝑘) are the Fourier coefficients at GP(𝑘), Δ𝑡 is the time
be highly approximated through the linear interpolation of step of computation.
For regional tidal models the values of 𝑎GP(𝑘) and 𝑏GP(𝑘) The Broyden-Fletcher-Goldfarb-Shanno (BFGS) meth-
can be obtained from large scale numerical models. It should od, which is a quasi-Newton conjugate-gradient algorithm,
be noted 𝑎GP(𝑘) and 𝑏GP(𝑘) are space dependent, and therefore has been widely used in the unconstrained inverse prob-
the variations of their values versus the grids along the open lems and is famous for its efficiency [19, 20]. The limited-
boundary will constitute two curves (curve 𝑎 and curve 𝑏) memory BFGS (L-BFGS) algorithm is an adaptation of the
similar to the one shown in Figure 1. The feature points for BFGS method to large problem. Zou et al. [20] concluded
this type of curve can be selected by computing the second- that among the tested quasi-Newton methods, the L-BFGS
order differential of each general point. The detailed selection method had the best performance. In this work L-BFGS
procedures are given as follows. method is employed to optimize the control variables,
namely, the OBCs. In order to perform inversion with L-
(1) Suppose the absolute values of second-order differen- BFGS, the gradients of cost function with respect to the
tials of general points GP(𝑘) are SD 𝑎(𝑘) for curve 𝑎 control variables in two strategies have to be calculated.
and SD 𝑏(𝑘) for curve 𝑏, respectively. For the general
points locating in the middle of curve a and curve b,
3.3.1. Gradients for Partly Space Varying Inversion Strategy.
that is, 2 ⩽ 𝑘 ⩽ 𝑁 − 1, SD 𝑎(𝑘) and SD 𝑏(𝑘) can be
In the first inversion strategy (partly space varying OBC),
computed as
feature points for open boundary curves are selected and
󵄨󵄨 𝑎 󵄨 the OBCs at general points can be linearly interpolated from
󵄨 GP(𝑘+1) − 2𝑎GP(𝑘) + 𝑎GP(𝑘−1) 󵄨󵄨󵄨
SD 𝑎 (𝑘) = 󵄨󵄨󵄨 󵄨󵄨 , feature points. Consequently, the gradients of cost function
󵄨󵄨 2Δ𝑑 󵄨󵄨
(15) with respect to the Fourier coefficients at feature points
󵄨󵄨 𝑏 󵄨
󵄨 GP(𝑘+1) − 2𝑏GP(𝑘) + 𝑏GP(𝑘−1) 󵄨󵄨󵄨 𝑎𝑎FP(𝑗) and 𝑏𝑏FP(𝑗) (𝑎𝑎𝑗 and 𝑏𝑏𝑗 for simplicity, 1 ⩽ 𝑗 ⩽ 𝑀)
SD 𝑏 (𝑘) = 󵄨󵄨󵄨󵄨 󵄨󵄨 ,
󵄨󵄨 2Δ𝑑 󵄨󵄨󵄨 have to be computed in order to optimize the OBCs with L-
BFGS. The gradients are deduced from
where Δ𝑑 is the size of computation grids and equals 𝜕𝐿 𝜕𝐿
Δ𝑥 or Δ𝑦 according to the direction of open bound- = 0, = 0, 1 ≤ 𝑗 ≤ 𝑀, (17)
𝜕𝑎𝑎𝑗 𝜕𝑏𝑏𝑗
aries (Δ𝑥 for west-east direction and Δ𝑦 for north-
south direction). which yields
(2) Further define that the “maximum second-order dif-
𝑖(2)−1
ferential” for point GP(𝑘) is SD(𝑘). The value of SD(𝑘) 𝜕𝐽
is calculated as + ∑ 𝑤𝑘,1 ∑ 𝑇𝑘𝑛 cos (𝜔𝑛Δ𝑡) = 0,
𝜕𝑎𝑎1 𝑘=1 𝑛∈Ω 𝑇
SD (𝑘) = max [SD 𝑎 (𝑘) , SD 𝑏 (𝑘)] . (16) 𝑗 = 1,
(3) Define a threshold value of SD(𝑘), 2 ⩽ 𝑘 ⩽ 𝑁 − 1, 𝜕𝐽

𝑖(𝑗+1)
to be 𝑇SD . The points with larger values of SD(𝑘) + ∑ 𝑤 ∑ 𝑇𝑛 cos (𝜔𝑛Δ𝑡) = 0,

𝜕𝑎𝑎𝑗 𝑘=𝑖(𝑗−1)+1 𝑘,𝑗 𝑛∈Ω 𝑘
than 𝑇SD are selected as feature points. The value of 𝑇
𝑇SD is problem dependent and should be determined

2 ≤ 𝑗 ≤ 𝑀 − 1,
according to the specific requirement on the number
of control variables. 𝑁
𝜕𝐽
(4) It is easy to understand that the first and the last + ∑ 𝑤𝑘,𝑗 ∑ 𝑇𝑘𝑛 cos (𝜔𝑛Δ𝑡) = 0,
𝜕𝑎𝑎𝑀 𝑘=𝑖(𝑗−1)+1 𝑛∈Ω
general points GP(1) and GP(𝑁) are automatically 𝑇
selected as feature points indexed as FP(1) and 𝑗 = 𝑀,

FP(𝑀). (18)
𝑖(2)−1
𝜕𝐽
3.3. Inversion Strategies and Gradients. In this work two + ∑ 𝑤𝑘,1 ∑ 𝑇𝑘𝑛 cos (𝜔𝑛Δ𝑡) = 0,
𝜕𝑏𝑏1 𝑘=1 𝑛∈Ω
strategies for inverting the open boundary conditions with 𝑇
adjoint method are compared by carrying out semi-idealized 𝑗 = 1,

numerical experiments. In the first strategy the open bound-
ary curves are assumed to be partly space varying and 𝑖(𝑗+1)
𝜕𝐽
are generated by linearly interpolating the values at feature + ∑ 𝑤 ∑ 𝑇𝑛 cos (𝜔𝑛Δ𝑡) = 0,
points. The feature points are selected by calculating the 𝜕𝑏𝑏𝑗 𝑘=𝑖(𝑗−1)+1 𝑘,𝑗 𝑛∈Ω 𝑘
𝑇
second-order derivatives of discrete curves and the values at
selected feature points are taken as control variables to be 2 ≤ 𝑗 ≤ 𝑀 − 1,
estimated. The advantage is that most of the variations of 𝑁
the curves can be reproduced by the minimum number of 𝜕𝐽
+ ∑ 𝑤 ∑ 𝑇𝑛 cos (𝜔𝑛Δ𝑡) = 0,
points. In the second strategy, the OBC is assumed to be fully 𝜕𝑏𝑏𝑀 𝑘=𝑖(𝑀−1)+1 𝑘,𝑗 𝑛∈Ω 𝑘
𝑇
space varying and the values at every open boundary point
are taken as control variables. 𝑗 = 𝑀,
Figure 2: The bathymetric map of the Bohai, Yellow, and East China Seas (contour) and the position of 𝑇/𝑃 satellite tracks (dot), tidal gauge
stations (plus), and open boundaries (open circle). The numbers are the water depth in meter.
where for simplicity, 1 ⩽ 𝑘 ⩽ 𝑁) have to be computed. The gradients

are deduced from
𝑔𝜇𝑘𝑛 𝜕𝐿 𝜕𝐿
𝑇𝑘𝑛 = − = 0, = 0, 1 ≤ 𝑘 ≤ 𝑁, (20)
Δ𝑥 𝜕𝑎𝑎𝑘 𝜕𝑏𝑏𝑘
(for GP (𝑘) on the right of the area calculated) , which yields

𝑗
𝑔𝜇𝑘 𝜕𝐽
+ ∑ 𝑇𝑛 cos (𝜔𝑛Δ𝑡) = 0, 1 ≤ 𝑘 ≤ 𝑁,
𝑇𝑘𝑛 = 𝑙
𝜕𝑎𝑎𝑘 𝑛∈Ω 𝑘
Δ𝑥 𝑇
(21)
(for GP (𝑘) on the left of the area calculated) , 𝜕𝐽
+ ∑ 𝑇𝑛 cos (𝜔𝑛Δ𝑡) = 0, 1 ≤ 𝑘 ≤ 𝑁,
𝜕𝑏𝑏𝑘 𝑛∈Ω 𝑘
𝑔]𝑛𝑘 𝑇
𝑇𝑘𝑛 = −
Δ𝑦 where 𝑇𝑘𝑛 can also be computed by using (19).
(for GP (𝑘) under the area calculated) ,
4. Numerical Experiments and Results
𝑔]𝑛 Analysis
𝑇𝑘𝑛 = 𝑘
Δ𝑦
4.1. Model Settings. The computing area in the present study
(for GP (𝑘) above the area calculated) , is the Bohai Sea, the Yellow Sea, and the East China Sea
(19) (BYECS), typical marginal shelf seas. The spatial resolution
for the model is 1/12∘ × 1/12∘ . 𝑇/𝑃 altimeter data and
tidal gauge data are assimilated into the tidal model. The
where 𝜇 and ] are the adjoint variables of west-east velocity bathymetry map of the BYECS, the position of 𝑇/𝑃 satellite
component 𝑢 and north-south velocity component V, respec- tracks, tidal gauge stations, and the open boundaries are
tively. The values of 𝜇 and ] are computed by running the shown in Figure 2. Since the purpose of this paper is to dis-
adjoint model. cuss the inversion of OBCs, the bottom friction coefficients
are fixed in all the experiments.
The numerical experiments in this work are semi-
3.3.2. Gradients for Fully Space Varying Inversion Strategy. In idealized. Specifically, the coastline, the number, and location
the second strategy, the OBC is assumed to be fully space of the observations are real. On the contrary, the values of
varying and the values at every open boundary points (i.e., open boundary conditions and observations are artificial. The
general points) are taken as control variables. Consequently, prescribed open boundary curves are generated by different
the gradients of cost function with respect to the Fourier number of feature points. Apparently, the complexity of open
coefficients at general points 𝑎𝑎GP(𝑘) and 𝑏𝑏GP(𝑘) (𝑎𝑎𝑘 and 𝑏𝑏𝑘 boundary curves is in direct proportion to the number of
feature points. For the semi-idealized experiments, only the out to calibrate the inversion ability of adjoint model and
location of real observations (satellite altimetry and tidal compare the effectiveness of two strategies developed in
gauge stations) is used and the values of “observations” Section 3. The prescribed distributions of artificial Fourier
are obtained by running the dynamic forward model with coefficients at 173 grid points along the eastern open bound-
prescribed open boundary conditions. The advantage of this ary are inverted. The prescribed distributions (PDs) are
kind of experiments is that we can obtain a thorough under- designed to be characterized by different numbers of feature
standing of the “observations.” The “observations” generated points. PDs 1–7 are characterized by 2, 6, 10, 14, 18, 22,
by the model can be accurate and we can control the quality and 26 feature points, respectively. The twin experiments are
of the “observations” by adding artificial error. In addition, correspondingly indexed with SEa 1–7 for inversion strategy
because the other factors are real, the conclusions based 1 and SEb 1–7 for inversion strategy 2.
on these semi-idealized experiments can be more useful for The prescribed and inverted distributions of open bound-
referring. ary curves in SEa 1–4 and SEb 1–4 are shown in Figure 3.
The semi-idealized numerical experiments are run as The prescribed and inverted distributions of open boundary
follows. First a distribution of artificial Fourier coefficients curves in SEa 5-6 and SEb 5-6 are shown in Figure 4. The
is prescribed and taken as “true values” of open boundary feature points for prescribed distributions have also been
conditions. Then the forward tidal model is run using the indicated in Figures 3 and 4. Table 1 gives the error statistics
“true values” and the simulation results recorded at grid for the experiments in this section. The 𝐿 2 norm of the
points of 𝑇/𝑃 satellite tracks and tidal gauge stations are taken gradients of cost function with respect to the control variables
as the “observations.” Having obtained the “observations”, an versus the iteration steps for the experiments using inversion
initial value (taken as zero in this work) of Fourier coefficients strategies 1 and 2 are presented in Figures 4(c) and 4(d),
is assigned to run the forward model. The differences between respectively. The decrease in data misfit (i.e., cost function)
simulated values and “observations” will function as the calculated from (3) versus the iteration steps is shown in
external force to drive the adjoint model. The optimized
Figure 5. Note that the values of data misfit and 𝐿 2 norm of
Fourier coefficients can be obtained through the backward
gradients have been normalized by their values at the first
integration of the adjoint equations. The inverse integral
iteration step.
time of the adjoint equations is equal to a period of 𝑀2
tide. With the procedures repeated above, the parameters For strategy 1, the values of data misfit can sharply
will be optimized continuously and the difference between decrease by about 4 orders for all the experiments in about
simulated values and “observations” will be diminished. 30 iteration steps. For strategy 2, the values of data misfit
Meanwhile, the difference between the prescribed and the can sharply decrease by about 5 orders for SEb 1–5 and by 4
inverted parameters will also be decreased. orders for SEb 6-7 in about 60 iteration steps. The decrease
The iteration of optimization will terminate once the in data misfit provides another proof for the inversion
following criterion is achieved [21]: ability of the adjoint model and strategies in this work.
Correspondingly, the 𝐿 2 norms of gradients also decrease by
‖𝐺‖ < eps × max (1, ‖𝑋‖) , (22)
at least 2 orders for inversion strategy 1 and by 3 orders for
where ‖𝐺‖ is the 𝐿 2 norm of the gradients of cost func- inversion strategy 2, which demonstrates that the gradients
tion with respect to the control variables (i.e., the Fourier calculated in Section 3.3 can work well with L-BFGS method.
coefficients at feature points), eps is a positive variable that From the decrease in data misfit and gradient it seems
determines the accuracy with which the solution is to be as if the effect of inversion strategy 2 is better than that
found, and ‖𝑋‖ is the 𝐿 2 norm of control variables. Both the of strategy 1. However, the differences between prescribed
values of ‖𝐺‖ and ‖𝑋‖ vary along the iterations. For a correct and inverted distributions shown in Table 1 indicate that the
adjoint model and a reasonable method, ‖𝐺‖ will gradually inversion results of strategy 1 are much better than those of
decrease versus the iteration steps and the inverted values strategy 2. This inconsistency will be explained in Section 4.3.
of control variables must gradually approach the prescribed One can find that the adjoint model combined with inversion
“true values”. When using L-BFGS, the number of corrections strategy 1 can reproduce the prescribed distributions of
used in the BFGS update is taken as 5 (usually between 3 and Fourier coefficients perfectly for SEa 1-2 or almost perfectly
7, see Alekseev et al. [19]). In the minimization algorithm, the for SEa 3-4. For SEa 5-6 the inversion is acceptable but
control variables should be scaled to similar magnitudes on
largely deviates from perfection. The major trend of the
the order of unity because within the optimization algorithm
inversion is quite obvious that the effect of inversion is in
convergence, tolerances, and other criteria are based on an
inverse proportion to the number of feature points which
implicit definition of small and large [22]. Zou et al. [20]
characterizes the complexity of open boundary curves. The
also proved that the efficiency could be greatly improved by
a simple scaling. In twin experiments we use 10 to scale the inverted open boundary curves shown in Figures 3 and 4 also
Fourier coefficients [4]. prove that the inversion using strategy 1 is better than that
using strategy 2.
4.2. Modeling Results
4.2.2. Effects of Data Noises. As we know, the real obser-
4.2.1. Effects of Complexity of Open Boundary Curves. In vations either from satellite altimetry or from tidal gauge
this section, the semi-idealized experiments (SE) are carried stations contain errors (or noises). In this section the effects
1 1
0.5 0.5
Value (m)
Value (m)
0 0
−0.5 −0.5
−1 −1
0 50 100 150 200 0 50 100 150 200
Open boundary grid points Open boundary grid points
FCa of PD 1 Inverted FCb in SEa 1 FCa of PD 2 Inverted FCb in SEa 2

Inverted FCa in SEa 1 Inverted FCb in SEb 1 Inverted FCa in SEa 2 Inverted FCb in SEb 2
Inverted FCa in SEb 1 Feature points Inverted FCa in SEb 2 Feature points
FCb of PD 1 FCb of PD 2
(a) (b)
1 1
0.5 0.5
Value (m)
Value (m)
0 0
−0.5 −0.5
−1 −1
0 50 100 150 200 0 50 100 150 200

(c) (d)
Figure 3: The prescribed and inverted distributions of open boundary curves in SEa 1–4 and SEb 1–4. The feature points are indicated by
open circles.
of the noises are studied. To do this, we replace each “obser- When 𝑃 was increased to 20%, the value of this difference
vation” 𝜁̂𝑖,𝑗
𝑛 𝑛 ̂𝑛
by (1 + 𝑝𝑟𝑖,𝑗 𝑛
)𝜁𝑖,𝑗 , where 𝑟𝑖,𝑗 are uniform random is also increased to 0.0562 (Table 5). However, for strategy
numbers lying in [−1, 1] and 𝑝 is a factor determining the 1 the values of this difference are just 0.0011, 0.0011, 0.0032
maximum percentage error. The maximum percentage errors and 0.0043 under 𝑃 value of 5%, 10%, 15%, and 20%. Similar
for each prescribed distribution (PDs 1–7) are assigned to 5%, results can be found from the inversion results of other
10%, 15%, and 20%. The corresponding inversion experiments distributions. This phenomenon indicates that the effect of
are then indexed with SE𝑥 i.1, SE𝑥 i.2, SE𝑥 i.3, and SE𝑥 i.4, ill-posedness of inverse problem will be amplified in the
respectively, where 1 ⩽ 𝑖 ⩽ 7 and 𝑥 = 𝑎 or 𝑏. The error conditions that observations contain noises. In addition, the
statistics for the experiments with 𝑃 values of 5%, 10%, 15%, parameter estimation problems with more control variables
and 20% are exhibited in Tables 2, 3, 4, and 5, respectively. will be much more sensitive to data noise and the negative
The figures are omitted because they are similar to those in effect of noises can be restricted by reducing the number of
Section 4.2.1. control variables.
One can find the noises in artificial observations will
significantly and negatively influence the inversion of open 4.3. Discussions
boundary conditions. It is clear that the inversion using
strategy 2 is much more sensitive to the noise than that using 4.3.1. Rationality of the Adjoint Method (Suggested by an
strategy 1. For example, when the simplest distribution PD 1 Anonymous Reviewer). The motivation of the present work
is inverted, the difference between prescribed and inverted is to take the open boundary condition as an example to
values will sharply increase from 0.0101 (Table 1) to 0.0238 investigate the performance of the adjoint method when
(Table 2) for strategy 2 even with a small value of error 5%. applied to ocean modeling and the ill-posedness of relevant
1 1
0.5 0.5
Value (m)
Value (m)
0 0
−0.5 −0.5
−1 −1
0 50 100 150 200 0 50 100 150 200

(a) (b)
0 0
−1
−2
Log of values
Log of values
−2
−4
−3
−6
−4
−8 −5
0 10 20 30 40 50 60 70 0 50 100 150 200
Iteration steps Iteration steps
Gradient in SEa 1 Gradient in SEa 5 Gradient in SEb 1 Gradient in SEb 5

Gradient in SEa 4 Gradient in SEb 4
(c) (d)
Figure 4: (a), (b) The prescribed and inverted distributions of open boundary curves in SEa 5-6 and SEb 5-6. The feature points are indicated
by open circles. (c), (d) The 𝐿 2 norm of the gradients of cost function with respect to the control variables versus the iteration steps for
strategies 1 and 2.
0 0
Log of values
Log of values
−2 −1
−4 −2
−6 −3
−8 −4
−10 −5
−12 −6
0 10 20 30 40 50 60 70 0 20 40 60 80 100 120 140 160 180 200
Iteration steps Iteration steps
Data misfit in SEa 1 Data misfit in SEa 5 Data misfit in SEb 1 Data misfit in SEb 5
Data misfit in SEa 4 Data misfit in SEb 4
(a) (b)
Figure 5: Data misfit versus the iteration steps for strategy 1 (a) and strategy 2 (b).
Table 1: Error statistics for SEa 1–7 and SEb 1–7.

𝐾3a 𝐾4a (m)
Exp. 𝐾1a 𝐾2a
Before After Before After
Inversion strategy 1
SEa 1 2 0.00 4979.7808 0.0000 0.3500 0.0000
SEa 2 6 0.00 4229.2929 0.0000 0.3332 0.0000
SEa 3 10 0.00 4549.4140 0.1565 0.3055 0.0059
SEa 4 14 0.00 3966.8884 0.1393 0.3121 0.0091
SEa 5 18 0.00 3546.1967 1.0772 0.3014 0.0334
SEa 6 22 0.00 3319.5297 0.7163 0.3066 0.0451
SEa 7 26 0.00 3776.7236 1.2877 0.3124 0.0737
SEb 1 2 0.00 4979.7808 0.0057 0.3500 0.0101
SEb 2 6 0.00 4229.2929 0.0054 0.3332 0.0125
SEb 3 10 0.00 4549.4140 0.0132 0.3055 0.0152
SEb 4 14 0.00 3966.8884 0.0111 0.3121 0.0194
SEb 5 18 0.00 3546.1967 0.0225 0.3014 0.0472
SEb 6 22 0.00 3319.5297 0.4051 0.3066 0.0662
SEb 7 26 0.00 3776.7236 1.0224 0.3124 0.0783
a
𝐾1 is the number of feature points for PDs 1–7 prescribed in semi-idealized experiments. 𝐾2 is the value of maximum percentage error. 𝐾3 is the data misfit
before and after assimilation. 𝐾4 is the mean absolute difference between prescribed and inverted Fourier coefficients.
Table 2: Error statistics for SEa 1.1–7.1 and SEb 1.1–7.1.

𝐾3a 𝐾4a (m)
Exp. 𝐾1a 𝐾2a
SEa 1.1 2 0.05 5060.1284 4.3569 0.3500 0.0011
SEa 2.1 6 0.05 4306.6660 3.5968 0.3332 0.0007
SEa 3.1 10 0.05 4600.6445 3.9834 0.3055 0.0082
SEa 4.1 14 0.05 4019.1911 3.2996 0.3121 0.0093
SEa 5.1 18 0.05 3614.2876 4.0757 0.3014 0.0443
SEa 6.1 22 0.05 3370.5825 3.4881 0.3066 0.0491
SEa 7.1 26 0.05 3838.0024 4.3227 0.3124 0.0740
SEb 1.1 2 0.05 5060.1284 4.2224 0.3500 0.0238
SEb 2.1 6 0.05 4306.6660 3.4525 0.3332 0.0250
SEb 3.1 10 0.05 4600.6445 3.6353 0.3055 0.0332
SEb 4.1 14 0.05 4019.1911 3.0429 0.3121 0.0337
SEb 5.1 18 0.05 3614.2876 3.0501 0.3014 0.0482
SEb 6.1 22 0.05 3370.5825 2.7539 0.3066 0.0736
SEb 7.1 26 0.05 3838.0024 3.2047 0.3124 0.0833
a
inverse problem. The inverse problems in ocean models are a problem urgent to be solved. Data assimilation methods
often quite complex. The ocean modeling is not just to solve have been used widely to solve this problem. Among all
the partial differential equations which might also be solved data assimilation methods, the adjoint data assimilation
by some simple methods like the method of characteristics. method is one of the most effective and powerful approaches
A reasonable ocean model should also be related to the developed over the past three decades. It is an advanced
field observations (satellite altimetry and tidal gauges in this data assimilation method and has the advantage of directly
work). In order to realize a more accurate simulation of assimilating various observations distributed in time and
ocean dynamics, how to organically combine the numerical space into the numerical model while maintaining dynamical
ocean model with available observations has already become and physical consistency with the model. The adjoint method

𝐾3a 𝐾4a (m)
Exp. 𝐾1a 𝐾2a
SEa 1.2 2 0.10 5096.1191 17.4292 0.3500 0.0011
SEa 2.2 6 0.10 4329.9121 14.4080 0.3332 0.0013
SEa 3.2 10 0.10 4621.0439 15.3755 0.3055 0.0115
SEa 4.2 14 0.10 4041.4563 12.8185 0.3121 0.0132
SEa 5.2 18 0.10 3633.7822 13.0238 0.3014 0.0438
SEa 6.2 22 0.10 3388.0535 11.9751 0.3066 0.0540
SEa 7.2 26 0.10 3861.5273 13.5251 0.3124 0.0753
SEb 1.2 2 0.10 5096.1191 16.7203 0.3500 0.0343
SEb 2.2 6 0.10 4329.9121 13.8206 0.3332 0.0340
SEb 3.2 10 0.10 4621.0439 14.4797 0.3055 0.0456
SEb 4.2 14 0.10 4041.4563 12.1758 0.3121 0.0485
SEb 5.2 18 0.10 3633.7822 11.9745 0.3014 0.0645
SEb 6.2 22 0.10 3388.0535 11.4183 0.3066 0.0846
SEb 7.2 26 0.10 3861.5273 12.1905 0.3124 0.0902
a

𝐾3a 𝐾4a (m)
Exp. 𝐾1a 𝐾2a
SEa 1.3 2 0.15 5140.9389 39.2166 0.3500 0.0032
SEa 2.3 6 0.15 4360.3886 32.3847 0.3332 0.0018
SEa 3.3 10 0.15 4649.1435 34.2411 0.3055 0.0135
SEa 4.3 14 0.15 4070.1625 28.6868 0.3121 0.0168
SEa 5.3 18 0.15 3659.3095 27.8828 0.3014 0.0446
SEa 6.3 22 0.15 3411.1008 26.0982 0.3066 0.0665
SEa 7.3 26 0.15 3891.3386 28.8848 0.3124 0.0771
SEb 1.3 2 0.15 5140.9389 37.8465 0.3500 0.0449
SEb 2.3 6 0.15 4360.3886 31.0404 0.3332 0.0430
SEb 3.3 10 0.15 4649.1435 32.1405 0.3055 0.0552
SEb 4.3 14 0.15 4070.1625 27.2928 0.3121 0.0569
SEb 5.3 18 0.15 3659.3095 26.6717 0.3014 0.0700
SEb 6.3 22 0.15 3411.1008 25.1515 0.3066 0.0913
SEb 7.3 26 0.15 3891.3386 27.4780 0.3124 0.0963
a
might be complicated and expensive for some simple prob- in Section 1). It has been widely applied to meteorological
lems. However, the inverse problems in ocean modeling are and oceanographic data assimilation, sensitivity studies, and
often quite complex in contrast with those simple problems. parameter estimation.
As is known, one advantage of the numerical method over
theoretical analysis lies in the disposal of nonlinear terms.
The ocean numerical models are usually strongly nonlinear, 4.3.2. Analysis on Ill-Posedness. From the statistics shown in
increasing the complexity of the relevant inverse problem. Tables 1–5, we can find an interesting phenomenon. Define
Therefore, the increased complexity of the inverse problem the data misfits after assimilation to be 𝑉1dm for inversion
makes the adjoint method effective. The adjoint method has strategy 1 and 𝑉2dm for inversion strategy 2. Further define
been proved to be effective and powerful in ocean and atmo- the differences between prescribed and inverted control
sphere problems by many works (see the references listed variables to be 𝑉1cv for inversion strategy 1 and 𝑉2cv for

𝐾3a 𝐾4a (m)
Exp. 𝐾1a 𝐾2a
SEa 1.4 2 0.20 5194.4404 69.7209 0.3500 0.0043
SEa 2.4 6 0.20 4398.0703 57.5559 0.3332 0.0025
SEa 3.4 10 0.20 4684.9663 61.7102 0.3055 0.0169
SEa 4.4 14 0.20 4105.4169 50.8808 0.3121 0.0207
SEa 5.4 18 0.20 3690.9194 48.2412 0.3014 0.0458
SEa 6.4 22 0.20 3439.8129 45.3626 0.3066 0.0711
SEa 7.4 26 0.20 3927.5261 50.4111 0.3124 0.0792
SEb 1.4 2 0.20 5194.4404 67.1112 0.3500 0.0562
SEb 2.4 6 0.20 4398.0703 55.1859 0.3332 0.0493
SEb 3.4 10 0.20 4684.9663 57.6774 0.3055 0.0637
SEb 4.4 14 0.20 4105.4169 48.3631 0.3121 0.0644
SEb 5.4 18 0.20 3690.9194 47.1181 0.3014 0.0755
SEb 6.4 22 0.20 3439.8129 43.9302 0.3066 0.0978
SEb 7.4 26 0.20 3927.5261 48.6330 0.3124 0.1011
a
inversion strategy 2. The values of 𝑉𝑖cv (𝑖 = 1, 2) and 𝑉𝑖dm (𝑖 = between prescribed and inverted control variables can also
1, 2) for all the experiments are plotted in Figure 6. We can reach a value of zero. With more control variables and larger
find 𝑉1dm are larger than or comparable with 𝑉2dm while data noises, the inversion results will not be exactly equal to
𝑉1cv are greatly smaller than 𝑉2cv . Consequently, for all the the prescribed distributions. In the work of Smedstad and
experiments except SEa 1 and SEa 2, without loss of generality, O’Brien [12] where the spatially distributed phase speed in
we can obtain an equatorial Pacific Ocean model was estimated, they could
not produce the exact values either, even in the condition
𝑉1cv < 𝑉2cv , 𝑉1dm > 𝑉2dm . (23) that perfect observations were available at every grid of the
model. Zhang and Lu [4] put forward the similar viewpoint
It is easy to understand that small values of 𝑉𝑖cv (𝑖 = 1, 2) and it also occurs in the parameter estimation of internal
indicate more accurate control variables, and small values of tidal model [23–25]. With identical twin experiments, the
𝑉𝑖dm (𝑖 = 1, 2) mean small differences between simulated and “observations” are perfect in the sense that they are produced
observed results. In this work, the open boundary conditions by the model and thus are consistent with the model physics.
are the only parameters for estimation and other parameters From the results of this paper and previous works, we
are fixed all the time. Instead of formula (23), we should have can conclude that ill-posedness has happened in other 33
expected experiments and the effects of ill-posedness will be amplified
𝑉1cv < 𝑉2cv , so 𝑉1dm < 𝑉2dm , (24) by increasing the number of control variables and data noises.
Formula (23) obtained in this work provides a concrete
which means a better parameter estimation drives a more evidence that ill-posedness of inverse problem can generate
accurate simulation. In other words, what we want are small poor parameter inversion results while producing an unreal
values of 𝑉dm and what we need are small values of 𝑉cv . “good data fitting”. For a specific problem, it is necessary and
Formulas (23) and (24) exactly indicate an inconsistency helpful to perform identical semi-idealized experiments in
between the effects of parameter estimation and observation order to find the optimal choices for the number of control
restricted data reproduction. variables and inversion strategy.
For PDs 1–7 the numbers of feature points are 2, 6, 10,
14, 18, 22, and 26, respectively. It should be noted that at 5. Conclusions
each feature point the Fourier coefficients include 𝑎 and 𝑏.
Therefore the numbers of control variables for inversion are In this work, two strategies for inverting the open boundary
doubled, that is, 4, 12, 20, 28, 36, 44, and 52, respectively. conditions with adjoint method are compared by carrying out
There are a total of 35 semi-idealized experiments in this semi-idealized numerical experiments. In the first strategy,
work. Among these experiments, only SEa 1 and SEa 2 can the open boundary curves are assumed to be partly space
realize a perfect inversion of control variables. Here we define varying and are generated by linearly interpolating the values
perfect inversion as follows: the data misfit between observed at feature points. The feature points are selected by calculating
and simulated values can decrease to zero and the difference the second-order derivatives of discrete curves and the values
70
60
50
40
Values
30
20
10
0
SE1
SE7
SE1.1
SE7.1
SE1.2
SE7.2
SE1.3
SE7.3
SE1.4
SE7.4
Index of experiments
V1dm for inversion strategy 1
V2dm for inversion strategy 2
(a)
0.12
0.1
0.08
Values
0.06
0.04
0.02
0
SE1
SE7
SE1.1
SE7.1
SE1.2
SE7.2
SE1.3
SE7.3
SE1.4
SE7.4
Index of experiments
V1cv for inversion strategy 1

V2cv for inversion strategy 2
(b)
Figure 6: (a) The values of 𝑉𝑖dm (𝑖 = 1, 2) versus the index of experiments. (b) The values of 𝑉𝑖cv (𝑖 = 1, 2) versus the index of experiments.
at selected feature points are taken as control variables to be support for this research was provided by the National Nat-
estimated. The advantage is that most of the variations of the ural Science Foundation of China through Grants 41206001
curves can be reproduced by the minimum number of points. and 41076006, the Major State Basic Research Development
In the second strategy, the OBC is assumed to be fully space Program of China through Grant 2013CB956500, the Nat-
varying and the values at every open boundary points are ural Science Foundation of Jiangsu Province through Grant
taken as control variables. BK2012315, the Priority Academic Program Development of
A series of semi-idealized experiments are carried out to Jiangsu Higher Education Institutions, and the Fundamental
calibrate the inversion ability of adjoint model and compare Research Funds for the Central Universities 201261006.
the effectiveness of two inversion strategies. The results
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http://dx.doi.org/10.1155/2013/136483
Research Article
Decoupling the Stationary Navier-Stokes-Darcy System
with the Beavers-Joseph-Saffman Interface Condition
Yong Cao,1 Yuchuan Chu,1 Xiaoming He,2 and Mingzhen Wei3

1
Department of Mechanical Engineering & Automation, Harbin Institute of Technology, Shenzhen Graduate School,
Shenzhen, Guangdong 518055, China
2
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA
3
Department of Geological Science and Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA
Correspondence should be addressed to Xiaoming He; hex@mst.edu
Received 5 April 2013; Accepted 31 July 2013
Academic Editor: R. K. Bera
Copyright © 2013 Yong Cao et al. This is an open access article distributed under the Creative Commons Attribution License, which
This paper proposes a domain decomposition method for the coupled stationary Navier-Stokes and Darcy equations with the
Beavers-Joseph-Saffman interface condition in order to improve the efficiency of the finite element method. The physical interface
conditions are directly utilized to construct the boundary conditions on the interface and then decouple the Navier-Stokes and
Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the
features of the proposed method.
1. Introduction domains by directly utilizing the given physical interface

conditions.
The Stokes-Darcy model has been extensively studied in the
The rest of paper is organized as follows. In Section 2, we
recent years due to its wide range of applications in many
introduce the Navier-Stokes-Darcy model with the Beavers-
natural world problems and industrial settings, such as the
Joseph-Saffman interface condition. In Section 3, we recall
subsurface flow in karst aquifers, oil flow in vuggy porous
the coupled weak formulation and the corresponding coupled
media, industrial filtrations, and the interaction between
finite element method for the Navier-Stokes-Darcy model.
surface and subsurface flows [1–8]. Since the problem domain
In Section 4, a parallel domain decomposition method and
naturally consists of two different physical subdomains,
its finite element discretization are proposed to decouple
several different numerical methods have been developed to
the Navier-Stokes-Darcy system by using the Robin-type
decouple the Stokes and Darcy equations [6, 9–26]. For other
boundary conditions constructed from the physical interface
works on the numerical methods and analysis of the Stokes-
conditions. Finally, in Section 5, we present a numerical
Darcy model, we refer the readers to [27–45].
example to illustrate the features of the proposed method.
Recently the more physically valid Navier-Stokes-Darcy
model has attracted scientists’ attention, and several coupled
finite element methods have been studied for it [46–51]. On 2. Stationary Navier-Stokes-Darcy Model
the other hand, the advantages of the domain decomposi-
tion methods (DDMs) in parallel computation and natural In this section we introduce the following coupled Navier-
preconditioning have motivated the development of different Stokes-Darcy model on a bounded domain Ω = Ω𝑚 ∪ Ω𝑐 ⊂
DDMs for solving the Stoke-Darcy model [6, 10–18, 21, 22]. In R𝑑 , (𝑑 = 2, 3); see Figure 1. In the porous media region Ω𝑚 ,
this paper, we will develop a domain decomposition method the flow is governed by the Darcy system
for the Navier-Stokes-Darcy model based on Robin boundary
conditions constructed from the interface conditions. This 𝑢⃗
𝑚 = −K∇𝜙𝑚 ,
physics-based DDM is different from the traditional ones in (1)
the sense that they focus on decomposing different physical ∇ ⋅ 𝑢⃗
𝑚 = 𝑓𝑚 .
homogeneous Dirichlet boundary condition except on Γ, that

is, 𝜙𝑚 = 0 on the boundary 𝜕Ω𝑚 /Γ and 𝑢⃗ 𝑐 = 0 on the
boundary 𝜕Ω𝑐 /Γ.
Ωm
3. Coupled Weak Formulation and
Finite Element Method
Γ In this section we will recall the coupled weak formulation
and the corresponding coupled finite element method for the
Ωc Navier-Stokes-Darcy model with Beavers-Joseph-Saffman
condition. Let (⋅, ⋅)𝐷 denote the 𝐿2 inner product on the
domain 𝐷 (𝐷 = Ω𝑐 or Ω𝑚 ) and let ⟨⋅, ⋅⟩ denote the 𝐿2 inner
product on the interface Γ or the duality pairing between
1/2 1/2
Figure 1: A sketch of the porous median domain Ω𝑚 , fluid domain
(𝐻00 (Γ))󸀠 and 𝐻00 (Γ) [5]. Define the spaces
Ω𝑐 , and the interface Γ. 𝑑 𝜕Ω𝑐
𝑋𝑐 = {V⃗∈ [𝐻1 (Ω𝑐 )] | V⃗= 0 on },
Γ
Here, 𝑢⃗
𝑚 is the fluid discharge rate in the porous media, K is 𝑄𝑐 = 𝐿2 (Ω𝑐 ) , (8)
the hydraulic conductivity tensor, 𝑓𝑚 is a sink/source term,
and 𝜙𝑚 is the hydraulic head defined as 𝑧 + (𝑝𝑚 /𝜌𝑔), where
𝜕Ω𝑚
𝑝𝑚 denotes the dynamic pressure, 𝑧 the height, 𝜌 the density, 𝑋𝑚 = {𝜓 ∈ 𝐻1 (Ω𝑚 ) | 𝜓 = 0 on },
and 𝑔 the gravitational acceleration. We will consider the Γ
following second-order formulation, which eliminates 𝑢⃗ 𝑚 in the bilinear forms
the Darcy system:
𝑎𝑚 (𝜙𝑚 , 𝜓) = (K∇𝜙𝑚 , ∇𝜓)Ω𝑚 ,
−∇ ⋅ (K∇𝜙𝑚 ) = 𝑓𝑚 . (2)
𝑎𝑐 (𝑢⃗ ⃗ = 2](D (𝑢⃗
𝑐 , V) ⃗Ω ,
𝑐 ) , D (V)) (9)
In the fluid region Ω𝑐 , the fluid flow is assumed to be 𝑐
governed by the Navier-Stokes equations:

𝑏𝑐 (V,⃗𝑞) = −(∇ ⋅ V,⃗𝑞)Ω𝑐 ,
𝑢⃗ ⃗
𝑐 ⋅ ∇𝑢⃗
𝑐 − ∇ ⋅ T (𝑢⃗
𝑐 , 𝑝𝑐 ) = 𝑓𝑐 , (3)
and the projection onto the tangent space on Γ:
∇ ⋅ 𝑢⃗
𝑐 = 0, (4) 𝑑−1
𝑃𝜏 𝑢⃗= ∑ (𝑢⃗⋅ 𝜏𝑗 ) 𝜏𝑗 . (10)
where 𝑢⃗ 𝑐 is the fluid velocity, 𝑝𝑐 is the kinematic pressure, 𝑗=1
𝑓𝑐⃗is the external body force, ] is the kinematic viscosity of
the fluid, T(𝑢⃗ 𝑐 , 𝑝𝑐 ) = 2]D(𝑢⃗ 𝑐 ) − 𝑝𝑐 I is the stress tensor, and With these notations, the weak formulation of the cou-
𝑇
D(𝑢⃗ 𝑐 ) = (∇ 𝑢 ⃗
𝑐 + ∇ 𝑢⃗
𝑐 )/2 is the deformation tensor. pled Navier-Stokes-Darcy model with BJS interface condition
Let Γ = Ω𝑚 ∩ Ω𝑐 denote the interface between the fluid is given as follows [46–51]: find (𝑢⃗
𝑐 , 𝑝𝑐 , 𝜙𝑚 ) ∈ 𝑋𝑐 × 𝑄𝑐 × 𝑋𝑚
and porous media regions. On the interface Γ, we impose the such that
following three interface conditions: (𝑢⃗
𝑐 ⋅ ∇𝑢⃗ ⃗Ω + 𝑎𝑐 (𝑢⃗
𝑐 , V) 𝑐
⃗+ 𝑏𝑐 (V,⃗𝑝𝑐 )
𝑐 , V)
𝑐 ⋅ 𝑢⃗
𝑢⃗ 𝑐 = −𝑢⃗ ⃗,
𝑚 ⋅ 𝑛𝑚 (5)
− 𝑏𝑐 (𝑢⃗
𝑐 , 𝑞) + 𝑎𝑚 (𝜙𝑚 , 𝜓)
−𝑢⃗
𝑐 ⋅ (T (𝑢⃗
𝑐 , 𝑝𝑐 ) ⋅ 𝑛𝑐⃗) = 𝑔 (𝜙𝑚 − 𝑧) , (6)
+ ⟨𝑔𝜙𝑚 , V⃗⋅ 𝑛𝑐⃗⟩ − ⟨𝑢⃗
𝑐 ⋅ 𝑛𝑐⃗, 𝜓⟩
𝛼]√d (11)
− 𝜏𝑗 ⋅ (T (𝑢⃗
𝑐 , 𝑝𝑐 ) ⋅ 𝑛𝑐⃗) = 𝜏𝑗 ⋅ 𝑢⃗
𝑐, (7) 𝛼]√d
√trace (∏) + ⟨𝑃𝜏 𝑢⃗ ⃗
𝑐 , 𝑃𝜏 V⟩
√trace (∏)
where 𝑛𝑐⃗and 𝑛𝑚 ⃗ denote the unit outer normal to the fluid
and the porous media regions at the interface Γ, respectively, = (𝑓𝑚 , 𝜓)Ω𝑚 + (𝑓𝑐⃗, V)⃗Ω
𝑐
𝜏𝑗 (𝑗 = 1, . . . , 𝑑 − 1) denote mutually orthogonal unit
tangential vectors to the interface Γ, and ∏ = K]/𝑔. The third + ⟨𝑔𝑧, V⃗⋅ 𝑛𝑐⃗⟩ , ∀ (V,⃗𝑞, 𝜓) ∈ 𝑋𝑐 × 𝑄𝑐 × 𝑋𝑚 .
condition (7) is referred to as the Beavers-Joseph-Saffman
(BJS) interface condition [52–55]. Assume that we have in hand regular subdivisions of Ω𝑚
In this paper, for simplification, we assume that the and Ω𝑐 into finite elements with mesh size ℎ. Then one can
hydraulic head 𝜙𝑚 and the fluid velocity 𝑢⃗ 𝑐 satisfy the define finite element spaces 𝑋𝑚ℎ ⊂ 𝑋𝑚 , 𝑋𝑐ℎ ⊂ 𝑋𝑐 and
𝑄𝑐ℎ ⊂ 𝑄𝑐 . We assume that 𝑋𝑐ℎ and 𝑄𝑐ℎ satisfy the inf-sup Let us first consider the following Robin condition for
condition [56, 57] the Darcy system: for a given constant 𝛾𝑝 > 0 and a given
function 𝜂𝑝 defined on Γ,
𝑏 (V⃗, 𝑞 )
inf sup 󵄩󵄩 𝑐 󵄩󵄩 ℎ󵄩󵄩 ℎ󵄩󵄩 > 𝛾, (12)
0 ≠𝑞ℎ ∈𝑄𝑐ℎ 0 ≠V⃗∈𝑋 󵄩V⃗
ℎ
󵄩 󵄩𝑞 󵄩
𝑐ℎ 󵄩 ℎ 󵄩1 󵄩 ℎ 󵄩0
𝛾𝑝 K∇𝜙̂𝑚 ⋅ 𝑛𝑚
⃗ + 𝑔𝜙̂𝑚 = 𝜂𝑝 , on Γ. (14)
where 𝛾 > 0 is a constant independent of ℎ. This condition Then, the corresponding weak formulation for the Darcy part
is needed in order to ensure that the spatial discretizations of is given by the following: for 𝜂𝑝 ∈ 𝐿2 (Γ), find 𝜙̂𝑚 ∈ 𝑋𝑚 such
the Navier-Stokes equations used here are stable. See [56, 57] that
for more details of finite element spaces 𝑋𝑚ℎ , 𝑋𝑐ℎ , and 𝑄𝑐ℎ
𝑔𝜙̂
that satisfy (12). One example is the Taylor-Hood element pair 𝑎𝑚 (𝜙̂𝑚 , 𝜓) + ⟨ 𝑚 , 𝜓⟩
that we use in the numerical experiments; for that pair, 𝑋𝑐ℎ 𝛾𝑝
consists of continuous piecewise quadratic polynomials and (15)
𝑄𝑐ℎ consists of continuous piecewise linear polynomials. 𝜂𝑝
Then a coupled finite element method with Newton = (𝑓𝑚 , 𝜓)Ω𝑚 + ⟨ , 𝜓⟩ , ∀𝜓 ∈ 𝑋𝑚 .
𝛾𝑝
iteration for the coupled Navier-Stokes-Darcy model is given
as follows [46]: find (𝑢⃗𝑐,ℎ , 𝑝𝑐,ℎ , 𝜙𝑚,ℎ ) ∈ 𝑋𝑐ℎ × 𝑄𝑐ℎ × 𝑋𝑚ℎ in the Second, we can propose the following two Robin-type
following procedure. conditions for the Navier-Stokes equations: for a given
0 ⃗ defined on
constant 𝛾𝑓 > 0 and given functions 𝜂𝑓 and 𝜂𝑓𝜏
(1) The initial value 𝑢⃗
𝑐,ℎ is chosen. Γ,
(2) For 𝑚 = 0, 1, 2, . . . , 𝑀, solve
̂⃗, 𝑝̂ ) ⋅ 𝑛⃗) + 𝛾 𝑢
𝑛𝑐⃗⋅ (T (𝑢 ̂ ⋅ 𝑛⃗= 𝜂 ,
𝑐 𝑐 𝑐 𝑓 ⃗
𝑐 𝑐 𝑓 on Γ,
𝑚+1 𝑚 𝑚 𝑚+1
(𝑢⃗
𝑐,ℎ ⋅ 𝑐,ℎ , V⃗
∇𝑢⃗ ℎ )Ω + (𝑢⃗
𝑐,ℎ ⋅ 𝑐,ℎ , V⃗
∇𝑢⃗ ℎ )Ω (16)
𝛼]√d
𝑐 𝑐
̂⃗, 𝑝 ) ⋅ 𝑛⃗) =
−𝑃𝜏 (T (𝑢 ̂⃗,
𝑃𝜏 𝑢 on Γ.
𝑐 𝑐 𝑐 𝑐
𝑚+1
+ 𝑎𝑐 (𝑢⃗ 𝑚+1 √trace (∏)
𝑐,ℎ , V⃗
ℎ ) + 𝑏𝑐 (V⃗
ℎ , 𝑝𝑐,ℎ )
𝑚+1 𝑚+1
Then, the corresponding weak formulation for the
− 𝑏𝑐 (𝑢⃗
𝑐,ℎ , 𝑞ℎ ) + 𝑎𝑚 (𝜙𝑚,ℎ , 𝜓ℎ ) Navier-Stokes equation is given by the following: for 𝜂𝑓 ∈
̂⃗, 𝑝̂ ) ∈ 𝑋 × 𝑄 such that
𝐿2 (Γ), find (𝑢
𝑚+1 𝑚+1 𝑐 𝑐 𝑐 𝑐
+ ⟨𝑔𝜙𝑚,ℎ , V⃗
ℎ ⋅ 𝑛𝑐⃗⟩ − ⟨𝑢⃗
𝑐,ℎ ⋅ 𝑛𝑐⃗, 𝜓ℎ ⟩
̂⃗⋅ ∇𝑢
(𝑢 ̂⃗, V)⃗ + 𝑎 (𝑢
̂⃗, V)⃗ + 𝑏 (V,⃗𝑝̂ )
(13) 𝑐 𝑐 Ω 𝑐 𝑐 𝑐 𝑐
𝛼]√d
𝑐
𝑚+1
+ 𝑐,ℎ , 𝑃𝜏 V⃗
⟨𝑃𝜏 𝑢⃗ ℎ⟩
√trace (∏) ̂⃗, 𝑞) + 𝛾 ⟨𝑢
− 𝑏𝑐 (𝑢 ̂⃗⋅ 𝑛⃗, V⃗⋅ 𝑢⃗⟩
𝑐 𝑓 𝑐 𝑐 𝑐
𝑚 𝑚 (17)
= (𝑢⃗ 𝑐,ℎ , V⃗
𝑐,ℎ ⋅ ∇𝑢⃗ ℎ )Ω + (𝑓𝑚 , 𝜓ℎ )Ω𝑚 𝛼]√d ̂⃗, 𝑃 V⟩
+ ⟨𝑃𝜏 𝑢 𝑐 𝜏 ⃗
𝑐
√trace (∏)
+ (𝑓𝑐⃗, V⃗
ℎ )Ω + ⟨𝑔𝑧, V⃗
ℎ ⋅ 𝑛𝑐⃗⟩ ,
𝑐
= (𝑓𝑐⃗, V)⃗Ω + ⟨𝜂𝑓 , V⃗⋅ 𝑛𝑐⃗⟩ , ∀ (V,⃗𝑞) ∈ 𝑋𝑐 × 𝑄𝑐 .
𝑐
∀ (V⃗
ℎ , 𝑞ℎ , 𝜓ℎ ) ∈ 𝑋𝑐ℎ × 𝑄𝑐ℎ × 𝑋𝑚ℎ .
Our next step is to show that, for appropriate choices of
𝑚+1 𝑚+1 𝑀+1 𝛾𝑓 , 𝛾p , 𝜂𝑓 , and 𝜂𝑝 , the solutions of the coupled system (11) are
(3) Set 𝑢⃗
𝑐,ℎ = 𝑢⃗ ⃗
𝑐,ℎ , 𝑝𝑐,ℎ = 𝑝𝑐,ℎ , and 𝜙𝑚,ℎ = 𝜙𝑚,ℎ . equivalent to the solutions of the decoupled equations (15)
and (17), and hence we may solve the latter system instead
4. Physics-Based Domain of the former.
Decomposition Method
Lemma 1. Let (𝜙𝑚 , 𝑢⃗ 𝑐 , 𝑝𝑐 ) be the solution of the coupled
The coupled finite element method may end up with a Navier-Stokes-Darcy system (11) and let (𝜙̂𝑚 , 𝑢 ̂⃗, 𝑝̂ ) be the
𝑐 𝑐
huge algebraic system, which combines all parts from the solution of the decoupled Navier-Stokes and Darcy equations
Navier-Stokes equations, Darcy equation, and interface con- (15) and (17) with Robin boundary conditions at the interface.
ditions together into one sparse matrix. Hence it is often ̂⃗, 𝑝̂ ) = (𝜙 , 𝑢⃗, 𝑝 ) if and only if 𝛾 , 𝛾 , 𝜂 , 𝜂⃗ ,
Then, (𝜙̂𝑚 , 𝑢𝑐 𝑐 𝑚 𝑐 𝑐 𝑓 𝑝 𝑓 𝑓𝜏
impractical to directly apply this method to large-scale real and 𝜂𝑝 satisfy the following compatibility conditions:
world applications. In order to develop a more efficient
numerical method in this section, we will directly utilize the ̂⃗⋅ 𝑛⃗+ 𝑔𝜙̂ ,
three physical interface conditions to construct a physics- 𝜂𝑝 = 𝛾𝑝 𝑢 𝑐 𝑐 𝑚 (18)
based parallel domain decomposition method to decouple
the Navier-Stokes and Darcy equations. ̂⃗⋅ 𝑛⃗− 𝑔𝜙̂ + 𝑔𝑧.
𝜂𝑓 = 𝛾𝑓 𝑢 (19)
𝑐 𝑐 𝑚
10−1 100
10−2 10−1
10−3
10−2
10−4
10−3
10−5
10−4
10−6
10−5
−7
10
10−6
10−8
10−9 10−7
0 5 10 15 20 0 5 10 15 20
𝛾p = 1/4𝛾f 𝛾p = 1/4𝛾f
𝛾p = 𝛾f 𝛾p = 𝛾f
𝛾p = 4𝛾f 𝛾p = 4𝛾f
(a) (b)
Figure 2: Convergence for the velocity of the free flow (a) and the hydraulic head of the porous medium flow (b) versus the iteration counter
𝑚 for the parallel DDM with BJS interface condition.
Table 1: Errors of the finite element method for the steady Navier-Stokes-Darcy model with BJS interface condition.
ℎ ‖𝑢⃗ ⃗0
ℎ − 𝑢‖ ‖𝑢⃗ ⃗1
ℎ − 𝑢‖ ‖𝑝ℎ − 𝑝‖0 ‖𝜙ℎ − 𝜙‖0 |𝜙ℎ − 𝜙|1
1/8 1.367 × 10−3 6.147 × 10−2 8.002 × 10−3 6.940 × 10−4 2.452 × 10−2
1/16 1.687 × 10−4 1.577 × 10−2 8.559 × 10−4 8.687 × 10−5 6.187 × 10−3
1/32 2.086 × 10−5 3.978 × 10−3 9.506 × 10−5 1.089 × 10−5 1.553 × 10−3
1/64 2.594 × 10−6 9.974 × 10−4 1.121 × 10−5 1.363 × 10−6 3.890 × 10−4
1/128 3.235 × 10−7 2.496 × 10−4 1.363 × 10−6 1.705 × 10−7 9.733 × 10−5
Proof. Adding (15) and (17) together, we obtain the following: For the necessity of the lemma, we pick 𝜓 = 0 and V⃗such
given 𝜂𝑝 , 𝜂𝑓 ∈ 𝐿2 (Γ), find (𝜙̂𝑚 , 𝑢̂𝑓 , 𝑝̂𝑐 ) ∈ 𝑋m × 𝑋𝑐 × 𝑄𝑐 such that 𝑃𝜏 V⃗= 0 in (11) and (20); then by subtracting (20) from
that (11), we get
𝑓 ⋅ 𝑛𝑐⃗+ 𝑔𝜙𝑚 − 𝑔𝑧, V⃗

⟨𝜂𝑓 − 𝛾𝑓 V⃗ ⋅ 𝑛𝑐⃗⟩ = 0,
(𝑢 ̂⃗, V)⃗ + 𝑎 (𝑢
̂⃗⋅ ∇𝑢 ̂⃗, V)⃗ + 𝑏 (V,⃗𝑝̂ ) (21)
𝑐 𝑐 Ω 𝑐 𝑐 𝑐 𝑐
𝑐 ∀V⃗∈ 𝑋𝑐 with 𝑃𝜏 V⃗= 0,
̂⃗, 𝑞) + 𝑎 (𝜙̂ , 𝜓) + 𝛾 ⟨𝑢
− 𝑏𝑐 (𝑢 ̂⃗⋅ 𝑛⃗, V⃗⋅ 𝑛⃗⟩
𝑐 𝑚 𝑚 𝑓 𝑐 𝑐 𝑐 which implies (19). The necessity of (18) can be derived in a
similar fashion.
𝑔𝜙̂𝑚 𝛼]√d ̂⃗, 𝑃 V⟩ As for the sufficiency of the lemma, by substituting
+⟨ , 𝜓⟩ + ⟨𝑃𝜏 𝑢 𝑐 𝜏 ⃗ the compatibility conditions (18)-(19), we easily see that
𝛾𝑝 √trace (∏) (20)
̂⃗, 𝑝̂ ) solves the coupled Navier-Stokes-Darcy system
(𝜙̂𝑚 , 𝑢 𝑐 𝑐
(11), which completes the proof.
= (𝑓𝑚 , 𝜓)Ω𝑚 + (𝑓𝑐⃗, V)⃗Ω + ⟨𝜂𝑓 , V⃗⋅ 𝑛𝑐⃗⟩
𝑐
𝜂𝑝 Now we use the decoupled weak formulation constructed

+⟨ , 𝜓⟩ , ∀ (V,⃗𝑞, 𝜓) ∈ 𝑋𝑚 × 𝑋𝑐 × 𝑄𝑐 . above to propose a physics-based parallel domain decompo-
𝛾𝑝 sition method with Newton iteration as follows.
101 102
100
100
−1
10
10−2
10−2
10−3
10−4
10−4
10−5
10−6
10−6
10−7 10−8
0 5 10 15 20 0 5 10 15 20
𝛾p = 1/4𝛾f 𝛾p = 1/4𝛾f
𝛾p = 𝛾f 𝛾p = 𝛾f
(a) (b)
Figure 3: Convergence for the pressure of the free flow (a) and 𝜂𝑓 (b) versus the iteration counter 𝑚 for the parallel DDM with BJS interface
condition.
(1) Initial values 𝜂𝑝0 and 𝜂𝑓0 are guessed. They may be 𝑘,𝑚
= (𝑢⃗
𝑐
𝑘,𝑚
⋅ ∇𝑢⃗ ⃗Ω + ⟨𝜂𝑓𝑘 , V⃗⋅ 𝑛𝑐⃗⟩ + (𝑓𝑐⃗, V)⃗Ω ,
𝑐 , V)
𝑐 𝑐
taken to be zero. (23)
(2) For 𝑘 = 0, 1, 2, . . ., independently solve the Darcy ∀ (V,⃗𝑞, 𝜓) ∈ 𝑋𝑐 × 𝑄𝑐 × 𝑋𝑚 .
and Navier-Stokes equations constructed above. More 𝑘 𝑘,𝑀+1
𝑘 (iii) Set 𝑢⃗
𝑐 = 𝑢⃗ and 𝑝𝑐𝑘 = 𝑝𝑐𝑘,𝑀+1 .
precisely, 𝜙𝑚 ∈ 𝑋𝑚 is computed from 𝑐
(3) 𝜂𝑝𝑘+1 and 𝜂𝑓𝑘+1 are updated in the following manner:

𝑘 𝑔𝜙𝑘 𝜂𝑝𝑘
𝑎𝑚 (𝜙𝑚 , 𝜓) + ⟨ 𝑚 , 𝜓⟩ = ⟨ , 𝜓⟩ + (𝑓𝑚 , 𝜓)Ω𝑚 , 𝛾𝑓 𝛾𝑓
𝛾𝑝 𝛾𝑝 (22) 𝜂𝑓𝑘+1 = 𝜂𝑝𝑘 − (1 + 𝑘
) 𝑔𝜙𝑚 + 𝑔𝑧,
𝛾𝑝 𝛾𝑝
∀𝜓 ∈ 𝑋𝑚 , (24)
𝜂𝑝𝑘+1 = −𝜂𝑓𝑘 + (𝛾𝑓 + 𝛾𝑝 ) 𝑢⃗
𝑘
𝑐 ⋅ 𝑛𝑐⃗+ 𝑔𝑧.
𝑘 𝑘
and 𝑢⃗𝑐 ∈ 𝑋𝑐 and 𝑝𝑐 ∈ 𝑄𝑐 are computed from the Then the corresponding domain decomposition finite ele-
following Newton iteration.
ment method is proposed as follows.
0 0
(i) Initial value 𝑢⃗ 𝑘,0
is chosen for the Newton (1) Initial values 𝜂𝑝,ℎ and 𝜂𝑓,ℎ are guessed. They may be
𝑐
iteration. For instance, it may be taken to be taken to be zero.
0,0 𝑘,0 𝑘−1
𝑢⃗𝑐 = 0 and 𝑢⃗ 𝑐 = 𝑢⃗ 𝑐 for 𝑘 = 1, 2, . . .. (2) For 𝑘 = 0, 1, 2, . . ., independently solve the Darcy and
(ii) For 𝑚 = 0, 1, 2, . . . , 𝑀, solve Navier-Stokes equations with the Robin boundary
conditions on the interface, which are constructed
𝑘
𝑘,𝑚+1
(𝑢⃗ 𝑘,𝑚
⋅ ∇𝑢⃗ 𝑘,𝑚
⃗Ω + (𝑢⃗ 𝑘,𝑚+1
⋅ ∇𝑢⃗ , V)⃗Ω previously. More precisely, 𝜙𝑚,ℎ ∈ 𝑋𝑚ℎ is computed
𝑐 𝑐 , V) 𝑐 𝑐
𝑐 𝑐 from
𝑘,𝑚+1
+ 𝑎𝑐 (𝑢⃗
𝑐 , V)⃗ + 𝑏𝑐 (V,⃗𝑝𝑐𝑘,𝑚+1 ) 𝑘
𝑘
𝑔𝜙𝑚,ℎ
𝑎𝑚 (𝜙𝑚,ℎ , 𝜓ℎ ) + ⟨ , 𝜓ℎ ⟩
𝑘,𝑚+1 𝑘,𝑚+1 𝛾𝑝
− 𝑏𝑐 (𝑢⃗
𝑐 , 𝑞) + 𝛾𝑓 ⟨𝑢⃗
𝑐 ⋅ 𝑛𝑐⃗, V⃗⋅ 𝑛𝑐⃗⟩
(25)
𝑘
𝛼]√d 𝜂𝑝,ℎ
𝑘,𝑚+1
+ ⟨𝑃𝜏 𝑢⃗
𝑐 ⃗
, 𝑃𝜏 V⟩ =⟨ , 𝜓ℎ ⟩ + (𝑓𝑚 , 𝜓ℎ )Ω𝑚 , ∀𝜓ℎ ∈ 𝑋𝑚ℎ ,
√trace (∏) 𝛾𝑝
10−1 100
10−2 10−1
10−3
10−2
10−4
10−3
10−5
10−4
−6
10
10−5
10−7
10−6
10−8
10−9 10−7
0 5 10 15 20 0 5 10 15 20
(a) (b)
Figure 4: Geometric convergence rate of the velocity of the free flow (a) and the hydraulic head of the porous medium flow (b) for the parallel
DDM with BJS interface condition.
𝑘 𝑘
and 𝑢⃗𝑐,ℎ ∈ 𝑋𝑐ℎ and 𝑝𝑐,ℎ ∈ 𝑄𝑐ℎ are computed from the 5. Numerical Example
following Newton iteration.
Example 1. Consider the model problem (2)–(6) with the
𝑘,0 BJS interface condition (7) on Ω = [0, 𝜋] × [−1, 1] with
(i) Initial value 𝑢⃗
is chosen for the Newton
𝑐,ℎ
iteration. For instance, it may be taken to be Ω𝑚 = [0, 𝜋] × [0, 1] and Ω𝑐 = [0, 𝜋] × [−1, 0]. Choose
𝑢⃗0,0 𝑘,0
⃗ 𝑘−1
⃗ (𝛼]√d/√trace(∏)) = 1, ] = 1, 𝑔 = 1, 𝑧 = 0, and K = 𝐾I,
𝑐,ℎ = 0 and 𝑢𝑐,ℎ = 𝑢𝑐,ℎ for 𝑘 = 1, 2, . . ..
(ii) For 𝑚 = 0, 1, 2, . . . , 𝑀, solve where I is the identity matrix and 𝐾 = 1. The boundary
condition data functions and the source terms are chosen
𝑘,𝑚+1 𝑘,𝑚 𝑘,𝑚 𝑘,𝑚+1
(𝑢⃗
𝑐,ℎ 𝑐,ℎ , V⃗
⋅ ∇𝑢⃗ ℎ )Ω + (𝑢⃗
𝑐,ℎ ⋅ ∇𝑢⃗
𝑐,ℎ , V⃗
ℎ )Ω
such that the exact solution is given by
𝑐 𝑐
𝑘,𝑚+1 𝑘,𝑚+1 𝑘,𝑚+1

+ 𝑎𝑐 (𝑢⃗ , V⃗
ℎ) + 𝑏𝑐 (V⃗
ℎ , 𝑝𝑐 ) − 𝑏𝑐 (𝑢⃗ , 𝑞ℎ )
𝑐,ℎ 𝑐,ℎ 𝜙𝑚 = (𝑒𝑦 − 𝑒−𝑦 ) sin (𝑥) 𝑒𝑡 ,
𝑘,𝑚+1
+ 𝛾𝑓 ⟨𝑢⃗
𝑐,ℎ ⋅ 𝑛𝑐⃗, V⃗
ℎ ⋅ 𝑛𝑐⃗⟩
𝐾
𝑢⃗
𝑐 = [ sin (2𝜋𝑦) cos (𝑥) 𝑒𝑡 ,
𝛼]√d 𝑘,𝑚+1
𝜋
+ ⟨𝑃𝜏 𝑢⃗
𝑐,ℎ , 𝑃𝜏 V⃗
ℎ⟩ (28)
√trace (∏) 𝐾 𝑇
(−2𝐾 + 2 sin2 (𝜋𝑦)) sin (𝑥) 𝑒𝑡 ] ,
𝑘,𝑚
= (𝑢⃗ 𝑘,𝑚 𝑘 ⃗ ) , 𝜋
𝑐,ℎ , V⃗
𝑐,ℎ ⋅ ∇𝑢⃗ ℎ )Ω + ⟨𝜂𝑓,ℎ , V⃗
ℎ ⋅ 𝑛𝑐⃗⟩ + (𝑓𝑐 , V⃗
𝑐
ℎ Ω
𝑐
𝑝𝑐 = 0.
∀ (V⃗
ℎ , 𝑞ℎ , 𝜓ℎ ) ∈ 𝑋𝑐ℎ × 𝑄𝑐ℎ × 𝑋𝑚ℎ .
(26)
𝑘 𝑘,𝑚+1 𝑘 𝑘,𝑀+1
We divide Ω𝑚 and Ω𝑐 into rectangles of height ℎ = 1/𝑁 and
(iii) Set 𝑢⃗
𝑐,ℎ = 𝑢⃗
𝑐,ℎ and 𝑝𝑐,ℎ = 𝑝𝑐,ℎ . width 𝜋ℎ, where 𝑁 is a positive integer, and then subdivide
𝑘+1 𝑘+1
each rectangle into two triangles by drawing a diagonal.
(3) 𝜂𝑝,ℎ and 𝜂𝑓,ℎ are updated in the following manner: The Taylor-Hood element pair is used for the Navier-Stokes
equations, and the quadratic finite element is used for the
𝑘+1
𝛾𝑓 𝑘
𝛾𝑓 𝑘
𝜂𝑓,ℎ = 𝜂𝑝,ℎ − (1 + ) 𝑔𝜙𝑚,ℎ + 𝑔𝑧, second-order formulation of the Darcy equation.
𝛾𝑝 𝛾𝑝
(27)
For the coupled finite element method of the steady
𝑘+1 𝑘 𝑘
𝜂𝑝,ℎ = −𝜂𝑓,ℎ + (𝛾𝑓 + 𝛾𝑝 ) 𝑢⃗
𝑐,ℎ ⋅ 𝑛𝑐⃗+ 𝑔𝑧. Navier-Stokes-Darcy model with BJS interface condition,
100 10−1
10−1 10−2
10−2 10−3
10−3 10−4
10−4 10−5
10−5 10−6
10−6 10−7
10−7 10−8
0 5 10 15 20 0 5 10 15 20
(a) (b)
Figure 5: Geometric convergence rate of the pressure of the free flow (a) and 𝜂𝑓 (b) versus the iteration counter 𝑚 for the parallel DDM with
BJS interface condition.
Table 2: 𝐿2 errors in velocity and hydraulic head for the parallel DDM with BJS interface condition.
𝐿2 velocity errors 𝑒(𝑖)/𝑒(𝑖 − 4) 𝐿2 hydraulic head errors 𝑒(𝑖)/𝑒(𝑖 − 4)

𝑒(0) 2.342 × 10−2 6.338 × 10−1
𝑒(4) (𝑖 = 4) 1.225 × 10−3 0.0523 3.337 × 10−2 0.0527
𝑒(8) (𝑖 = 8) 6.450 × 10−5 0.0527 1.756 × 10−3 0.0526
𝑒(12) (𝑖 = 12) 3.395 × 10−6 0.0526 9.246 × 10−5 0.0527
𝑒(16) (𝑖 = 16) 1.787 × 10−7 0.0526 4.868 × 10−6 0.0527
𝑒(20) (𝑖 = 20) 9.409 × 10−9 0.0527 2.562 × 10−7 0.0526
Table 1 provides errors for different choices of ℎ. Using linear For the parallel DDM with ] = 1, 𝐾 = 1, 𝛾𝑓 = 0.3, and
regression, the errors in Table 1 satisfy ℎ = 1/32, Figures 2 and 3 show the 𝐿2 errors of hydraulic
󵄩󵄩󵄩𝑢⃗ 󵄩󵄩 ≈ 0.714ℎ3.011 , 󵄨󵄨󵄨𝑢⃗ 󵄨󵄨 1.987 head, velocity, pressure, and 𝜂𝑓 . We can see that the parallel
󵄩 𝑐,ℎ − 𝑢⃗
𝑐󵄩󵄩0 󵄨 𝑐,ℎ − 𝑢⃗ 𝑐 󵄨󵄨1 ≈ 3.867ℎ ,
domain decomposition method is convergent for 𝛾𝑓 ≤ 𝛾𝑝 .
󵄩󵄩 󵄩 Moreover, Figures 4 and 5 show that a smaller 𝛾𝑓 /𝛾𝑝 leads to
󵄩󵄩𝑝𝑐,ℎ − 𝑝𝑐 󵄩󵄩󵄩0 ≈ 5.123ℎ
3.129
,
faster convergence.
󵄩󵄩 󵄩 󵄨󵄨 󵄨 Then Tables 2 and 3 list some 𝐿2 errors in velocity,
󵄩󵄩𝜙𝑚,ℎ − 𝜙𝑚 󵄩󵄩󵄩0 ≈ 0.354ℎ 󵄨󵄨𝜙𝑚,ℎ − 𝜙𝑚 󵄨󵄨󵄨1 ≈ 1.556ℎ
2.998 1.995
, .
(29) hydraulic head, pressure, and 𝜂𝑓 for the parallel domain
decomposition method with 𝛾𝑓 = 0.3 and 𝛾𝑝 = 1.2. The data
These rates of convergence are consistent with the approxi- in these two tables indicate the geometric convergence rate
mation capability of the Taylor-Hood element and quadratic 4
√𝛾𝑓 /𝛾𝑝 since all the error ratios are less than (√𝛾𝑓 /𝛾𝑝 ) =
element, which is third order with respect to 𝐿2 norm of 𝑢⃗𝑐
and 𝜙𝑚 , second order with respect to 𝐻1 norm of 𝑢⃗ (√1/4)4 = 0.0625.
𝑐 and
𝜙𝑚 , and second order with respect to 𝐿2 norms of 𝑝𝑐 . In Finally, for the preconditioning feature of the domain
particular, the third-order convergence rate of 𝑝𝑐 observed decomposition method, Table 4 shows the number of itera-
above, which is better than the approximation capability tions 𝑀 is independent of the grid size ℎ. Here, we set 𝛾𝑆 =
of the linear element, is mainly due to the special analytic 0.3, 𝛾𝐷 = 1.2, ] = 1, and 𝐾 = 1. Let 𝜙ℎ𝑘 , 𝑢⃗𝑘 𝑘
ℎ , and 𝑝ℎ denote
𝑘 𝑘 𝑘
solution 𝑝 = 0. the finite element solutions of 𝜙𝐷, 𝑢⃗
𝑆 , and 𝑝𝑆 at the 𝑘th step
Table 3: 𝐿2 errors in pressure and 𝜂𝑓 for the parallel DDM with BJS interface condition.
𝐿2 velocity errors 𝑒(𝑖)/𝑒(𝑖 − 4) 𝐿2 hydraulic head errors 𝑒(𝑖)/𝑒(𝑖 − 4)

𝑒(0) 7.268 × 10−1 5.668 × 10−2
𝑒(4) (𝑖 = 4) 3.826 × 10−2 0.0526 2.752 × 10−3 0.0486
𝑒(8) (𝑖 = 8) 2.014 × 10−3 0.0526 1.399 × 10−4 0.0508
𝑒(12) (𝑖 = 12) 1.060 × 10−4 0.0526 7.233 × 10−6 0.0517
𝑒(16) (𝑖 = 16) 5.579 × 10−6 0.0526 3.767 × 10−7 0.0521
𝑒(20) (𝑖 = 20) 2.937 × 10−7 0.0526 1.969 × 10−8 0.0523
Table 4: The iteration counter 𝑀 versus the grid size ℎ for both [5] Y. Cao, M. Gunzburger, F. Hua, and X. Wang, “Coupled
the parallel Robin-Robin domain decomposition method with BJS Stokes-Darcy model with Beavers-Joseph interface boundary
interface condition. condition,” Communications in Mathematical Sciences, vol. 8,
no. 1, pp. 1–25, 2010.
ℎ 1/8 1/16 1/32 1/64
𝑀 19 19 19 19 [6] M. Discacciati, Domain decomposition methods for the coupling
of surface and groundwater flows [Ph.D. thesis], École Polytech-
nique Fédérale de Lausanne, Lausanne, Switzerland, 2004.
of the domain decomposition algorithm. The criterion used [7] V. J. Ervin, E. W. Jenkins, and S. Sun, “Coupled generalized
to stop the iteration, that is, to determine the value 𝑀, is nonlinear Stokes flow with flow through a porous medium,”
𝑘 𝑘−1 𝑘 𝑘−1 𝑘 𝑘−1 SIAM Journal on Numerical Analysis, vol. 47, no. 2, pp. 929–952,
‖𝑢⃗ℎ − 𝑢⃗
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−5 2009.
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[8] M. Moraiti, “On the quasistatic approximation in the Stokes-
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6. Conclusions Mathematical Analysis and Applications, vol. 394, no. 2, pp. 796–
808, 2012.
In this paper, a parallel physics-based domain decomposition
[9] I. Babuška and G. N. Gatica, “A residual-based a posteriori error
method is proposed for the stationary Navier-Stokes-Darcy
estimator for the Stokes-Darcy coupled problem,” SIAM Journal
model with the BJS interface condition. This method is based on Numerical Analysis, vol. 48, no. 2, pp. 498–523, 2010.
on the Robin boundary conditions constructed from the
three physical interface conditions. Moreover, it is convergent [10] Y. Cao, M. Gunzburger, X. He, and X. Wang, “Robin-Robin
with geometric convergence rates if the relaxation parameter domain decomposition methods for the steady-state Stokes-
Darcy system with the Beavers-Joseph interface condition,”
is selected properly. The number of iteration steps is inde-
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advantage of the domain decomposition methods. [11] Y. Cao, M. Gunzburger, X.-M. He, and X. Wang, “Parallel,
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Acknowledgments Computation. In press.
This work is partially supported by DOE Grant DE- [12] W. Chen, M. Gunzburger, F. Hua, and X. Wang, “A parallel
Robin-Robin domain decomposition method for the Stokes-
FE0009843, National Natural Science Foundation of China
Darcy system,” SIAM Journal on Numerical Analysis, vol. 49, no.
(11175052). 3, pp. 1064–1084, 2011.
[13] M. Discacciati, “Iterative methods for Stokes/Darcy coupling,”
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http://dx.doi.org/10.1155/2013/597807
Research Article
A New Integro-Differential Equation for Rossby Solitary Waves
with Topography Effect in Deep Rotational Fluids
Hongwei Yang,1 Qingfeng Zhao,1 Baoshu Yin,2,3 and Huanhe Dong1

1
Information School, Shandong University of Science and Technology, Qingdao 266590, China
2
Institute of Oceanology, China Academy of Sciences, Qingdao 266071, China
3
Key Laboratory of Ocean Circulation and Wave, Chinese Academy of Sciences, Qingdao 266071, China
Correspondence should be addressed to Baoshu Yin; baoshuyin@126.com
Received 7 May 2013; Accepted 2 September 2013
Copyright © 2013 Hongwei Yang et al. This is an open access article distributed under the Creative Commons Attribution License,
From rotational potential vorticity-conserved equation with topography effect and dissipation effect, with the help of the multiple-
scale method, a new integro-differential equation is constructed to describe the Rossby solitary waves in deep rotational fluids.
By analyzing the equation, some conservation laws associated with Rossby solitary waves are derived. Finally, by seeking the
numerical solutions of the equation with the pseudospectral method, by virtue of waterfall plots, the effect of detuning parameter
and dissipation on Rossby solitary waves generated by topography are discussed, and the equation is compared with KdV equation
and BO equation. The results show that the detuning parameter 𝛼 plays an important role for the evolution features of solitary
waves generated by topography, especially in the resonant case; a large amplitude nonstationary disturbance is generated in the
forcing region. This condition may explain the blocking phenomenon which exists in the atmosphere and ocean and generated by
topographic forcing.
1. Introduction intermediate long-wave (ILW) model [12, 13]. Many math-

ematicians solved the above models by all kinds of method
Among the many wave motions that occur in the ocean and and got a series of results [14–19]. We note that most of the
atmosphere, Rossby waves play one of the most important
previous researches about solitary waves were carried out in
roles. They are largely responsible for determining the ocean’s
the zonal area and could not be applied directly to the spheri-
response to atmospheric and other climate changes [1]. In
cal earth, and little attention had been focused on the solitary
the past decades, the research on nonlinear Rossby solitary
waves in the rotational fluids [20]. Furthermore, as everyone
waves had been given much attention in the mathematics and
physics, and some models had been constructed to describe knows the real oceanic and atmospheric motion is a forced
this phenomenon. Based upon the pioneering work of Long and dissipative system. Topography effect as a forcing factor
[2] and Benney [3] on barotropic Rossby waves, there had has been studied by many researchers [21–25]; on the other
been remarkably exciting developments [4–11] and formed hand, dissipation effect must be considered in the oceanic
classical solitary waves theory and algebraic solitary waves and atmospheric motion; otherwise, the motion would grow
theory. The so-called classical solitary waves indicate that the explosively because of the constant injecting of the external
evolution of solitary waves is governed by the Korteweg-de forcing energy. Our aim is to construct a new model to
Vries (KdV) type model, while the behavior of solitary waves describe the Rossby solitary waves in rotational fluid with
is governed by the Benjamin-Ono (BO) model, it is called topography effect and dissipation effect. It has great difference
algebraic solitary waves. After the KdV model and BO model, from the previous researches.
a more general evolution model for solitary waves in a finite- In this paper, from rotational potential vorticity-con-
depth fluid was given by Kubota, and the model was called served equation with topography effect and dissipation effect,
with the help of the multiple-scale method, we will first con- Substituting (2), (3), and (4) into (1) leads to the following
struct a new model to describe Rossby solitary waves in deep equation for the perturbation stream function 𝜓:
rotational fluids. Then we will analyse the conservation rela-
𝜕 𝜕 1 𝜕𝜓 𝜕 1 𝜕𝜓 𝜕
tions of the model and derive the conservation laws of Rossby [ + (𝜔 − 𝑐 + 𝜀𝛼) + 𝜀( − )]
solitary waves. Finally, the model is solved by the pseudospec- 𝜕𝑡 𝜕𝜃 𝑟 𝜕𝑟 𝜕𝜃 𝑟 𝜕𝜃 𝜕𝑟
tral method [26]. Based on the waterfall plots, the effect of
1 𝜕 𝜕𝜓 1 𝜕2 𝜓
detuning parameter and dissipation on Rossby solitary waves ×[ (𝑟 ) + 2 2 + 𝜀𝐻 (𝑟, 𝜃)]
generated by topography are discussed, the model is com- 𝑟 𝜕𝑟 𝜕𝑟 𝑟 𝜕𝜃
pared with KdV model and BO model, and some conclusions (5)
1 𝑑 1 𝑑𝑟2 𝜔 𝜕𝜓
are obtained. + [𝛽 − ( )]
𝑟 𝑑𝑟 𝑟 𝑑𝑟 𝜕𝜃
2. Mathematics Model 1 𝜕 𝜕𝜓 1 𝜕2 𝜓
= −𝜀3/2 𝜆 [ (𝑟 ) + 2 2 ] .
According to [27], taking plane polar coordinates (𝑟, 𝜃), 𝑟 𝑟 𝜕𝑟 𝜕𝑟 𝑟 𝜕𝜃
pointing to lower latitude is positive and the positive rotation
In the domain [𝑟2 , ∞], the parameter 𝛽 is smaller than that
is counter-clockwise, and then the rotational potential vortic-
in the domain [𝑟1 , 𝑟2 ], and we assume 𝛽 = 0 for [𝑟2 , ∞]. Fur-
ity-conserved equation including topography effect and tur-
thermore, the turbulent dissipation and topography effect are
bulent dissipation is, in the nondimensional form, given by
absent in the domain and only consider the features of distur-
𝜕 1 𝜕Ψ 𝜕 1 𝜕Ψ 𝜕 bances generated. Substituting (2) and (3) into (1), we have the
[ +( − )]
𝜕𝑡 𝑟 𝜕𝑟 𝜕𝜃 𝑟 𝜕𝜃 𝜕𝑟 following governing equations:
1 𝜕 𝜕Ψ 1 𝜕2 Ψ 𝛽 𝜕Ψ 𝜕 𝜕 1 𝜕𝜓 𝜕 1 𝜕𝜓 𝜕
×[ (𝑟 ) + 2 2 + ℎ (𝑟, 𝜃)] + (1) [ + (𝜔1 − 𝑐 + 𝜀𝛼) + 𝜀( − )]
𝑟 𝜕𝑟 𝜕𝑟 𝑟 𝜕𝜃 𝑟 𝜕𝜃 𝜕𝑡 𝜕𝜃 𝑟 𝜕𝑟 𝜕𝜃 𝑟 𝜕𝜃 𝜕𝑟
(6)
1 𝜕 𝜕Ψ 1 𝜕2 Ψ 1 𝜕 𝜕𝜓 1 𝜕2 𝜓
= −𝜆 0 [ (𝑟 ) + 2 2 ] + 𝑄, ×[ (𝑟 ) + 2 2 ] = 0.
𝑟 𝜕𝑟 𝜕𝑟 𝑟 𝜕𝜃 𝑟 𝜕𝑟 𝜕𝑟 𝑟 𝜕𝜃
where Ψ is the dimensionless stream function; 𝛽 = For (5), we introduce the following stretching transforma-
(𝜔0 /𝑅0 ) cos 𝜙0 (𝐿2 /𝑈), in which 𝑅0 is the Earth’s radius, 𝜔0 tions:
is the angular frequency of the Earth’s rotation, 𝜙0 is the
latitude, 𝐿 and 𝑈 are the characteristic horizontal length Θ = 𝜀1/2 𝜃, 𝑟 = 𝑟, 𝑇 = 𝜀3/2 𝑡, (7)
and velocity scales, ℎ(𝑟, 𝜃) expresses the topography effect, and the perturbation expansion of 𝜓 is in the following form:
𝜆 0 [(1/𝑟)(𝜕/𝜕𝑟)(𝑟(𝜕Ψ/𝜕𝑟))+(1/𝑟2 )(𝜕2 Ψ/𝜕𝜃2 )]denotes the vor-
ticity dissipation which is caused by the Ekman boundary 𝜓 = 𝜓1 (Θ, 𝑟, 𝑇) + 𝜀𝜓2 (Θ, 𝑟, 𝑇) + ⋅ ⋅ ⋅ . (8)
layer and 𝜆 0 is a dissipative coefficient, 𝑄 is the external
source, and the form of 𝑄 will be given in the latter. Substituting (7) and (8) into (5), comparing the same power of
In order to consider weakly nonlinear perturbation on a 𝜀 term, we can obtain the 𝜀1/2 equation:
rotational flow, we assume L𝜓1 = 0, (9)
𝑟
Ψ = ∫ (Ω (𝑟) − 𝑐 + 𝜀𝛼) 𝑟𝑑𝑟 + 𝜀𝜓 (𝑟, 𝜃, 𝑡) , (2) where the operator L is defined as
where 𝛼 is a small disturbance in the basic flow and reflects 1 𝜕 𝜕 𝜕 𝑑 1 𝑑𝑟2 𝜔
the proximity of the system to a resonate state; 𝑐 is a constant, L= {(𝜔 − 𝑐) [ (𝑟 )] + [𝛽 − ( )]} .
𝑟 𝜕Θ 𝜕𝑟 𝜕𝑟 𝑑𝑟 𝑟 𝑑𝑟
which is regarded as a Rossby waves phase speed; 𝜓 denotes (10)
disturbance stream function; Ω(𝑟) expresses the rotational
angular velocity. In order to consider the role of nonlinearity, Assume the perturbation at boundary 𝑟 = 𝑟1 does not exist,
we assume the following type of rotational angular velocity: that is,
𝜔 (𝑟) 𝑟1 ≤ 𝑟 ≤ 𝑟2 , 𝜓1 = 𝜓2 = ⋅ ⋅ ⋅ = 0, (11)
Ω (𝑟) = { (3)
𝜔1 𝑟 > 𝑟2 , and the perturbation at boundary 𝑟 = 𝑟2 is determined by (6).
where 𝜔1 is constant and 𝜔(𝑟) is a function of 𝑟. For simplicity, For the linear solution to be separable, assuming the solution
𝜔(𝑟) is assumed to be smooth across 𝑟 = 𝑟2 . of (9) in the form:
In the domain [𝑟1 , 𝑟2 ], in order to achieve a balance among 𝜓1 = 𝐴 (Θ, 𝑇) 𝜙 (𝑟) , (12)
topography effect, turbulent dissipation, and nonlinearity
and to eliminate the derivative term of dissipation, we assume thus 𝜙(𝑟) should satisfy the following equation:
2 3/2
ℎ (𝑟, 𝜃) = 𝜀 𝐻 (𝑟, 𝜃) , 𝜆0 = 𝜀 𝜆, 𝑑 𝑑𝜙 (𝑟) 𝑑 1 𝑑𝑟2 𝜔
(4) (𝜔 − 𝑐) (𝑟 ) + [𝛽 − ( )] 𝜙 (𝑟) = 0.
1 𝜕 2 𝑑𝑟 𝑑𝑟 𝑑𝑟 𝑟 𝑑𝑟
𝑄 = 𝜀3/2 𝜆 [𝑟 (𝜔 − 𝑐0 + 𝜀𝛼)] . (13)
𝑟 𝜕𝑟
On the other hand, we proceed to the 𝜀3/2 equation: Taking the derivative with respect to 𝑟 for both sides of (19)
leads to
𝜔 − 𝑐 𝜕3 𝜓1
L𝜓2 + 𝜕𝜓̃
𝑟2 𝜕Θ3
𝜕𝑟
𝜕 𝜕 1 𝜕𝜓1 𝜕 1 𝜕𝜓1 𝜕
+( +𝛼 + − + 𝜆) (14) 𝑟2 2𝜋 [2𝑟2 𝑟 − (𝑟2 + 𝑟22 ) cos (𝜌 − 𝜌󸀠 )] 󸀠
𝜕𝑇 𝜕Θ 𝑟 𝜕𝑟 𝜕Θ 𝑟 𝜕Θ 𝜕𝑟 ̃ 󸀠
= ∫ 𝜓 (𝜌 , 𝑟2 , 𝑇, 𝜀) 𝑑𝜌 .
𝜋 0 2
[𝑟22 − 2𝑟2 𝑟 cos (𝜌 − 𝜌󸀠 ) + 𝑟2 ]
1 𝜕 𝜕𝜓 𝜕𝐻 (𝑟, Θ)
×[ (𝑟 1 )] + (𝜔 − 𝑐) = 0. (20)
𝑟 𝜕𝑟 𝜕𝑟 𝜕Θ
Because the solution of (5) matches smoothly with the solu-
Multiplying the both sides of (14) by 𝑟𝜙/(𝜔 − 𝑐) and integrat- tion of (6) at 𝑟 = 𝑟2 , we obtain
ing it with respect to 𝑟 from 𝑟1 to 𝑟2 , employing the boundary
conditions (11), we get 𝜓1 (Θ, 𝑟2 , 𝑇) + 𝜀𝜓2 (Θ, 𝑟2 , 𝑇) = 𝜓̃ (𝜌, 𝑟2 , 𝑇, 𝜀) + 𝑂 (𝜀2 ) , (21)
𝜕 𝜕 𝑑𝜙 󵄨󵄨
𝑟 (𝜙 𝜓2 − 𝜓2 )󵄨󵄨󵄨
󵄨 𝜕𝜓1 𝜕𝜓 𝜕𝜓̃
𝜕Θ 𝜕𝑟 𝑑𝑟 󵄨󵄨𝑟=𝑟2 (Θ, 𝑟2 , 𝑇) + 𝜀 2 (Θ, 𝑟2 , 𝑇) = (𝜌, 𝑟2 , 𝑇, 𝜀) + 𝑂 (𝜀2 ) .
𝜕𝑟 𝜕𝑟 𝜕𝑟
(22)
𝜕𝐴 𝑟2 𝜙3 𝑑
+𝐴 ∫ From (21), we have
𝜕Θ 𝑟1 𝜔 − 𝑐 𝑑𝑟
𝛽 − (𝑑/𝑑𝑟) ((1/𝑟) (𝑑𝑟2 𝜔/𝑑𝑟)) 𝐴 (Θ, 𝑇) 𝜙 (𝑟2 ) = 𝜓̃ (𝜌, 𝑟2 , 𝑇, 𝜀) , 𝜓2 (Θ, 𝑟2 , 𝑇) = 0. (23)

×[ ] 𝑑𝑟
𝜙 (𝜔 − 𝑐) Substituting (23) into (20) leads to
𝜙 𝑑 𝑟2 𝑑𝜙
𝜕𝐴 𝜕𝐴 𝜕𝜓̃ 𝜕2 J (𝐴 (Θ, 𝑇))
+( +𝛼 + 𝜆𝐴) ∫ (𝑟 ) 𝑑𝑟 (𝜌, 𝑟2 , 𝑇, 𝜀) = 𝜀𝜙 (𝑟2 ) , (24)
𝜕𝑇 𝜕Θ 𝑟1 𝜔 − 𝑐 𝑑𝑟 𝑑𝑟 𝜕𝑟 𝜕Θ2
𝜕3 𝐴 𝑟2 𝜙2 𝑟2
𝜕𝐻 2𝜋
+ ∫ 𝑑𝑟 + ∫ 𝑟𝜙 𝑑𝑟 = 0. where J(𝐴(Θ, 𝑇)) = (𝑟2 /2𝜋) ∫0 𝐴(Θ󸀠 , 𝑇) ln | sin((Θ −
3
𝜕Θ 𝑟1 𝑟 𝜕Θ
𝑟1
Θ󸀠 )/2)|𝑑Θ󸀠 . Then, based on (22) and (24), we get
(15)
𝜕𝜓2 𝜕2 J (𝐴 (Θ, 𝑇))
In (15), if the boundary conditions on 𝜙 and 𝜓2 are known, 𝜙󸀠 (𝑟2 ) = 0, (Θ, 𝑟2 , 𝑇) = 𝜙 (𝑟2 ) .
𝜕𝑟 𝜕Θ2
the equation governing the amplitude 𝐴 will be determined. (25)
Assuming the solution of (5) matches smoothly with the solu-
tion of (6) at 𝑟 = 𝑟2 , we can solve (6) to seek the solution at Substituting the boundary conditions (23) and (25) into (15)
𝑟 = 𝑟2 . yields
For (6), we adopt the transformations in the forms:
𝜕𝐴 𝜕𝐴 𝜕𝐴 𝜕3 𝐴
3/2 +𝛼 + 𝑎1 𝐴 + 𝑎2 3
𝜌 = 𝜃, 𝑟 = 𝑟, 𝑇=𝜀 𝑡, (16) 𝜕𝑇 𝜕Θ 𝜕Θ 𝜕Θ
(26)
and the perturbation function is shown 𝜓; ̃ then by substitut- 𝜕3 𝜕𝐺
+ 𝑎3 3 J (𝐴 (Θ, 𝑇)) + 𝜆𝐴 = .
ing (16) into (6), we can get the 𝜀0 equation: 𝜕Θ 𝜕Θ
Equation (26) can be rewritten as follows:
𝜕 1 𝜕 𝜕𝜓̃ 1 𝜕2 𝜓̃
(𝜔1 − 𝑐) [ (𝑟 ) + 2 2 ] = 0. (17)
𝜕𝜌 𝑟 𝜕𝑟 𝜕𝑟 𝑟 𝜕𝜌 𝜕𝐴 𝜕𝐴 𝜕𝐴 𝜕3 𝐴
+𝛼 + 𝑎1 𝐴 + 𝑎2 3
𝜕𝑇 𝜕Θ 𝜕Θ 𝜕Θ
It is easy to find that (17) can reduce to (27)
𝜕2 𝜕𝐺
+ 𝑎3 2 H (𝐴 (Θ, 𝑇)) + 𝜆𝐴 = ,
1 𝜕 𝜕𝜓̃ 1 𝜕2 𝜓̃ 𝜕Θ 𝜕Θ
(𝑟 ) + 2 2 = 0, 𝑟 ≥ 𝑟2 ,
𝑟 𝜕𝑟 𝜕𝑟 𝑟 𝜕𝜌 (18) 2𝜋
where H(𝐴(Θ, 𝑇)) = (𝑟2 /4𝜋) ∫0 𝐴(Θ󸀠 , 𝑇)cot((Θ−Θ󸀠 )/2)𝑑Θ󸀠
𝑟2 𝑟
𝜓̃ 󳨀→ 0, 𝑟 󳨀→ ∞. and 𝑎 = ∫𝑟 (𝜙/(𝜔 − 𝑐))(𝑑/𝑑𝑟)(𝑟(𝑑𝜙/𝑑𝑟))𝑑𝑟, 𝑎1 = ∫𝑟 2 (𝜙3 /(𝜔 −
1 1
𝑐))(𝑑/𝑑𝑟)[𝛽 − (𝑑/𝑑𝑟)((1/𝑟)(𝑑𝑟2 𝜔/𝑑𝑟))/𝜙(𝜔 − 𝑐)]𝑑𝑟/𝑎, 𝑎2 =
Obviously, the solution of (18) is 𝑟 𝑟
∫𝑟 2 (𝜙2 /𝑟)𝑑𝑟/𝑎, 𝑎3 = 𝑟2 𝜙2 (𝑟2 )/𝑎, 𝐺 = ∫𝑟 2 𝑟𝜙𝐻𝑑𝑟/𝑎. Equation
1 1
2 2 󸀠 (27) is an integro-differential equation and 𝜆𝐴 expresses
1 2𝜋 (𝑟 − 𝑟2 ) 𝜓̃ (𝜌 , 𝑟2 , 𝑇, 𝜀)
𝜓̃ (𝜌, 𝑟, 𝑇, 𝜀) = ∫ 𝑑𝜌󸀠 . (19) dissipation effect and has the same physical meaning with the
2𝜋 0 𝑟22 − 2𝑟2 𝑟 cos (𝜌 − 𝜌󸀠 ) + 𝑟2 term 𝜕2 𝐴/𝜕Θ2 in Burgers equation. When 𝑎3 = 𝜆 = 𝐻 = 0,
the equation degenerates to the KdV equation. When 𝑎2 = 𝑎3

+ 𝑎3 (𝐴2 − H (𝐴 Θ )) (H (𝐴))ΘΘ
𝜆 = 𝐻 = 0, the equation degenerates to the so-called rota- 𝑎1
tional BO equation. Here we call (27) forced rotational KdV-
𝑎3
BO-Burgers equation. As we know, the forced rotational KdV- + 𝜆 (𝐴2 − H (𝐴 Θ )) 𝐴 = 0.
BO-Burgers equation as a governing model for Rossby soli- 𝑎1
tary waves is first derived in the paper. (30)
Then taking the derivative of (27) with respect to Θ and mul-

3. Conservation Laws tiplying (−(2𝑎2 /𝑎1 )𝐴 Θ + (𝑎3 /𝑎1 )H(𝐴)) lead to
In this section, the conservation laws are used to explore some 𝑎
2𝑎2
features of Rossby solitary waves. In [7], Ono presented four (− 𝐴 Θ + 3 H (𝐴)) 𝐴 Θ𝑇
conservation laws of BO equation, and we extend Ono’s work 𝑎1 𝑎1
to investigate the following questions: Has the rotational 2𝑎2 𝑎
KdV-BO-Burgers equation also conservation laws without + [𝛼𝐴 ΘΘ + 𝑎1 (𝐴𝐴 Θ )Θ ] (− 𝐴 Θ + 3 H (𝐴))
𝑎1 𝑎1
dissipation effect? Has it four conservation laws or more?
How to change of these conservation quantities in the pres- 2𝑎2 𝑎
+ 𝑎2 𝐴 ΘΘΘΘ (− 𝐴 + 3 H (𝐴)) (31)
ence of dissipation effect? 𝑎1 Θ 𝑎1
In this section, topography effect is ignored; that is, 𝐻 is
taken zero in (27). Based on periodicity condition, we assume 2𝑎2 𝑎
+ 𝑎3 (− 𝐴 + 3 H (𝐴)) H(𝐴)ΘΘΘ
that the values of 𝐴, 𝐴 Θ , 𝐴 ΘΘ , 𝐴 ΘΘΘ at Θ = 0 equal that at 𝑎1 Θ 𝑎1
Θ = 2𝜋. Then integrating (27) with respect to Θ over (0, 2𝜋),
we are easy to obtain the following conservation relation: 2𝑎2 𝑎
+ 𝜆𝐴 (− 𝐴 + 3 H (𝐴)) = 0.
𝑎1 Θ 𝑎1
2𝜋 2𝜋
𝑄1 = ∫ 𝐴𝑑Θ = exp (−𝜆𝑇) ∫ 𝐴 (Θ, 0) 𝑑Θ. (28) Adding (30) to (31), by virtue of the property of operator H:
0 0
2𝜋 2𝜋
From (28), it is obvious that 𝑄1 decreases exponentially with H(𝐴)ΘΘ = H (𝐴 ΘΘ ) , ∫ 𝑢HV𝑑Θ = − ∫ VH𝑢𝑑Θ,
the evolution of time 𝑇 and the dissipation coefficient 𝜆. By 0 0
(32)
analogy with the KdV equation, 𝑄1 is regarded as the mass
of the solitary waves. This shows that the dissipation effect we have
causes the mass of solitary waves decrease exponentially.
When dissipation effect is absent, the mass of the solitary 1 𝑎 𝑎
( 𝐴3 − 2 𝐴2Θ + 3 𝐴 Θ H (𝐴))
waves is conserved. 3 𝑎1 𝑎1 𝑇
In what follows, (27) has another simple conservation
law, which becomes clear if we multiply (27) by 𝐴(Θ, 𝑇) and 1 𝑎 𝑎
+ 𝛼[ 𝐴3 − 2 𝐴2Θ + 3 H (𝐴) 𝐴 Θ ]
carry the integration; by using the property of the operator 3 𝑎1 𝑎1 Θ
2𝜋
H : ∫0 𝑓(Θ)H(𝑓(Θ))𝑑Θ = 0, then we get
𝑎1 4 𝑎2
+( 𝐴 ) + 3 [𝐻 (𝐴) 𝐻(𝐴)ΘΘ ]Θ
2𝜋 2𝜋 4 Θ 2𝑎1
𝑄2 = ∫ 𝐴2 𝑑Θ = exp (−2𝜆𝑇) ∫ 𝐴2 (Θ, 0) 𝑑Θ. (29) 𝑎2 𝑎3
0 0 + (𝐴 ΘΘΘ 𝐻 (𝐴) − 2𝐴 Θ 𝐻(𝐴)ΘΘ )Θ
𝑎1
Similar to the mass 𝑄1 , 𝑄2 is regarded as the momentum of
2𝑎2 1
the solitary waves and is conserved without dissipation. The − (𝐴 Θ 𝐴 ΘΘΘ − 𝐴2ΘΘ ) + 𝑎2 (𝐴2 𝐴 ΘΘ − 2𝐴𝐴2Θ )Θ
momentum of the solitary waves also decreases exponentially 𝑎1 2 Θ
with the evolution of time 𝑇 and the increasing of dissipative 𝑎2 2 𝑎3
coefficient 𝜆 in the presence of dissipation effect. Further- + 𝑎3 [𝐴(𝐴H (𝐴))Θ ]Θ + 𝜆 (𝐴3 − 𝐴 + 𝐴 𝐻 (𝐴))
more, the rate of decline of momentum is faster than the rate 𝑎1 Θ 𝑎1 Θ
of mass. = 0.
Next, we multiply (27) by (𝐴2 −(𝑎3 /𝑎1 )H(𝐴 Θ )) and obtain (33)
1 𝑎 2𝜋
( 𝐴3 ) − 3 H (𝐴 Θ ) 𝐴 𝑇 Taking 𝑄3 = ∫0 ((1/3)𝐴3 −(𝑎2 /𝑎1 )𝐴2Θ +(𝑎3 /𝑎1 )𝐴 Θ H(𝐴))𝑑Θ,
3 𝑇 𝑎1
we are easy to see that when the dissipation effect is absent,
𝑎3 that is, 𝜆 = 0, 𝑄3 is a conserved quantity and regarded as
+ (𝛼 + 𝑎1 𝐴) 𝐴 Θ (𝐴2 − H (𝐴 Θ )) the energy of the solitary waves. So we can conclude that the
𝑎1
energy of solitary waves is conserved without dissipation. By
𝑎3 analysing (33), we can find the decreasing trend of energy of
+ 𝑎2 𝐴 ΘΘΘ (𝐴2 − H (𝐴 Θ ))
𝑎1 solitary waves.
Finally, let us consider a quantity related to the phase of 𝐹−1 {𝑖V𝐹𝐻}, and so on. Combined with a leap-frog time step,
solitary waves: (27) would be approximated by
2𝜋 𝐴 (𝑋, 𝑇 + Δ𝑇) − 𝐴 (𝑋, 𝑇 − Δ𝑇) + 𝑖𝛼𝐹−1 {V𝐹𝐴} Δ𝑇

̃4 = 𝑑 ∫ Θ𝐴𝑑Θ,
𝑄 (34)
𝑑𝑇 0 + 𝑖𝑎1 𝐴𝐹−1 {V𝐹𝐴} Δ𝑇 − 𝑎2 𝑖𝐹−1 {V3 𝐹𝐴} Δ𝑇 (38)
−1 2 −1
− 𝑎3 𝐹 {V 𝐹H (𝐴)} Δ𝑇 + 𝜆𝐴 = 𝑖𝐹 {V𝐹𝐺} Δ𝑇.
and we can get 𝑑𝑄 ̃4 /𝑑𝑇 = 0 without dissipation. According
[7], we present the velocity of the center of gravity for The computational cost for (38) is six fast Fourier transforms
the ensemble of such waves 𝑄4 = 𝑄 ̃4 /𝑄1 ; by employing per time step.
̃
𝑑𝑄1 /𝑑𝑇 = 0 and 𝑑𝑄4 /𝑑𝑇 = 0, we have 𝑑𝑄4 /𝑑𝑇 = 0, which Once the zonal flow Ω(𝑟) and the topography function
shows that the velocity of the center of gravity is conserved 𝐻(𝑟, Θ) as well as dissipative coefficient 𝜆 are given, it is easy
without dissipation. to get the coefficients of (27) by employing (13). In order to
After the four conservation relations are given, we can simplify the calculation and to focus attention on the time
proceed to seek the fifth conservation quantity. In fact, after evolution of the solitary waves with topography effect and
tedious calculation, we can also verify that dissipation effect and to show the difference among the KdV
model, BO model, and rotational KdV-BO model, we take
2𝜋 𝑎1 = 1, 𝑎2 = −1, and 𝑎3 = −1. As an initial condition, we take
1 3𝑎 9𝑎 𝑎
𝑄5 = ∫ ( 𝐴4 − 2 𝐴𝐴2Θ + 22 𝐴2ΘΘ + 3 𝐴2 H (𝐴)) 𝑑Θ 𝐴(𝑋, 0) = 0. In the present numerical computation, the
0 4 𝑎1 𝑎1 4𝑎1 2
topography forcing is taken as 𝐺 = 𝑒−[30(Θ−𝜋)] /4 .
(35)
4.1. Effect of Detuning Parameter 𝛼 and Dissipation. In
is also conservation quantity. According the idea, we can
Figure 1, we consider the effect of detuning parameter 𝛼 on
obtain the sixth conservation quantity 𝑄6 and the seventh
solitary waves. The evolution features of solitary waves gen-
conservation quantity 𝑄7 . . ., so we can guess that, similar to
erated by topography are shown in the absence of dissipation
the KdV equation, the rotational KdV-BO-Burgers equation
with different detuning parameter 𝛼. It is easy to find from
without dissipation also owns infinite conservation laws, but
these waterfall plots that the detuning parameter 𝛼 plays an
it needs to be verified in the future.
important role for the evolution features of solitary waves
generated by topography.
4. Numerical Simulation and Discussion When 𝛼 > 0 (Figure 1(a)), a positive stationary solitary
wave is generated in the topographic forcing region, and
In this section, we will take into account the generation and a modulated cnoidal wave-train occupies the downstream
evolution feature of Rossby solitary waves under the influence region. There is no wave in the upstream region. A flat buffer
of topography and dissipation, so we need to seek the solu- region exists between the solitary wave in the forcing region
tions of forced rotational KdV-BO-Burgers equation. But we and modulated cnoidal wave-train in the downstream. With
know that there is no analytic solution for (27), and here we the detuning parameter 𝛼 decreasing, the amplitudes of both
consider the numerical solutions of (27) by employing the solitary wave in the forcing region and modulated cnoidal
pseudospectral method. wave-train in the downstream region increase and the modu-
The pseudo-spectral method uses a Fourier transform lated cnoidal wave-train closes to the forcing region gradually
treatment of the space dependence together with a leap-forg and the flat buffer region disappears slowly.
scheme in time. For ease of presentation the spatial period is Up to 𝛼 = 0 (Figure 1(b)), the resonant case forms. In
normalized to [0, 2𝜋]. This interval is divided into 2𝑁 points, this case, a large amplitude nonstationary disturbance is gen-
and then Δ𝑇 = 𝜋/𝑁. The function 𝐴(𝑋, 𝑇) can be trans- erated in the forcing region. To some degree, this condition
formed to the Fourier space by may explain the blocking phenomenon which exists in the
atmosphere and ocean and generated by topographic forcing.
1 2𝑁−1 As 𝛼 < 0, from Figure 1(c) we can easy to find that a
̂ (V, 𝑇) = 𝐹𝐴 =
𝐴 ∑ 𝐴 (𝑗Δ𝑋, 𝑇) 𝑒−𝜋𝑖𝑗V/𝑁, negative stationary solitary wave is generated in the forcing
√2𝑁 𝑗=0 (36) region, and this is great difference with the former two condi-
tions. Meanwhile, there are both wave-trains in the upstream
V = 0, ±1, . . . , ±𝑁.
and downstream region. The amplitude and wavelength of
wave-train in the upstream region are larger than those in
The inversion formula is the downstream regions. Similar to Figure 1(b) and unlike
Figure 1(a), the wave-trains in the upstream and downstream
̂= 1 ̂ (V, 𝑇) 𝑒𝜋𝑖𝑗V/𝑁.
𝐴 (𝑗Δ𝑋, 𝑇) = 𝐹−1 𝐴 ∑𝐴 (37) regions connect to the forcing region and the flat buffer region
√2𝑁 V disappears.
Figure 2 shows the solitary waves generated by topogra-
These transformations can use Fast Fourier Transform algo- phy in the presence of dissipation with dissipative coefficient
rithm to efficiently perform. With this scheme, 𝜕𝐴/𝜕𝑋 can be 𝜆 = 0.3 and detuning parameter 𝛼 = 2.5. The conditions of
evaluated as 𝐹−1 {𝑖V𝐹𝐴}, 𝜕3 𝐴/𝜕𝑋3 as −𝑖𝐹−1 {V3 𝐹𝐴}, 𝜕𝐻/𝜕𝑋 as 𝛼 = 0 and 𝛼 < 0 are omitted. Compared to Figure 1(a), we will
0.5 1.5
15
1
A(𝜃, T)
15
A(𝜃, T)
10
0 10 0.5
T
T
5 0 5
−0.5 0 −0.5 0
0 𝜋/4 𝜋/2 3𝜋/4 𝜋 5𝜋/4 3𝜋/2 7𝜋/4 2𝜋 0 𝜋/4 𝜋/2 3𝜋/4 𝜋 5𝜋/4 3𝜋/2 7𝜋/4 2𝜋
𝜃 𝜃
(a) 𝛼 = 2.5 (b) 𝛼 = 0
0.4
0.2
15
A(𝜃, T)
0 10
T
−0.2 5
−0.4 0
0 𝜋/4 𝜋/2 3𝜋/4 𝜋 5𝜋/4 3𝜋/2 7𝜋/4 2𝜋
𝜃
(c) 𝛼 = −2.5
Figure 1: Solitary waves generated by topography in the absence of dissipation.
4.2. Comparison of KdV Model, BO Model, and KdV-BO

Model. We know that the rotation KdV-BO equation reduces
to the KdV equation as 𝑎3 = 0 and to the BO equation as
𝑎2 = 0, so, in this subsection by comparing Figure 1(a) with
0.4 Figure 3, we will look for the difference of solitary waves
which is described by KdV-BO model, KdV model, and BO
0.2 15 model. The role of detuning parameter 𝛼 and dissipation
A(𝜃, T)
effect has been studied in the former subsection, so here we

0 10 only consider the condition of 𝜆 = 0, 𝛼 = 2.5.
T
−0.2
At first, we can find that a positive solitary wave is all gen-
5
erated in the forcing region in Figures 1(a), 3(a) and 3(b), but
−0.4 0 it is stationary in Figures 1(a) and 3(a), and is nonstationary in
0 𝜋/4 𝜋/2 3𝜋/4 𝜋 5𝜋/4 3𝜋/2 7𝜋/4 2𝜋 Figure 3(b). By surveying carefully we find that the amplitude
𝜃 of stationary wave in the forcing region in Figure 1(a) is larger
than that in Figure 3(a). Additionally, a modulated cnoidal
Figure 2: Solitary waves generated by topography in the presence of wave-train is excited in the downstream region in Figures 1(a)
dissipation (𝜆 = 0.3, 𝛼 = 2.5). and 3(a), and in both downstream and upstream region in
Figure 3(b). The amplitude of modulated cnoidal wave-train
in downstream region in Figure 3(b) is the largest and in
find that there is also a solitary wave generated in the forcing Figure 1(a) is the smallest among the three models. Further-
region, but because of dissipation effect the amplitude of soli- more, in Figure 3(a) the wave number of modulated cnoidal
tary wave in the forcing region decreases as the dissipative wave-train is more than that in Figures 1(a) and 3(b). In
coefficient 𝜆 increases (Figures omitted) and time evolution. a word, by the above analysis and comparison, it is easy
Meanwhile, the modulated cnoidal wave-train in the down- to find that Figure 1(a) is similar to Figure 3(a) and has
stream region is dissipated. When 𝜆 is big enough, the mod- great difference with Figure 3(b). This indicates that the
ulated cnoidal wave-train in the downstream region disap- term 𝑎2 (𝜕3 𝐴/𝜕Θ3 ) plays more important role than the term
pears. 𝑎3 (𝜕2 /𝜕Θ2 )H(𝐴) in rotational KdV-BO equation.
0.4 1
0.2 15 15
0.5
A(𝜃, T)
A(𝜃, T)
0 10 10
0
T
T
−0.2 5 5
−0.4 0 −0.5 0
0 𝜋/4 𝜋/2 3𝜋/4 𝜋 5𝜋/4 3𝜋/2 7𝜋/4 2𝜋 0 𝜋/4 𝜋/2 3𝜋/4 𝜋 5𝜋/4 3𝜋/2 7𝜋/4 2𝜋
𝜃 𝜃
(a) KdV model (𝑎3 = 0, 𝛼 = 2.5, 𝜆 = 0) (b) BO model (𝑎2 = 0, 𝛼 = 2.5, 𝜆 = 0)
Figure 3: Comparison of KdV model, Bo model, and KdV-BO model.
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http://dx.doi.org/10.1155/2013/614874
Research Article
A One Step Optimal Homotopy Analysis Method for
Propagation of Harmonic Waves in Nonlinear Generalized
Magnetothermoelasticity with Two Relaxation Times
under Influence of Rotation
S. M. Abo-Dahab,1,2 Mohamed S. Mohamed,1,3 and T. A. Nofal1,4

1
Mathematics Department, Faculty of Science, Taif University, P.O. Box 888, Saudi Arabia
2
Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt
3
Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt
4
Mathematics Department, Faculty of Science, Minia University, Minia, Egypt
Correspondence should be addressed to Mohamed S. Mohamed; m s mohamed2000@yahoo.com
Received 1 May 2013; Revised 2 June 2013; Accepted 4 June 2013
Copyright © 2013 S. M. Abo-Dahab et al. This is an open access article distributed under the Creative Commons Attribution
cited.
The aim of this paper is to apply OHAM to solve numerically the problem of harmonic wave propagation in a nonlinear
thermoelasticity under influence of rotation, thermal relaxation times, and magnetic field. The problem is solved in one-dimensional
elastic half-space model subjected initially to a prescribed harmonic displacement and the temperature of the medium. The HAM
contains a certain auxiliary parameter which provides us with a simple way to adjust and control the convergence region and rate
of convergence of the series solution. This optimal approach has a general meaning and can be used to get fast convergent series
solutions of the different type of nonlinear fractional differential equation. The displacement and temperature are calculated for the
models with the variations of the magnetic field, relaxation times, and rotation. The results obtained are displayed graphically to
show the influences of the new parameters.
1. Introduction the main topic of research in many scientific publications.

Wave generation in nonlinear thermoelasticity problems has
In the past recent years, much attention has been devoted gained a considerable interest for its utilitarian aspects in
to simulate some real-life problems which can be described understanding the nature of interaction between the elastic
by nonlinear coupled differential equations using reliable and thermal fields as well as the system of PDEs for its appli-
and more efficient methods. Nonlinear partial differential cations. A lot of applications were paid on existence, unique-
equations are useful in describing various phenomena in ness, and stability of the solution of the problem, see [5–7].
disciplines. The nonlinear coupled systems of partial differ- Recently, much attention has been devoted to numerical
ential equations often appear in the study of circled fuel methods, which do not require discretization of space-time
reactor, high-temperature hydrodynamics, and thermoelas- variables or linearization of the nonlinear equations, among
ticity problems, see [1–4]. From the analytical point of the homotopy analysis methods. Since most of the nonlinear
view, a lot of work has been done for such systems. With FDEs cannot be solved exactly, approximate and numerical
the rapid development of nanotechnology, there appears an methods must be used. Some of the recent analytical meth-
ever increasing interest of scientists and researchers in this ods for solving nonlinear problems include the homotopy
field of science. Nanomaterials, because of their exceptional analysis method HAM [8–14]. The HAM, first proposed in
mechanical, physical, and chemical properties, have been 1992 by Liao [8], has been successfully applied to solve many
problems in physics and science. This method is applied to where 𝑁 is a nonlinear operator for this problem, 𝑥 and
solve linear and nonlinear systems. 𝑡 denote independent variables, and 𝑢(𝑥, 𝑡) is an unknown
The homotopy perturbation method HPM has the merits function.
of simplicity and easy execution. The homotopy perturbation By means of the HAM, one first constructs the following
method was first proposed by He [15]. Unlike the traditional zero-order deformation equation:
numerical methods, the HPM does not need discretization
and linearization. Most perturbation methods assume that (1 − 𝑞) L (𝜙 (𝑥, 𝑡; 𝑞) − 𝑢0 (𝑥, 𝑡)) = 𝑞ℎ𝐻 (𝑡) 𝑁 [𝜙 (𝑥, 𝑡; 𝑞)] ,
a small parameter exists, but most nonlinear problems have (2)
no small parameter at all. Many new methods have been
proposed to eliminate the small parameter. Recently, the where 𝑞 ∈ [0, 1] is the embedding parameter, ℎ ≠ 0 is an
applications of homotopy theory among scientists appeared, auxiliary parameter, 𝐻(𝑡) ≠0 is an auxiliary function, L is
and the homotopy theory became a powerful mathematical an auxiliary linear operator, and 𝑢0 (𝑥, 𝑡) is an initial guess.
tool, when it is successfully coupled with perturbation theory. Obviously, when 𝑞 = 0 and 𝑞 = 1, it holds
Recently, Gepreel et al. [16] investigated the homotopy
𝜙 (𝑥, 𝑡; 0) = 𝑢0 (𝑥, 𝑡) , 𝜙 (𝑥, 𝑡; 1) = 𝑢 (𝑥, 𝑡) . (3)
perturbation method and variational iteration method for
harmonic waves propagation in nonlinear magnetothermoe- Liao [8, 9] expanded 𝜙(𝑥, 𝑡; 𝑞) in Taylor series with
lasticity with rotation. Abd-Alla and Abo-Dahab [17] inves- respect to the embedding parameter 𝑞, as follows:
tigated the effect of rotation and initial stress on an infinite
generalized magnetothermoelastic diffusion body with a ∞
spherical cavity. Abo-Dahab and Mohamed [18] studied the 𝜙 (𝑥, 𝑡; 𝑞) = 𝑢0 (𝑥, 𝑡) + ∑ 𝑢𝑚 (𝑥, 𝑡) 𝑞𝑚 , (4)
influence of magnetic field and hydrostatic initial stress on 𝑚=1
reflection phenomena of P (Primary) and SV (Shear Vertical) where

waves from a generalized thermoelastic solid half space.
󵄨
1 𝜕 𝜙 (𝑥, 𝑡; 𝑞) 󵄨󵄨󵄨
𝑚
Abd-Alla and Mahmoud [19] investigated the magnetother-
𝑢𝑚 (𝑥, 𝑡) = 󵄨󵄨 . (5)
moelastic problem in rotating non-homogeneous orthotropic 𝑚! 𝜕𝑞𝑚 󵄨󵄨
󵄨𝑞=0
hollow cylinder under the hyperbolic heat conduction model.
Abd-Alla et al. [20] studied the thermal stresses effect in a Assume that the auxiliary linear operator, the initial guess,
non-homogeneous orthotropic elastic multilayered cylinder. the auxiliary parameter ℎ, and the auxiliary function 𝐻(𝑡) are
Abd-Alla et al. [21] studied the generalized magnetother- selected such that the series (4) is convergent at 𝑞 = 1; then
moelastic Rayleigh waves in a granular medium under the we have from (4)
influence of a gravity field and initial stress. Abd-Alla and
∞
Abo-Dahab [22] investigated the time-harmonic sources in a
generalized magnetothermoviscoelastic continuum with and 𝑢 (𝑥, 𝑡) = 𝑢0 (𝑥, 𝑡) + ∑ 𝑢𝑚 (𝑥, 𝑡) . (6)
𝑚=1
without energy dissipation.
In the present paper, investigation is devoted for solving Let us define the vector
numerically the problem of harmonic wave propagation in
a nonlinear thermoelasticity under influence of magnetic 𝑢⃗
𝑛 (𝑥, 𝑡) = {𝑢0 (𝑥, 𝑡) , 𝑢1 (𝑥, 𝑡) , 𝑢2 (𝑥, 𝑡) , . . . , 𝑢𝑛 (𝑥, 𝑡)} . (7)
field, thermal relaxation times, and rotation. The problem is
solved in one-dimensional elastic half-space model subjected Differentiating (2) 𝑚 times with respect to 𝑞, then setting
initially to a prescribed harmonic displacement and the 𝑞 = 0 and dividing then by 𝑚!, we have the following 𝑚th-
temperature of the medium. order deformation equation:
The HAM contains a certain auxiliary parameter which
provides us with a simple way to adjust and control the L (𝑢𝑚 (𝑥, 𝑡) − 𝜘𝑚 𝑢𝑚−1 (𝑥, 𝑡)) = ℎ𝐻 (𝑡) R𝑚 (𝑢⃗
𝑚−1 ) , (8)
convergence region and rate of convergence of the series solu-
where
tion. The ℎ-curve of the third-order approximate solutions
is displayed graphically to show the interval that the exact 1 𝜕𝑚−1 N [𝜙 (𝑥, 𝑡; 𝑞)] 󵄨󵄨󵄨󵄨
and approximate solutions take the same values. The displace- R𝑚 (𝑢⃗
𝑚−1 ) = 󵄨󵄨 ,
(𝑚 − 1)! 𝜕𝑞𝑚−1 󵄨󵄨
ment and temperature are calculated for the methods with 󵄨𝑞=0
(9)
the variations of the magnetic field and rotation. The results
0, 𝑚 ≤ 1,
obtained are displayed graphically to show the influences of 𝜘𝑚 = {
the new parameters. 1, 𝑚 > 1.
Applying the integral operator on both sides of (8), we

2. A One-Step Optimal Homotopy Analysis have
Method for PDEs 𝑡
𝑢𝑚 (𝑥, 𝑡) = 𝜘𝑚 𝑢𝑚−1 (𝑥, 𝑡) + ℎ ∫ 𝐻 (𝑡) R𝑚 (𝑢⃗
𝑚−1 ) 𝑑𝑡, (10)
To describe the basic ideas of the HAM, we consider the 0
following general nonlinear differential equation:
where the 𝑚th-order deformation equation (8) can be easily
𝑁 [𝑢 (𝑥, 𝑡)] = 0, (1) solved, especially by means of symbolic computation software
such as Mathematica, Maple, and MathLab. The convergence initial conditions (15) and the boundary conditions (16) by
of the homotopy analysis method for solving these equations choosing the linear operators
is discussed in [23].
Abbasbandy and Jalili [24] and Turkyilmazoglu [25–29]
applied the homotopy analysis method to nonlinear ODEs 𝜕2 𝜙1 (𝑥, 𝑡; 𝑞)
and suggested the so-called optimization method to find out L1 [𝜙1 (𝑥, 𝑡; 𝑞)] = ,
𝜕𝑡2
the optimal convergence control parameters by minimum (17)
of the square residual error integrated in the whole region 𝜕𝜙 (𝑥, 𝑡; 𝑞)
L2 [𝜙2 (𝑥, 𝑡; 𝑞)] = 2 ,
having physical meaning. Their approach is based on the 𝜕𝑡
square residual error.
Let Δ(ℎ) denote the square residual error of the governing
equation (1) and express it as with the property L1 [𝑐1 + 𝑐2 𝑡] = 0, L2 [𝑐3 ], where 𝑐𝑖 , (𝑖 =
1, 2, 3) are the integral constants and the nonlinear operators
2
are defined as
𝑢𝑛 (𝑡)]) 𝑑Ω,
Δ (ℎ) = ∫ (𝑁 [̃ (11)
Ω
𝜕2 𝜙1 𝜕𝜙1 𝜕2 𝜙1
where 𝑁1 [𝜙1 , 𝜙2 ] = (1 + 𝜎1 ) + Ω −
𝜕𝑡2 𝜕𝑡 𝜕𝑥2
𝑚
𝑢̃𝑚 (𝑡) = 𝑢0 (𝑡) + ∑ 𝑢𝑘 (𝑡) ; 𝜕𝜙1 𝜕𝜙 2 𝜕𝜙
(12) × (1 − 𝜎2 + 2𝛾 + 3𝛿( 1 ) ) − 𝛽1 2
𝑘=1 𝜕𝑥 𝜕𝑥 𝜕𝑥
the optimal value of ℎ is given by a nonlinear algebraic 𝜕𝜙2 𝜕 𝜕𝜙1

+ 𝑖𝜔𝜏2 𝛽1 − 𝛽2 (𝜙 ),
equation: 𝜕𝑥 𝜕𝑥 2 𝜕𝑥
𝜕 𝜕𝜙
𝑑Δ (ℎ) 𝑁2 [𝜙1 , 𝜙2 ] = ((1 − 𝑖𝜔𝜏1 ) 𝜙2 − 𝑎 (1 − 𝑖𝜔𝜏1 𝛿) 1
= 0. (13) 𝜕𝑡 𝜕𝑥
𝑑ℎ
1 𝜕𝜙 2 𝜕 𝜕𝜙 𝜕𝜙
3. Application of HAM on − 𝑏( 1 ) − (1 + 𝛼 1 ) 2 ) .
2 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑥
the Nonlinear Magnetothermoelastic (18)
with Rotation Equations
In this section, we use the homotopy analysis method to cal- Choosing 𝐻𝑖 (𝑡) = 1 for 𝑖 = 1, 2, the zeroth-order deformation
culate the approximate solutions of the following nonlinear equations are
magnetothermoplastic model with rotation equations
(1 + 𝜎1 ) 𝑢𝑡𝑡 + Ω𝑢𝑡 − 𝑢𝑥𝑥 (1 − 𝜎2 + 2𝛾𝑢𝑥 + 3𝛿𝑢𝑥2 ) (1 − 𝑞) L1 [𝜙1 (𝑥, 𝑡; 𝑞) − 𝑢0 (𝑥, 𝑡)]
− 𝛽1 (1 − 𝑖𝜔𝜏2 ) 𝜃𝑥 − 𝛽2 (𝜃𝑢𝑥 )𝑥 = 0, = 𝑞ℎ1 𝑁1 [𝜙1 (𝑥, 𝑡; 𝑞) , 𝜙2 (𝑥, 𝑡; 𝑞)] ,

(19)
1 (14) (1 − 𝑞) L2 [𝜙2 (𝑥, 𝑡; 𝑞) − V0 (𝑥, 𝑡)]
(𝜃 (1 − 𝑖𝜔𝜏1 ) − 𝑎𝑢𝑥 (1 − 𝑖𝜔𝛿𝜏1 ) − 𝑏𝑢𝑥2 )
2 𝑡
= 𝑞ℎ2 𝑁2 [𝜙1 (𝑥, 𝑡; 𝑞) , 𝜙2 (𝑥, 𝑡; 𝑞)] ,
− [(1 + 𝛼𝑢𝑥 ) 𝜃𝑥 ]𝑥 = 0,
where 𝜎1 , 𝜎2 , Ω, 𝛾, 𝛽1 , 𝛽2 , 𝑎, 𝑏, and 𝛼 are arbitrary constants where

with the initial conditions
𝜙1 (𝑥, 𝑡; 0) = 𝑢0 (𝑥, 𝑡) , 𝜙1 (𝑥, 𝑡; 1) = 𝑢 (𝑥, 𝑡) ,
𝑢 (𝑥, 0) = 𝜃 (𝑥, 0) = 𝐴 (1 − cos (𝑥)) , (20)
(15) 𝜙2 (𝑥, 𝑡; 0) = V0 (𝑥, 𝑡) , 𝜙2 (𝑥, 𝑡; 1) = 𝜃 (𝑥, 𝑡) .
𝑢𝑡 (𝑥, 0) = 𝜃𝑡 (𝑥, 0) = 0,
where 𝐴 is an arbitrary constant and the boundary conditions Then, the 𝑚th-order deformation equations become
𝑢 (0, 𝑡) = 𝜃 (0, 𝑡) = 0,
(16) L1 [𝑢𝑚 (𝑥, 𝑡) − 𝜘𝑚 𝑢𝑚−1 (𝑥, 𝑡)] = ℎ1 R1𝑚 (𝑢⃗ ⃗
𝑢𝑡 (0, 𝑡) = 𝜃𝑡 (0, 𝑡) = 0. 𝑚−1 , 𝜃𝑚−1 ) ,
L2 [𝜃𝑚 (𝑥, 𝑡) − 𝜘𝑚 𝜃𝑚−1 (𝑥, 𝑡)] = ℎ2 R2𝑚 (𝑢⃗ ⃗

To demonstrate the effectiveness of the method, we consider 𝑚−1 , 𝜃𝑚−1 ) ,
the system of nonlinear initial-value problem (14) with the (21)

where In this case, where 𝑢0 and 𝜃0 are constants, the general

solution of (23) is taking the following form:
R1𝑚 (𝑢⃗ ⃗
𝑚−1 , 𝜃𝑚−1 ) ∞
𝑢 (𝑥, 𝑡) = 𝑢0 (𝑥, 𝑡) + ∑ 𝑢𝑚 (𝑥, 𝑡) ,
𝜕2 𝑢𝑚−1 𝑚=1
= (24)
𝜕𝑡2 ∞
𝜃 (𝑥, 𝑡) = 𝜃0 (𝑥, 𝑡) + ∑ 𝜃𝑚 (𝑥, 𝑡) .
1
+ (Ω(𝑢𝑚−1 )𝑡 − (𝑢𝑚−1 )𝑥𝑥 + 𝜎2 (𝑢𝑚−1 )𝑥𝑥 𝑚=1
1 + 𝜎1
𝑚−1 The problems above can be readily solved by symbolic

− 2𝛾 ∑ (𝑢𝑗 )𝑥𝑥 (𝑢𝑚−1−𝑗 )𝑥 computation packages such as Mathematica. Thereupon,
𝑗=0 successive solving of these problems yields
𝑚−1 𝑖
− 3𝛿 ∑ ∑(𝑢𝑗 )𝑥 (𝑢𝑖−𝑗 )𝑥 (𝑢𝑚−1−𝑖 )𝑥𝑥 𝑢0 (𝑥, 0) = 𝐴 (1 − cos (𝑥)) ,
𝑖=0 𝑗=0
𝜃0 (𝑥, 0) = 𝐴 (1 − cos (𝑥)) ,
𝑚−1
− 𝛽2 ∑ (𝜃𝑗 )𝑥 (𝑢𝑚−1−𝑗 )𝑥 − 𝛽1 (𝜃𝑚−1 )𝑥 𝐴ℎ𝑡2
𝑗=0 𝑢1 (𝑥, 𝑡) = (2𝐴𝛽2 (− cos (𝑥) + cos (2𝑥))
4 (1 + 𝜎1 )
𝑚−1
−𝛽2 ∑ (𝜃𝑗 ) (𝑢𝑚−1−𝑗 )𝑥𝑥 + 𝑖𝜔𝜏2 𝛽1 𝜃𝑚−1 ) , + cos (𝑥) (−2 − 3𝐴2 𝛿
𝑗=0
+ 3𝐴2 𝛿 cos (2𝑥)
R2𝑚 (𝑢⃗ ⃗
𝑚−1 , 𝜃𝑚−1 )
−4𝐴𝛾 sin (𝑥) + 2𝜎2 )
𝜕𝜃𝑚−1 1 +2𝑖 sin (𝑥) 𝛽1 (𝑖 + 𝜔𝜏2 )) ,
= + ( − 𝑎(𝑢𝑚−1 )𝑥𝑡
𝜕𝑡 1 − 𝑖𝜔𝜏1
𝐴ℎ𝑡 cos (𝑥) (1 + 2𝐴𝛼 sin (𝑥))
𝜃1 (𝑥, 𝑡) = ,
1 𝑚−1 −1 + 𝑖𝜔𝜏1
− 𝑏 ∑ (𝑢𝑗 )𝑥𝑡 (𝑢𝑚−1−𝑗 )𝑥𝑡
2 𝑗=0 𝑢2 (𝑥, 𝑡)
− (𝜃𝑚−1 )𝑥𝑥 𝐴ℎ𝑡2
=
𝑚−1 4 (1 + 𝜎1 )
− 𝛼 ∑ (𝜃𝑚−1−𝑗 )𝑥 (𝑢𝑗 )𝑥𝑥
𝑗=0 × (2𝐴𝛽2 (− cos (𝑥) + cos (2𝑥)) + cos (𝑥)
𝑚−1 × (−2 − 3𝐴2 𝛿 + 3𝐴2 𝛿 cos (2𝑥)

− 𝛼 ∑ (𝜃𝑚−1−𝑗 )𝑥𝑥 (𝑢𝑗 )𝑥
𝑗=0
−4𝐴𝛾 sin (𝑥) + 2𝜎2 )
− (𝜃𝑚−1 )𝑥𝑥 + 𝑎𝑖𝜔𝜏1 𝛿(𝑢𝑚−1 )𝑥𝑡 ) . +2𝑖 sin (𝑥) 𝛽1 (𝑖 + 𝜔𝜏2 ))

1
(22) + 2
384(1 + 𝜎1 )
For simplicity, we suppose ℎ1 = ℎ2 ; the system (21) has the × (𝐴ℎ2 𝑡4 (−135𝐴4 𝛿2 cos (5𝑥)
following general solutions:
+ 240𝐴3 𝛿𝛾 sin (4𝑥) − 16𝐴2
𝑡
× (2 cos(𝑥) − 5cos 2 (𝑥) + 3 cos (3𝑥))
𝑢𝑚 (𝑥, 𝑡) = 𝜘𝑚 𝑢𝑚−1 (𝑥, 𝑡) + ℎ∬ R1𝑚 (𝑢⃗ ⃗
𝑚−1 , 𝜃𝑚−1 ) 𝑑𝑡 𝑑𝑡,
0 × 𝛽22 − 96𝐴𝛾 sin (2𝑥)
𝑡
𝜃𝑚 (𝑥, 𝑡) = 𝜘𝑚 𝜃𝑚−1 (𝑥, 𝑡) + ℎ ∫ R2𝑚 (𝑢⃗ ⃗ × (1 + 2𝐴2 𝛿 − 𝜎2 ) + 3𝐴2 cos (3𝑥)
𝑚−1 , 𝜃𝑚−1 ) 𝑑𝑡.
0
(23) × (48𝛿 + 63𝐴2 𝛿2 + 32𝛾2 − 48𝛿𝜎2 )
𝐴ℎ (1 + ℎ) 𝑡 cos (𝑥) (1 + 2𝐴𝛼 sin (𝑥))

𝜃2 (𝑥, 𝑡) =
2
+ 8𝐴𝛽2 (− 2 (2 + 3𝐴 𝛿) cos (𝑥) −1 + 𝑖𝜔𝜏1
𝐴2 ℎ2 𝑡3 𝛼
+ 10 cos (2𝑥) +
6 (1 + 𝜎1 ) (𝑖 + 𝜔𝜏1 )
+ 𝐴 (3𝐴𝛿 (3 cos (2𝑥)
1
× (− 𝑖 (2𝐴𝛾 cos (𝑥) − 6𝐴𝛾 cos (3𝑥)
+ 6 cos (3𝑥) 2
−7 cos (4𝑥)) − 9𝐴2 𝛿 sin (4𝑥)
− 2𝛾 (sin (𝑥) + 2𝐴 (sin (𝑥) + sin (2𝑥)
+ 6 sin (2𝑥) −3 sin (3𝑥)) 𝛽2
−9 sin (3𝑥))) + 2 sin (2𝑥) (1 + 3𝐴2 𝛿 − 𝜎2 ))
+ 2 (2 cos (𝑥)
+ cos (2𝑥) 𝛽1 (𝑖 + 𝜔𝜏2 )
−5 cos (2𝑥)) 𝜎2 ) 1
+
8 (𝑖 + 𝜔𝜏1 )
− 2 cos (𝑥) (8 + 𝐴2
× (𝑖𝐴ℎ2 𝑡2
× (3𝛿 (8 + 9𝐴2 𝛿)
+16𝛾2 ) 𝑎
×(
1 + 𝜎1
+8𝜎2 (−2 − 3𝐴2 𝛿 + 𝜎2 ))
× ( − 4 sin (𝑥)
+ 4𝑖𝛽1 (4 sin (𝑥) + 𝐴 (8] cos (2𝑥)
+ 𝐴 (−8𝛾 cos (2𝑥) − 3𝐴𝛿 (sin (𝑥)
+ 3𝐴𝛿 (sin (𝑥) −3 sin (3𝑥)))
− 3 sin (3𝑥))) + 4 sin (𝑥) (𝐴 (−1 + 4 cos (𝑥))
− 4 sin (𝑥) (𝐴 (−1 +2 cos (𝑥)) ×𝛽2 + 𝜎2 )
×𝛽2 + 𝜎2 )) +4 cos (𝑥) 𝛽1 (1 − 𝑖𝜔𝜏2 ))
× (𝑖 + 𝜔𝜏0 ))) 𝑖𝑎𝜔𝛿1
+ (4 sin (3𝑥)
1 1 + 𝜎1
− 2
24 (1 + 𝜎1 ) (𝑖 + 𝜔𝜏1 ) + 𝐴 (−8] cos (𝑥)
× (𝐴ℎ2 𝑡3 (−2 − 3𝐴2 𝛿 + 3𝐴2 𝛿 cos (2𝑥) + 3𝐴𝛿 (sin (𝑥)

−3 sin (3𝑥)))
−4𝐴𝛾 sin (𝑥) + 2𝜎2 )
− 4 sin (𝑥) (𝐴 (−1
× (𝑖 + 𝜔𝜏1 ) + 2𝐴𝛽2
+4 cos (𝑥))
× (𝑖𝐴𝛼 sin (𝑥) (1 + 𝜎1 ) − 3𝐴𝑖𝛼 sin (3𝑥)
×𝛽2 + 𝜎2 )
× (1 + 𝜎1 ) + 2Ω cos (𝑥) (𝑖 + 𝜔𝜏1 )
+ 4𝑖 cos (𝑥) 𝛽1 (𝑖 + 𝜔𝜏2 ))
− 2𝑖 cos (2𝑥) (1 + Ω + 𝜎1 − 𝑖Ω𝜔𝜏1 ))
1
× 𝜏1 −
+ 4𝛽1 (𝑖 + 𝜔𝜏2 ) 𝑖 + 𝜔𝜏1
× (−2𝐴𝛼 cos (2𝑥) (1 + 𝜎1 ) × 4𝑖 ((1 + 𝐴2
+ sin (𝑥) (1 + Ω + 𝜎1 − 𝑖𝜔Ω𝜏1 ))) , + 𝛼2 cos (𝑥) + 𝐴𝛼
Table 1: The optimal values of ℎ at third-order approximate Table 2: The optimal values of ℎ at 5th-order approximate solutions
solutions of (14) when 𝑥 = 1.0, 𝑡 = 1.0, for 𝛿 = 1, 𝜎1 = 0.2, 𝜎2 = of (14) when 𝑥 = 1.0, 𝑡 = 1.0, for 𝛿 = 1, 𝜎1 = 0.2, 𝜎2 = 0.1, Ω =
0.1, Ω = 0.1, and 𝜔 = 0.02. 0.1, and 𝜔 = 0.02.
𝜏1 𝜏2 Optimal value of ℎ Minimum value 𝜏1 𝜏2 Optimal value of ℎ Minimum value
𝑢(1, 1) 𝑢(1, 1)
1 0 −0.95623 1.37312 × 10−3 1 0 −0.990321 1.37312 × 10−7
0 1 −0.97543 7.9675 × 10−7 0 1 −0.995339 7.9675 × 10−9
𝜃(1, 1) 𝜃(1, 1)
1 0 −0.95623 1.3242 × 10−2 1 0 −0.990321 1.3242 × 10−4
0 1 −0.97543 2.1432 × 10−5 0 1 −0.995339 2.1432 × 10−6
× (−3𝐴 × 𝛼 cos (3𝑥) is concluded that the displacement 𝑢(1, 1) and temperature
𝜃(1, 1) increase with increasing the values of ℎ to their
+5 sin (2𝑥))))) . maxima and then decrease with the high values of ℎ; also, it
is shown that 𝑢(1, 1) is convergent when −0.9 < ℎ < −0.6 and
(25) 𝜃(1, 1) is convergent when −0.6 < ℎ < −0.4.
Figures 2–7 show the variations of the radial displacement
Now we make calculations for the results obtained by and temperature with respect to axial 𝑥, respectively, for
the HAM using the Mathematica software package with the different values of the time 𝑡, rotation Ω, and sensitive parts
following arbitrary constants: of the magnetic fields 𝜎1 and 𝜎2 . In both figures, it is clear
that the radial displacement and temperature have a zero
𝑎 = 0.5, 𝐴 = 0.001, 𝑏 = 0.5, 𝛼 = 1, value only in a bounded region of space. It is observed from
Figure 3 that the displacement 𝑢 and the temperature 𝜃 start
𝛽1 = 0.5, 𝛽2 = 0.05, 𝛾 = 1, 𝛿 = 0.8, (26) from their maximum values, decrease, and increase periodi-
cally with an increase of the coordinate 𝑥; also, it is obvious
Ω = 1, 𝜎1 = 0.2, 𝜎2 = 0.1.
that their values take the minimum values and increases with
the increasing values of the time 𝑡. From Figure 4, one can
To investigate the influence of ℎ on the convergence of the
see that 𝑢 and 𝜃 decrease with an increase of the rotation Ω.
solution series given by the HAM, we first plot the so-called
It is shown that the components of the displacement 𝑢 and
ℎ-curves of 𝑢(1, 1) and 𝜃(1, 1). According to the ℎ-curves, it
the temperature 𝜃 start from the minimum values near zero,
is easy to discover the valid region of ℎ. We used 3 terms in
increase, and then decrease periodically with the coordinate
evaluating the approximate solution 𝑢(𝑥, 𝑡) ≅ ∑2𝑖=0 𝑢𝑖 (𝑥, 𝑡) 𝑥; it is clear also that there is a slight increase with an increase
and 𝜃(𝑥, 𝑡) ≅ ∑2𝑖=0 𝜃𝑖 (𝑥, 𝑡). Note that the solution series of the sensitive parts of the magnetic field (see, Figures 5 and
contains the auxiliary parameter ℎ which provides us with 6). It is shown that the increasing of the coordinate 𝑥 sensitive
a simple way to adjust and control the convergence of the an increasing and decreasing on them periodically due to
solution series. In general, by means of the so-called ℎ-curve appearance of the pairs (cos, sin) in the initial condition and
that is, a curve of a versus ℎ. As pointed by Liao [8] and the approximate solutions; it is also clear that the components
Turkyilmazoglu [25], the valid region of ℎ is a horizontal begin from their minimum values and increase absolutely
line segment. Therefore, it is straightforward to choose an with the variation of the time 𝑡. The variations of the rotation
appropriate range for ℎ which ensures the convergence of and magnetic field tend to slightly affect the displacement and
the solution series (Tables 1 and 2). We sketch the ℎ-curve of the temperature.
𝑢(1, 1) and 𝜃(1, 1) in Figure 1, which shows that the solution From Figures 7 and 8 (GL model), it is clear that the
series is convergent when −1.45 < ℎ < −0.5. displacement component and temperature if the rotation
and magnetic field are vanish, take larger values than the
4. Discussion corresponding values with the rotation and magnetic field
effects.
In order to gain physical insight, the temperature 𝑇 and radial Figures 9, 10, and 11 show the variations of the displace-
displacement 𝑢 have been discussed by assigning numerical ment and temperature with respect to the time 𝑡 with LS and
values to the parameter encountered in the problem in GL models; it is shown that the radial displacement and the
which the numerical results are displayed with the graphical temperature increase with an increase of 𝑡 that takes a slight
illustrations in 2D and 3D formats. The variations are shown change with the rotation Ω if 𝜎1 = 𝜎2 = 0, 𝜎1 if Ω = 𝜎2 = 0,
in Figures 1–15, with the view of illustrating the theoretical and 𝜎2 if Ω = 𝜎1 = 0.
results obtained in the preceding sections; a numerical result Figures 12 and 13 show clearly the variations of the
is calculated for the homotopy analysis method. displacement and temperature in the presence and absence of
Figures 1 and 2 display the ℎ-curve of the third-order the rotation and sensitive magnetic field; it is observed that
approximate solutions (14) when 𝑥 = 1.0, 𝑡 = 1.0; it 𝑢 and 𝜃 in presence of the parameters are smaller than the
0.00065
0.0006
0.0006
u(1, 1)
0.0004
𝜃(1, 1)
0.00055
0.0002
0.0005
0
−1.75 −1.5 −1.25 −1 −0.75 −0.5 −0.25 0 −1.5 −1.25 −1 −0.75 −0.5 −0.25 0 0.25
h h
(a) (b)
Figure 1: The ℎ-curve of the third-order approximate solutions of (14) when 𝑥 = 1.0, 𝑡 = 1.0; for LS model when 𝜏2 = 0, 𝜏1 = 1, 𝛿 = 1,
𝜎1 = 0.2, 𝜎2 = 0.1, Ω = 0.1, and 𝜔 = 0.02.
0.0006
0.0006
0.0005
0.0004
u(1, 1)
0.0004
𝜃(1, 1)
0.0003
0.0002
0.0002
0.0001
0 0
−2.5 −2 −1.5 −1 −0.5 0 0.5 −1.25 −1 −0.75 −0.5 −0.25 0 0.25
h h
(a) (b)
Figure 2: The ℎ-curve of the third-order approximate solutions of (14) when 𝑥 = 1.0, 𝑡 = 1.0; for GL model when 𝜏2 = 0.2, 𝜏1 = 0.1, 𝛿 = 0,
𝜎1 = 0.2, 𝜎2 = 0.1, Ω = 0.1, and 𝜔 = 0.02.
100 1
75 50 0.75 50
50 𝜃 0.5
u 40
25 40 0.25
0 0
30 30
0 0 t
t
20 20
5 5
x 10 10 10
10 x
0
15 0 15
(a) (b)
Figure 3: Variations of the displacement 𝑢 and the temperature 𝜃 for various values of the 𝑥-axis and time 𝑡 when 𝜏2 = 0, 𝜏1 = 0.1, 𝛿 = 1,
𝜎1 = 0.2, 𝜎2 = 0.1, Ω = 0.1, and 𝜔 = 0.02.
0.002
0.0015 0.0015
50 50
0.001
u 0.001 𝜃
0.0005 40 0.0005 40
0 0
30 30
0 0
Ω Ω
5 20 5 20
10 10 10 10
x x
150 150
(a) (b)
Figure 4: Variations of the displacement 𝑢 and the temperature 𝜃 for various values of the 𝑥-axis and rotation Ω when 𝑡 = 0.1, 𝜏2 = 0, 𝜏1 = 0.1,
𝛿 = 1, 𝜎1 = 0.2, 𝜎2 = 0.1, and 𝜔 = 0.02.
0.002
0.0015 50 0.0015
550
u 0.001 0.001
40 𝜃 40
0.0005 0.0005
0 30 0 30
0 𝜎1 0 𝜎1
20 20
5 5
10 10
x 10 x 10
0 0
15 15
(a) (b)
Figure 5: Variations of the displacement 𝑢 and the temperature 𝜃 for various values of the 𝑥-axis and magnetic field 𝜎1 when 𝑡 = 0.1, 𝜏2 = 0,
𝜏1 = 0.1, 𝛿 = 1, 𝜎2 = 0.1, Ω = 0.1, and 𝜔 = 0.02.
corresponding values in the absence of 𝜎1 , 𝜎2 , and Ω, but there it is concluded that takes large values for LS comparing with
is a slight change for LS and GL models. those in GL model, vice versa for the temperature.
Finally, Figures 14 and 15 show the variations of the dis- It is obvious that if the rotation and the sensitive part of
placement and the temperature with respect to the time 𝑡 for the magnetic field are neglected, the approximate solutions
different values of 𝛽 = 0.5, 0.005 and with and without rota- obtained by HAM agree with the results obtained by Sweilam
tion and magnetic field effects, respectively. It is obvious that and Khader [1], taking into consideration VIM. Finally,
the radial displacement and the temperature increase with an it is obvious that the displacement takes large values if
increase of 𝑡; the displacement increases with an increase of there are no rotation, thermal relaxation times, and sensitive
𝛽 parameter, but the temperature is not affected by 𝛽. Is also part of the magnetic field parameters compared with the
seen that the radial displacement and the temperature take corresponding value with the rotation and magnetic fields
large values with the rotation and magnetic field effects. Also, parameters.
0.002
0.0015
0.0015 5
50 550
u 0.001 0.001
40 𝜃 40
0.0005 0.0005
0 30 0 30
0 𝜎2 0 𝜎2
20 20
5 5
10 10
x 10 x 10
0
15 15
(a) (b)
Figure 6: Variations of the displacement 𝑢 and temperature 𝜃 for various values of the 𝑥-axis and magnetic field 𝜎2 when 𝑡 = 0.1, 𝜏2 = 0,
𝜏1 = 0.1, 𝛿 = 1, and 𝜔 = 0.02.
200 1
150 50 0.75 50
u 100 40 𝜃 0.5
40
50 0.25
0 30 0 30
0 t 0 20 t
20
5 5
10 10
10 x 10
x
0 0
15 15
(a) (b)
Figure 7: Variations of the displacement 𝑢 and the temperature 𝜃 for various values of the 𝑥-axis and time 𝑡 when Ω = 𝜎1 = 𝜎2 = 0, 𝜏2 = 0.2,
𝜏1 = 0.1, 𝛿 = 1, and 𝜔 = 0.02.
The results indicate that the effect of the rotation and the provides a quite successful in dealing with such problems.
magnetic field on the radial displacement and the tempera- This method gives numerical solutions in the elastic medium
ture is very pronounced. without any restrictions on the actual physical quantities that
appear in the governing equations of the considered problem.
Important phenomena are observed in these computations.
5. Conclusion
(i) The homotopy analysis method has been successfully
Due to the complicated nature of the governing equations of applied to obtain the numerical solutions of the non-
the magnetothermoelastic , the finished works in this field linear equation with initial conditions. The reliability
are unfortunately limited. The method used in this study of this method and reduction in computations give
200 1
150 0.75 50
50
100 0.5
u 40 𝜃 40
50 0.25
0 30 0 30
0 0 t
20 t 20
5 5
10 10
x 10 x 10
0 0
15 15
(a) (b)
Figure 8: Variations of the displacement 𝑢 and the temperature 𝜃 for various values of the 𝑥-axis and time 𝑡 when Ω = 𝜎1 = 𝜎2 = 0, 𝜏2 = 0.2,
𝜏1 = 0.1, 𝛿 = 0, and 𝜔 = 0.02.
0.002
0.0015 0.0015
50 50
0.001
u 0.001 𝜃
40 0.0005 40
0.0005
0 0
30 30
0 Ω 0
Ω
20 20
5 5
10 10
x 10 x 10
0 0
15 15
(a) (b)
Figure 9: Variations of the displacement 𝑢 and the temperature 𝜃 for various values of the 𝑥-axis and rotation Ω when 𝜎1 = 𝜎2 = 0, 𝑡 = 0.1,
𝜏2 = 0.2, 𝜏1 = 0.1, 𝛿 = 0, and 𝜔 = 0.02.
this method a wider applicability. HAM contains a other methods. The illustrative examples suggest that
certain auxiliary parameter ℎ which provides us with HAM is a powerful method for nonlinear problems in
a simple way to adjust and control the convergence science and engineering. Mathematica has been used
region and rate of convergence of the series solution. for computations in this paper.
It was also demonstrated that the Adomian decom-
(ii) It was found that for large values of time the large
position method, homotopy perturbation method,
and the generalization give numerical results. The
and variational iteration method are specialcases
case is quite different when we consider small values
of. HAM is clearly a very efficient and powerful
of rotation and magnetic field. The coupled theory
technique for finding the numerical solutions of
predicts infinite speeds of wave propagation. The
the proposed equation. It therefore provides more
solutions obtained in the context of generalized ther-
realistic series solutions that generally converge very
moelasticity theory, however, exhibit the behavior of
rapidly in real physical problems. HAM provides us
finite speeds of wave propagation.
with a convenient way of controlling the convergence
of approximation series, which is a fundamental (iii) By comparing Figures 1–15 for thermoelastic medium
qualitative difference in analysis between HAM and with presence and absence of the rotation and
0.002
0.0015
0.0015 50 50
u 0.001 0.001
40 𝜃 40
0.0005 0.0005
0 30 0 30
0 𝜎1 0 𝜎1
20 20
5 5
10 10
x 10 x 10
0 0
15 15
(a) (b)
Figure 10: Variations of the displacement 𝑢 and the temperature 𝜃 for various values of the 𝑥-axis and magnetic field 𝜎1 when Ω = 𝜎2 = 0,
𝑡 = 0.1, 𝜏2 = 0.2, 𝜏1 = 0.1, 𝛿 = 0, and 𝜔 = 0.02.
0.002
0.0015
0.0015 10 10
u 𝜃 0.001
0.001
8 8
0.0005 0.0005
0 6 0
6
0 𝜎2
𝜎2
4
5 5 4
2
x 10 x 10 2
0
15 15
(a) (b)
Figure 11: Variations of the displacement 𝑢 and the temperature 𝜃 for various values of the 𝑥-axis and magnetic field 𝜎2 when Ω = 𝜎1 = 0,
𝑡 = 0.1, 𝜏2 = 0.2, 𝜏1 = 0.1, 𝛿 = 0, and 𝜔 = 0.02.
magnetic field, it was found that they have the (iv) The results presented in this paper will be very help-
same behavior in both media. The effect of rotation ful for researchers concerned with material science,
and sensitive parts of the magnetic field is strongly designers of new materials, and low-temperature
effective in the displacement and temperature of the physicists, as well as for those working on the develop-
propagation of the harmonic waves propagation in ment of a theory of hyperbolic propagation of hyper-
nonlinear thermoelasticity. bolic thermoelastic. The study of the phenomenon of
250000
40
200000
30
150000
𝜃(100)
u(100)
20
100000
10
50000
0 0
0 100 200 300 400 500 0 100 200 300 400 500
t t
(a) (b)
Figure 12: The displacement as a function of time and the temperature without and with rotation and magnetic field (LS) at 𝑥 = 100, 𝑎 = 0.5,
𝛽1 = 0.5, 𝐴 = 0.001, 𝑏 = 0.5, 𝛼 = 1, 𝛽2 = 0.05, 𝛾 = 1, 𝛿 = 0.8, 𝜏2 = 0, 𝜏1 = 0.1, 𝛿 = 1, 𝜛 = 0.02 (Ω = 𝜎1 = 𝜎2 = 0, —), and (Ω = 0.1, 𝜎1 = 0.2,
𝜎2 = 0.1, - - -).
250000
40
200000
30
150000
𝜃(100)
u(100)
20
100000
10
50000
0 0
0 100 200 300 400 500 0 100 200 300 400 500
t t
(a) (b)
Figure 13: The displacement as a function of time and the temperature without and with rotation and magnetic field (GL) at 𝑥 = 100, 𝑎 = 0.5,
𝛽1 = 0.5, 𝐴 = 0.001, 𝑏 = 0.5, 𝛼 = 1, 𝛽2 = 0.05, 𝛾 = 1, 𝛿 = 0.8, 𝜏2 = 0.2, 𝜏1 = 0.1, 𝛿 = 0, 𝜛 = 0.02 (Ω = 𝜎1 = 𝜎2 = 0, —), and (Ω = 0.1, 𝜎1 = 0.2,
𝜎2 = 0.1, - - -).
250000
40
200000
30
150000
𝜃(100)
u(100)
20
100000
10
50000
0 0
0 100 200 300 400 500 0 100 200 300 400 500
t t
(a) (b)
Figure 14: The displacement as a function of time and the temperature for two values of 𝛽1 (LS) at 𝑥 = 100, 𝑎 = 0.5, 𝐴 = 0.001, 𝑏 = 0.5,
𝛼 = 1, 𝛽2 = 0.05, 𝛾 = 1, 𝛿 = 0.8, Ω = 0.1, 𝜎1 = 0.2, 𝜎2 = 0.1, 𝜏2 = 0, 𝜏1 = 0.1, 𝛿 = 1, 𝜛 = 0.02, — 𝛽1 = 0.5, and - - - 𝛽1 = 0.005.
70
200000
60
150000 50
40
𝜃(100)
u(100)
100000
30
20
50000
10
0 0
0 100 200 300 400 500 0 100 200 300 400 500
t t
(a) (b)
Figure 15: The displacement as a function of time and the temperature for two values of 𝛽1 (GL) at 𝑥 = 100, 𝑎 = 0.5, 𝐴 = 0.001, 𝑏 = 0.5,
𝛼 = 1, 𝛽2 = 0.05, 𝛾 = 1, 𝛿 = 0.8, Ω = 0.1, 𝜎1 = 0.2, 𝜎2 = 0.1, 𝜏2 = 0.2, 𝜏1 = 0.1, 𝛿 = 0, 𝜛 = 0.02, — 𝛽1 = 0.5, and - - - 𝛽1 = 0.005.
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http://dx.doi.org/10.1155/2013/858597
Research Article
The Effect of Boundary Slip on the Transient Pulsatile Flow of
a Modified Second-Grade Fluid
N. Khajohnsaksumeth,1 B. Wiwatanapataphee,2 and Y. H. Wu1

1
Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia
2
Department of Mathematics, Mahidol University, Faculty of Science, Bangkok 10400, Thailand
Correspondence should be addressed to N. Khajohnsaksumeth; jobscma@yahoo.com

and B. Wiwatanapataphee; benchawan.wiw@mahidol.ac.th
Received 20 May 2013; Accepted 9 August 2013
Copyright © 2013 N. Khajohnsaksumeth et al. This is an open access article distributed under the Creative Commons Attribution
cited.
We investigate the effect of boundary slip on the transient pulsatile fluid flow through a vessel with body acceleration. The Fahraeus-
Lindqvist effect, expressing the fluid behavior near the wall by the Newtonian fluid while in the core by a non-Newtonian fluid,
is also taken into account. To describe the non-Newtonian behavior, we use the modified second-grade fluid model in which the
viscosity and the normal stresses are represented in terms of the shear rate. The complete set of equations are then established and
formulated in a dimensionless form. For a special case of the material parameter, we derive an analytical solution for the problem,
while for the general case, we solve the problem numerically. Our subsequent analytical and numerical results show that the slip
parameter has a very significant influence on the velocity profile and also on the convergence rate of the numerical solutions.
1. Introduction taken into account under the same problem by Majhi et al.
[7, 10]. Later, Massoudi and Phuoc [11] used the (generalized)
In this paper, we study a fluid-structure interaction problem, second-grade fluid constitutive model to describe the shear
namely, the effect of boundary slip on the flow of a non- thinning and normal stress effect, and the behavior of blood
Newtonian fluid through microchannels. This problem has flow near the wall is modeled by the Newtonian fluid model,
many applications, and in this paper we particularly focus on while the behavior of the blood flow at the core is described
blood flow in the cardiovascular system. by the second-grade fluid model.
For the study of blood flow in arteries, two major types of In all of the above mentioned models, the so-called no-
constitutive models have been used. The first type of models slip boundary condition is used; namely, the velocity of
is based on the microcontinuum or the structured continuum flow relative to the solid is zero on the fluid-solid interface
theories [1–6] in which the balance laws are used to determine [12]. Although the no-slip condition is supported by many
the characteristics of blood motion. In the other type of experimental results, the existence of slip of a fluid on the
models, blood is considered as a suspension, and its flow is solid surface was also observed by many other researches
modeled by the non-Newtonian fluid mechanics. Due to the [13–20]. The Navier slip condition has been used by various
red blood cells (RBC) migration as shown experimentally, researchers to describe boundary slip and is a more general
blood has been modeled as a two-stage fluid by many boundary condition, in which the fluid velocity component
researchers [7–9]. The first stage is a peripheral layer which tangential to the solid surface, relative to the solid surface, is
is modeled as a Newtonian viscous fluid, while the other one proportional to the shear stress on the fluid-solid interface
is a centre core which is modeled as a non-Newtonian fluid. and the slip length. The surface characteristics constant, slip
The effect of body acceleration and pulsatile conditions were length, describes the “slipperiness” of the surface. Recently,
2.5
3 2
Dimensionless velocity
2.5
2 1.5
1.5
1
1
0.5
0
0.5
−0.5
5
4 1 0
Dim 3 0.8
ens 2 0.6
ion 0.4 r
less 1 0.2 nless
tim 0 0 im ensio −0.5
e D 0 1 2 3 4 5 6
Dimensionless time
r=0
r = 0.6
r=1
(a)
9 8.5
8.5
8 8
7.5
7 7.5
6.5
6
7
5.5
5
4 1 6.5
Dim 3 0.8
ens 2 0.6
ion 0.4
less 1 r
0.2 nless 6
tim
e 0 0 ensio
Dim 0 1 2 3 4 5 6
Dimensionless time
r=0
r = 0.6
r=1
(b)
Figure 1: The velocity profile in the small artery with radius 0.15 cm under two different slip parameter values: (a) 𝑙𝑏 = 0; (b) 𝑙𝑏 = 2. In the
figure, the 3D graphs show the variation of velocity as a function of time and location, while the 2D graphs show the variation of velocity with
time at three radial locations including the artery centre (𝑟 = 0), the interface of inner-outer layer (𝑟 = 0.6), and the arterial wall (𝑟 = 1).
we and many other researchers have investigated various flow this work include establishment of the underlying boundary
problems of Newtonian fluids with the traditional no-slip value problem for the problem, the derivation of an exact
and the Navier slip boundary conditions [12, 20–30], and it solution for a special case, and demonstration of the influence
is found that the boundary slip and the slip parameter have of the slip parameter on the flow profile and flow behavior.
significant influence on the flow of Newtonian fluids through The rest of the paper is organized as follows. In Section 2,
microchannels and tubes. we present the underlying boundary value problem for
Motivated by the above mentioned work, we extend the problem in dimensionless form. Then in Section 3, we
previous work on slip flows of Newtonian fluids [21, 22] to derive an exact solution for a special case. In Section 4, we
the case involving both Newtonian and non-Newtonian fluid investigate numerically the effect of the slip parameter for the
flow in the flow region. The new feature and contribution of general case. Finally, a conclusion is given in Section 5.
2.5
2
3
2.5
1.5
2
1.5
1
1
0.5 0.5
0
−0.5 0
5
4 1
Dim 3 0.8 −0.5
ens 2 0.6 0 1 2 3 4 5 6
ion 0.4
less 1 r
tim 0.2 nless Dimensionless time
e 0 0 ensio
Dim
r=0
r = 0.6
r=1
(a)
9
9 8.5
8.5
8
8
7.5 7.5
7
7
6.5
6 6.5
5.5
5 6
4 1
Dim 3 0.8
ens 0.6 5.5
ion 2
less 0.4 0 1 2 3 4 5 6
tim 1 r
0.2 nless
e 0 0
imensio Dimensionless time
D
r=0
r = 0.6
r=1
(b)
Figure 2: The velocity profile in the large artery with radius 0.50 cm under two different slip parameter values: (a) 𝑙𝑏 = 0; (b) 𝑙𝑏 = 2. In the
figure, the 3D graphs show the variation of velocity as a function of time and location, while the 2D graphs show the variation of velocity with
time at three radial locations including the artery centre (𝑟 = 0), the interface of inner-outer layer (𝑟 = 0.6), and the arterial wall (𝑟 = 1).
2. Mathematical Formulation where 𝜌 is the density of the fluid, 𝜕/𝜕𝑡 is the partial derivative
with respect to time, k is the velocity vector, b is the body
The flow of a fluid with no thermochemical and electromag- force vector, and T is the stress tensor.
netic effects can be described by the conservation equations The stress tensor is related to the velocity gradient by the
of mass and linear momentum; namely, constitutive equations. For a modified (generalized) second-
grade fluid [11, 31, 32], the constitutive equations can be
expressed by
𝜕𝜌
+ div (𝜌k) = 0, T = −𝑝I + Π𝑚/2 (𝜇A1 + 𝛼1 A2 + 𝛼2 A21 , ) , (2)
𝜕𝑡
(1)
𝜕k where 𝑚 is a material parameter, Π = (1/2) tr A21 is the
𝜌( + k ⋅ ∇k) = div T + 𝜌b, second invariant of A1 , 𝑝 is the fluid pressure, 𝜇 is the
𝜕𝑡
30 30
25 25
20 20
15 15
10 10
5 5
0 0
−5 −5
0 1 2 3 4 5 0 1 2 3 4 5
Dimensionless time Dimensionless time
lb = 0 lb = 6 lb = 0 lb = 6
lb = 2 lb = 8 lb = 2 lb = 8
lb = 4 lb = 4
(a) (b)
Figure 3: Diagrams showing the velocity profile on the arterial wall with five different slip parameters 𝑙𝑏 for two different artery radii (a)
𝑟 = 0.15 cm; (b) 𝑟 = 0.5 cm.
coefficient of viscosity, 𝛼𝑖 are material moduli (the normal where 𝐴 0 , 𝐴 1 , 𝜔𝑝 = 2𝜋𝑓𝑝 , and 𝑓𝑝 are the constant component
stress coefficients), and Ai are the kinematical tensors given of the pressure gradient, the amplitude of the pressure
by fluctuation (establishing the systolic and diastolic pressures),
the circular frequency, and the frequency of pulse rate,
A 1 = L + L𝑇 , respectively.
(3) The body acceleration 𝐺 can be approximated by
𝜕A1
A2 = + [grad (A1 )] k + A1 L + (L)𝑇 A1 ,
𝜕𝑡 𝐺 = 𝐴 𝑔 cos (𝜔𝑏 𝑡 + 𝜙) , (7)
in which 𝐿 is grad k and the superscript 𝑇 refers to matrix where 𝐴 𝑔 is the amplitude, 𝑓𝑏 = 𝜔𝑏 /2𝜋 is the frequency, and
transposition. 𝜙 is the lead angle of 𝐺 with respect to the action of the heart.
For the axially symmetrical blood flow through a circular Substituting (5)–(7) into (4), the blood flow equation for
tube of radius 𝑏, we can assume that k = V(𝑟, 𝑡)ez , where 𝑧 a modified second-grade fluid in the 𝑧-direction, in the inner
is the axial direction and 𝑟 is the radial direction. Under the and outer core, becomes
periodic body acceleration and a unsteady pulsatile pressure
gradient [7, 10], the momentum equation in the 𝑧-direction 𝜕V1
in the cylindrical polar coordinate (𝑟, 𝜃, 𝑧) is 𝜌1 = 𝐴 0 + 𝐴 1 cos 𝜔𝑝 𝑡 + 𝜌𝐴 𝑔 cos (𝜔𝑏 𝑡 + 𝜙)
𝜕𝑡
𝜕V 𝜕𝑝 1 𝜕 1 𝜕 󵄨󵄨 𝜕V 󵄨󵄨𝑚 𝜕V
󵄨 󵄨
𝜌 = − + 𝜌𝐺 + (𝑟𝑇𝑟𝑧 ) . (4) + (𝑟𝜇1 󵄨󵄨󵄨 1 󵄨󵄨󵄨 1
) , for 0 ≤ 𝑟 ≤ 𝑎,
𝜕𝑡 𝜕𝑧 𝑟 𝜕𝑟 𝑟 𝜕𝑟 󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟
(8)
The shear stress 𝑇𝑟𝑧 for a generalized second-grade fluid 𝜕V
can be expressed by 𝜌2 2 = 𝐴 0 + 𝐴 1 cos 𝜔𝑝 𝑡 + 𝜌𝐴 𝑔 cos (𝜔𝑏 𝑡 + 𝜙)
𝜕𝑡
󵄨󵄨 𝜕V 󵄨󵄨𝑚 𝜕V 1 𝜕 𝜕V
{ 󵄨󵄨 1 󵄨󵄨 1 + (𝑟𝜇2 2 ) , for 𝑎 ≤ 𝑟 ≤ 𝑏.
𝜇
{ 1 󵄨󵄨󵄨󵄨 𝜕𝑟 󵄨󵄨󵄨󵄨 𝜕𝑟
{ 0 ≤ 𝑟 ≤ 𝑎,
𝑟 𝜕𝑟 𝜕𝑟
𝑇𝑟𝑧 = { (5)
{
{ 𝜕V2
𝜇 𝑎 ≤ 𝑟 ≤ 𝑏. In order to completely define the problem, boundary
{ 2 𝜕𝑟 and initial conditions are required. In this work, the Navier
The approximate periodic form of the pressure gradient slip condition is applied. That is, on the solid-fluid interface
generated by the heart can be described by 𝑟 = 𝑏, the axial fluid velocity, relative to the solid surface, is
proportional to the shear stress on the interface. As the fluid
𝜕𝑝 layer near the wall is modeled as a Newtonian fluid in our
− = 𝐴 0 + 𝐴 1 cos 𝜔𝑝 𝑡, (6) model, the shear stress on the boundary is related to the shear
𝜕𝑧
2.5 10
2 8
1.5 6
1 4
0.5 2
0 0
30 30
1 1
Dim 20 0.8 Dim 20 0.8
ens 0.6 ens 10 0.6
ion 10 0.4 r ion 0.4 r
less 0.2 nless less 00.22 nless
tim 0 0 imensio tim
e 0 0
imensio
e D D
2.5
10
8
1.5
6
1
4
0.5
2
0
30 0
30
20 1
Dim 0.8 1
ens
ion 10 0.6 Dim 20 0.8
less 0.4 r ens 0.6
tim 0.2 nless ion 10
ss r
e 0 0 ensio less 0.4
ionle
Dim tim 0.2 ens
e 0 0 Di m
(a)
Figure 4: Velocity profiles in arteries with different radii 𝑟: (a) 𝑟 = 0.15 cm; (b) 𝑟 = 0.5 cm. In the figure, the graphs on the left column
correspond to 𝑙𝑏 = 0, while the graphs on the right column correspond to 𝑙𝑏 = 2.
Difference error of dimensionless velocity at r = 0
Difference error of dimensionless velocity at r = 0
5 5
0 0
−5 −5
−10 −10
−15 −15
−20 −20
−25 −25
−30 −30
0 10 20 30 40 50 0 10 20 30 40 50
lb = 0 lb = 6 lb = 0 lb = 6
lb = 2 lb = 8 lb = 2 lb = 8
lb = 4 lb = 4
(a) (b)
Figure 5: Diagrams showing the convergence of numerical solutions for different slip parameters and artery radii: (a) 𝑟 = 0.15 cm; (b)
𝑟 = 0.50 cm.
2.5
10
8
1.5
6
1
4
0.5
2
0
30 0
30
20 1
Dim 0.8 20 1
ens 0.6 Dim 0.8
ion 10 0.4 ens 0.6
less r ion 10
tim 0.2 nless less 0.4
e 0 0
Dim
ensio tim 0.2 less r
e 0 0 ension
Dim
(a)
3
10
2.5
8
2
6
1.5
1 4
0.5 2
0
30 0
30
20 1
Dim 0.8 20 1
ens 0.6 Dim 0.8
ion 10 0.4 ens 0.6
less r ion 10 0.4
0.2 nless
less r
tim less
e 0 0 imensio tim 0.2
sion
D e 0 0 men
Di
(b)
Figure 6: Velocity profiles in arteries with different slip parameters 𝑙𝑏 and radii 𝑟: (a) 𝑟 = 0.15 cm; (b) 𝑟 = 0.50 cm. In the Figure, the graphs
on the left column correspond to 𝑙𝑏 = 0, while the graphs on the right column correspond to 𝑙𝑏 = 2.
strain rate by 𝜎𝑟𝑧 = 𝜇2 (𝜕V/𝜕𝑧). Thus, the Navier slip condition On the interface between two different fluids, for contin-
can be written as uous and smooth behavior of the velocity and shear stresses,
we require
𝜕V2
V2 (𝑏, 𝑡) + 𝑙 (𝑏, 𝑡) = 0, (9)
𝜕𝑡 V1 (𝑎, 𝑡) = V2 (𝑎, 𝑡) ,
󵄨󵄨 𝜕V 󵄨󵄨𝑚 𝜕V 𝜕V (11)
󵄨 󵄨
where 𝑙 is the slip parameter. Moreover, we assume that the [𝜇1 󵄨󵄨󵄨 1 󵄨󵄨󵄨 1
] (𝑎, 𝑡) = [𝜇2 2 ] (𝑎, 𝑡) .
󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟 𝜕𝑟
slip parameter does not change along the axial direction.
On 𝑟 = 0, the symmetry condition is introduced:
The initial conditions are set to
𝜕V1
(0, 𝑡) = 0. (10) V1 (𝑟, 0) = 0 = V2 (𝑟, 0) , (12)
𝜕𝑟
2.5 9
8
2
7
1.5 6
5
1
4
0.5 3
2
0
1
−0.5 0
0 5 10 15 20 25 30 0 5 10 15 20 25 30
(a)
3 10
9
2.5
8
2
7
6
1.5
5
1
4
0.5 3
2
0
1
−0.5 0
0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16
r = 0 (centre) r = 0 (centre)
r = 0.6 (inner-outer layer interface) r = 0.6 (inner-outer layer interface)
r = 1 (wall) r = 1 (wall)
(b)
Figure 7: Velocity profiles at three arterial locations (𝑟1 , 𝑟2 , 𝑟3 ): for 𝑚 = −1/4 and under different slip parameters 𝑙𝑏 and artery radii (a)
𝑟 = 0.15 cm; (b) 𝑟 = 0.50 cm. In the Figure, the graphs on the left column correspond to 𝑙𝑏 = 0, while the graphs on the right column
correspond to 𝑙𝑏 = 2.
𝜌1 𝜇2
which is essential for the numerical scheme adopted to esti- 𝜌∗ = , 𝜇∗ = ,
mate the time at which the pulsatile steady state is achieved. 𝜌2 𝜇
To simplify the equations, we introduce the following 𝐴 0 𝑏2 𝑏2 𝜌1 𝐴 𝑔
nondimensional variables and parameters: 𝐶1 = , 𝐶2 = 𝜌1 𝐴 𝑔 = 𝐵,
𝜇𝑢0 𝜇𝑢0 𝐴0 1
𝜌1 𝜔𝑝 𝑏2 𝜌2 𝜔𝑝 𝑏2 𝜌2 𝜔𝑝 𝑏2 𝜌1 𝜌∗
𝑟 V 𝜔𝑝 𝐴 𝑏2 𝛼= , 𝛾= = =𝛼 ,
𝑟= , V= , 𝑡= 𝑡, 𝑢0 = 0 , 2𝜋𝜇 2𝜋𝜇𝜇∗ 2𝜋𝜇𝜇∗ 𝜌1 𝜇∗
𝑏 V0 2𝜋 𝜇2
2 𝜌2 𝐴 𝑔 𝑏2 𝜌1 𝜌∗
𝐴 𝜔 𝑎 𝑢 𝑚 ̂1 = 𝐴 0 𝑏 = 𝐶1 ,
𝐶 ̂2 =
𝐶 = 𝐶2 .
𝑒 = 1, 𝜔𝑟 = 𝑏 , 𝑟0 = , 𝑚𝑢 = 𝜇( 0 ) , 𝜇𝑢0 𝜇∗ 𝜇∗ 𝜇𝑢0 𝜇∗ 𝜌1 𝜇∗
𝐴0 𝜔𝑝 𝑏 𝑏
(13)
30 30
25 25
20 20
15 15
10 10
5 5
0 0
−5 −5
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
lb = 0 lb = 6 lb = 0 lb = 6
lb = 2 lb = 8 lb = 2 lb = 8
lb = 4 lb = 4
(a) (b)
Figure 8: Diagrams showing the convergence of numerical results of the fluid velocity on the wall to the steady state pulsatile velocity field
under various slip parameters 𝑙𝑏 for two different artery radii: (a) 𝑟 = 0.15 cm; (b) 𝑟 = 0.50 cm.
In terms of the nondimensional variables and parameters, 3. Analytical Solution

(8)–(12) can be written in the form of
For 𝑚 = 0, the model reduces to the linear model with
different viscosity in the peripheral layer and the centre core.
𝜕V1 In this case, (14) have the same form:
𝛼 = 𝐶1 (1 + 𝑒 cos 2𝜋𝑡) + 𝐶2 cos (2𝜋𝜔𝑟 𝑡 + 𝜙)
𝜕𝑡
𝜕V 1 𝜕V 𝜕2 V
1 𝜕 󵄨󵄨󵄨󵄨 𝜕V1 󵄨󵄨󵄨󵄨𝑚 𝜕V1 𝐿 (V) = 𝛽 − −
𝜕𝑡 𝑟 𝜕𝑟 𝜕𝑟2
+ [𝑟󵄨 󵄨 ] , for 0 ≤ 𝑟 ≤ 𝑟0 , (20)
𝑟 𝜕𝑟 󵄨󵄨󵄨 𝜕𝑟 󵄨󵄨󵄨 𝜕𝑟
= 𝐵1 (1 + 𝑒 cos (2𝜋𝑡)) + 𝐵2 cos (2𝜋𝜔𝑟 𝑡 + 𝜙) .
𝜕V
𝛾 2 = 𝐶1 (1 + 𝑒 cos 2𝜋𝑡) + 𝐶2 cos (2𝜋𝜔𝑟 𝑡 + 𝜙)
𝜕𝑡 By the superposition principle, if V0 , V1 , and V2 are
1 𝜕 𝜕V the solution of 𝐿(V) = 𝑓(𝑡), respectively, for 𝑓(𝑡) =
+ [𝑟 2 ] , for 𝑟0 ≤ 𝑟 ≤ 1. 𝐵1 𝑒0𝑡𝑖 , 𝐵1 𝑎𝑒2𝜋𝑡𝑖 , and 𝐵2 𝑒(2𝜋𝜔𝑟 𝑡+𝜙)𝑖 , then the complete solution
𝑟 𝜕𝑟 𝜕𝑟
(14) of (20) is V = ∑2𝑛=0 Re(V𝑛 ).
To determine V𝑛 , we solve
𝜕V𝑛 1 𝜕V𝑛 𝜕2 V𝑛
The boundary conditions and initial conditions, in dime- 𝛽 = 𝐷𝑛 𝑒𝑔𝑛 (𝑡)𝑖 + + 2, (21)
nsionless form, can be expressed by 𝜕𝑡 𝑟 𝜕𝑟 𝜕𝑟
where 𝑔0 (𝑡) = 0, 𝑔1 (𝑡) = 2𝜋𝑡, 𝑔2 (𝑡) = 2𝜋𝜔𝑟 𝑡 + 𝜙, 𝐷0 = 𝐵1 ,
𝜕V1 𝐷1 = 𝑎𝐵1 , and 𝐷2 = 𝐵2 . As (21) admits solutions of the
(0, 𝑡) = 0, (15) form V𝑛 = 𝑓𝑛 (𝑟)𝑒𝑔𝑛 (𝑡)𝑖 , we have from (21) that
𝜕𝑟
𝜕V2 𝛽𝑔𝑛󸀠 (𝑡) 𝑓𝑛 (𝑟) 𝑒𝑔𝑛 (𝑡)𝑖 𝑖
𝑏V2 (1, 𝑡) + 𝑙 (1, 𝑡) = 0, (16)
𝜕𝑟 (22)
1
V1 (𝑟0 , 𝑡) = V2 (𝑟0 , 𝑡) , (17) = 𝐷𝑛 𝑒𝑔𝑛 (𝑡)𝑖 + 𝑓𝑛󸀠 (𝑟) 𝑒𝑔𝑛 (𝑡)𝑖 + 𝑓𝑛󸀠󸀠 (𝑟) 𝑒𝑔𝑛 (𝑡)𝑖 .
𝑟
󵄨󵄨 𝜕V 󵄨󵄨𝑚 𝜕V 𝜕V
󵄨 󵄨 Dividing by 𝑒𝑔𝑛 (𝑡)𝑖 on both sides of (22), we obtain
[󵄨󵄨󵄨 1 󵄨󵄨󵄨 1
] (𝑟0 , 𝑡) = [𝜇∗ 2 ] (𝑟0 , 𝑡) , (18)
󵄨󵄨 𝜕𝑟 󵄨󵄨 𝜕𝑟 𝜕𝑟
1
V1 (𝑟, 0) = 0 = V2 (𝑟, 0) . (19) 𝛽𝑔𝑛󸀠 (𝑡) 𝑓𝑛 (𝑟) 𝑖 = 𝐷𝑛 + 𝑓𝑛󸀠 (𝑟) + 𝑓𝑛󸀠󸀠 (𝑟) . (23)
𝑟
For 𝑛 = 0, we get where 𝛽̂12 = −2𝜋𝛾𝑖, 𝛽̂22 = −2𝜋𝜔𝑟 𝛾𝑖, 𝛽12 = −2𝜋𝛼𝑖, and 𝛽22 =
−2𝜋𝜔𝑟 𝛼𝑖.
1 As 𝑑𝐽0 (𝑥)/𝑑𝑥 = −𝐽1 (𝑥) and 𝑑𝑌0 (𝑥)/𝑑𝑥 = −𝑌0 (𝑥), we have
𝑓0󸀠󸀠 (𝑟) + 𝑓0󸀠 (𝑟) = −𝐵1 , (24)
𝑟
𝜕V1 𝐶
which has the general solution: 𝑓0 (𝑟) = (𝑐1 +𝑐2 ln 𝑟)−(𝐵1 /4)𝑟2 . = Re (− 1 𝑟 − 𝑑1 𝛽1 𝐽1 (𝛽1 𝑟) 𝑒2𝜋𝑡𝑖
For 𝑛 = 1, we have 𝜕𝑟 2
1 −𝑑2 𝛽2 𝐽1 (𝛽2 𝑟) 𝑒(2𝜋𝜔𝑟 𝑡+𝜙)𝑖 ) ,

𝑓1󸀠󸀠 (𝑟) + 𝑓1󸀠 (𝑟) − 2𝜋𝛽𝑖𝑓1 (𝑟) = −𝑒𝐵1 . (25)
𝑟
𝜕V2 1 𝐶
2 = Re (̂𝑐2 − 1 𝑟
Let 𝛽1 = −2𝜋𝛽𝑖; then, 𝜕𝑟 𝑟 2
1 1 𝑒𝐵1 + [−𝑑̂1 𝛽̂1 𝐽1 (𝛽̂1 𝑟) − 𝑒̂1 𝛽̂1 𝑌1 (𝛽̂1 𝑟)] 𝑒2𝜋𝑡𝑖

𝑓󸀠󸀠
2 1
(𝑟) + 2
𝑓1󸀠 (𝑟) + 𝑓1 (𝑟) = − 2
. (26)
𝛽1 𝛽1 𝑟 𝛽1
+ [−𝑑̂2 𝛽̂2 𝐽1 (𝛽̂2 𝑟) − 𝑒̂2 𝛽̂2 𝑌1 (𝛽̂2 𝑟)] 𝑒(2𝜋𝜔𝑟 𝑡+𝜙)𝑖 ) .
Let 𝑟̂ = 𝛽1 𝑟; we have
(32)
𝑒𝐵1
𝑟̂2 𝑓1󸀠󸀠 (̂𝑟) + 𝑟̂𝑓1󸀠 2
(̂𝑟) + 𝑟̂ 𝑓1 (̂𝑟) = − 2
2
𝑟̂ . (27)
𝛽1 Obviously, V1 satisfies the boundary condition (15) automati-
cally. We now consider the boundary condition (16); namely,
The general solution of (27) is
𝑏 𝐶
𝑒𝐵1 Re [(𝑏̂𝑐1 + 𝑙̂𝑐2 − (𝑙 + ) 1 )
𝑓1 (𝑟) = 𝑑1 𝐽0 (𝛽1 𝑟) + 𝑒1 𝑌0 (𝛽1 𝑟) − 𝑖, (28) 2 2
2𝜋𝛽
where 𝑑1 and 𝑒1 are integration constants and 𝐽0 and 𝑌0 + ( (𝑏𝐽0 (𝛽̂1 ) − 𝑙𝛽̂1 𝐽1 (𝛽̂1 )) 𝑑̂1
denote the zero-order Bessel functions of the first kind and
the second kind, respectively.
𝑒𝑏𝐶1 𝑖 2𝜋𝑡𝑖
Similarly, for 𝑛 = 2, we have + (𝑏𝑌0 (𝛽̂1 ) − 𝑙𝛽̂1 𝑌1 (𝛽̂1 )) 𝑒̂1 − )𝑒
2𝜋𝛾 (33)
1
𝑓2󸀠󸀠 (𝑟) + 𝑓2󸀠 (𝑟) − 2𝛽𝜋𝜔𝑟 𝑓2 (𝑟) 𝑖 = −𝐵2 , (29)
𝑟 + (𝑑̂2 (𝑏𝐽0 (𝛽̂2 ) − 𝑙𝛽̂2 𝐽1 (𝛽̂2 ))
and the general solution is
+ 𝑒̂2 (𝑏𝑌0 (𝛽̂2 ) − 𝑙𝛽̂2 𝑌1 (𝛽̂2 ))
𝐵2
𝑓2 = 𝑑2 𝐽0 (𝛽2 𝑟) + 𝑒2 𝑌0 (𝛽2 𝑟) − 𝑖, (30)
2𝛽𝜔𝑟 𝜋 𝑏𝐶2
− 𝑖) 𝑒(2𝜋𝜔𝑟 𝑡+𝜙)𝑖 ] = 0.
2 2𝜋𝜔𝑟 𝛾
where 𝛽2 = −2𝜋𝛽𝜔𝑟 𝑖.
Because the boundness of V1 , V2 , 𝑐2 , 𝑒1 , and 𝑒2 are set to
Further, from boundary conditions (17) and (18), we have
zero, hence, from (14) and the solutions for (20), we have
𝐶1 2 𝑒𝐶1 𝑟02
V1 = Re {𝑐1 − 𝑟 + [𝑑1 𝐽0 (𝛽1 𝑟) − 𝑖] 𝑒2𝜋𝑡𝑖 ̂1 )
Re [(𝑐1 − 𝑐̂1 − 𝑐̂2 ln 𝑟0 − (𝐶1 − 𝐶 )
4 2𝜋𝛼 4
𝐶2
+ [𝑑2 𝐽0 (𝛽2 𝑟) − 𝑖] 𝑒(2𝜋𝜔𝑟 𝑡+𝜙)𝑖 } , + (𝑑1 𝐽0 (𝛽1 𝑟0 ) − 𝑑̂1 𝐽0 (𝛽̂1 𝑟0 ) − 𝑒̂1 𝑌0 (𝛽̂1 𝑟0 )
2𝜋𝜔𝑟 𝛼
𝐶1 2 𝑒𝑖
V2 = Re {̂𝑐1 + 𝑐̂2 ln 𝑟 − 𝑟 − (𝛾𝐶1 − 𝛼𝐶1 ) ) 𝑒2𝜋𝑡𝑖
4 2𝜋𝛼𝛾
(31)
̂1
𝑒𝐶
+ [𝑑̂1 𝐽0 (𝛽̂1 𝑟) + 𝑒̂1 𝑌0 (𝛽̂1 𝑟) − 𝑖] 𝑒2𝜋𝑡𝑖 + (𝑑2 𝐽0 (𝛽2 𝑟0 ) − 𝑑̂2 𝐽0 (𝛽̂2 𝑟0 )
2𝜋𝛾
𝑐2 𝑖 𝑖
+ [𝑑̂2 𝐽0 (𝛽̂2 𝑟) + 𝑒̂2 𝑌0 (𝛽̂2 𝑟) − ] 𝑒2 𝑌0 (𝛽̂2 𝑟0 ) − (𝛾𝐶2 − 𝛼𝐶2 )
−̂ )
2𝜋𝜔𝑟 𝛾 2𝜋𝜔𝑟 𝛾𝛼
× 𝑒(2𝜋𝜔𝑟 𝑡+𝜙)𝑖 } , × 𝑒(2𝜋𝜔𝑟 𝑡+𝜙)𝑖 ] = 0,

𝑟0 𝑐̂
Re [((𝜇∗ 𝐶1 − 𝐶1 ) − 𝜇∗ 2 ) Solving the above system of equations yields
2 𝑟0
𝑙 𝑟2
+ ( − 𝑑1 𝛽1 𝐽1 (𝛽1 𝑟0 ) + 𝑑̂1 𝜇∗ 𝛽̂1 𝐽1 (𝛽̂1 𝑟0 ) 𝑐1 = (ln 𝑟0 − ) ((𝜇∗ 𝐶1 − 𝐶1 ) 0∗ )
𝑏 2𝜇
𝑒1 𝜇∗ 𝛽̂1 𝑌1 (𝛽̂1 𝑟0 )) 𝑒2𝜋𝑡𝑖
+̂ 2
𝑙 1 − 𝑟0 𝐶1 𝑟2
+( + ) + 𝐶1 0 ,
+ (−𝑑2 𝛽2 𝐽1 (𝛽2 𝑟0 ) + 𝑑̂2 𝜇∗ 𝛽̂2 𝐽1 (𝛽̂2 𝑟0 ) 𝑏 2 2 4
𝑙 𝑟2 𝑙 1 𝐶
𝑒2 𝜇∗ 𝛽̂2 𝑌1 (𝛽̂2 𝑟0 ))
+̂ 𝑐̂1 = − ((𝜇∗ 𝐶1 − 𝐶1 ) 0∗ ) + ( + ) 1 ,
𝑏 2𝜇 𝑏 2 2
× 𝑒(2𝜋𝜔𝑟 𝑡+𝜙)𝑖 ] = 0. 𝑟02
𝑐̂2 = (𝜇∗ 𝐶1 − 𝐶1 ) ,
(34) 2𝜇∗
As (33)-(34) must be satisfied for any instant of time 𝑡, 𝑑1 = 𝜇∗ [ (𝐽1 (𝛽̂1 𝑟0 ) 𝑌0 (𝛽̂1 𝑟0 ) − 𝐽0 (𝛽̂1 𝑟0 ) 𝑌1 (𝛽̂1 𝑟0 ))
we require that the constant terms and the coefficients of the
exponential terms all vanish; namely, 𝑒𝑏𝛽̂1 𝐶1 𝑖
× + (𝛾𝐶1 − 𝛼𝐶1 )
2𝜋𝛾
𝑏 𝐶
𝑏̂𝑐1 + 𝑙̂𝑐2 − (𝑙 + ) 1 = 0, × [𝐽1 (𝛽̂1 𝑟0 ) (𝑏𝑌0 (𝛽̂1 ) − 𝑙𝛽̂1 𝑌1 (𝛽̂1 ))
2 2
𝑟02 −𝑌1 (𝛽̂1 𝑟0 ) (𝑏𝐽0 (𝛽̂1 ) − 𝑙𝛽̂1 𝐽1 (𝛽̂1 ))]
𝑐1 − 𝑐̂1 − 𝑐̂2 ln 𝑟0 − (𝐶1 − 𝐶1 ) = 0,
4
𝛽̂1 𝑒𝑖
𝑟 𝑐̂ × ] / (𝑏𝐽0 (𝛽̂1 ) − 𝑙𝛽̂1 𝐽1 (𝛽̂1 ))
(𝜇 𝐶1 − 𝐶1 ) 0 − 𝜇∗ 2 = 0,
∗
2𝜋𝛾𝛼
2 𝑟0
× (𝛽1 𝐽1 (𝛽1 𝑟0 ) 𝑌0 (𝛽̂1 𝑟0 ) − 𝜇∗ 𝛽̂1 𝑌1 (𝛽̂1 𝑟0 ) 𝐽0 (𝛽1 𝑟0 ))
𝑑̂1 (𝑏𝐽0 (𝛽̂1 ) − 𝑙𝛽̂1 𝐽1 (𝛽̂1 )) + 𝑒̂1 (𝑏𝑌0 (𝛽̂1 ) − 𝑙𝛽̂1 𝑌1 (𝛽̂1 ))
+ (𝑏𝑌0 (𝛽̂1 ) − 𝑙𝛽̂1 𝑌1 (𝛽̂1 ))
𝑒𝑏𝐶1
− 𝑖 = 0,
2𝜋𝛾 × (𝜇∗ 𝛽̂1 𝐽0 (𝛽1 𝑟0 ) 𝐽1 (𝛽̂1 𝑟0 ) − 𝛽1 𝐽1 (𝛽1 𝑟0 ) 𝐽0 (𝛽̂1 𝑟0 )) ,
𝑒𝐶1 𝑑̂1 = [(𝛽1 𝐽1 (𝛽1 𝑟0 ) 𝑌0 (𝛽̂1 𝑟0 )
𝑑1 𝐽0 (𝛽1 𝑟0 ) − 𝑑̂1 𝐽0 (𝛽̂1 𝑟0 ) − 𝑒̂1 𝑌0 (𝛽̂1 𝑟0 ) − 𝑖
2𝜋𝛼
𝑒𝐶 −𝜇∗ 𝛽̂1 𝑌1 (𝛽̂1 𝑟0 ) 𝐽0 (𝛽1 𝑟0 ))
+ 1 𝑖 = 0,
2𝜋𝛾 (𝛾𝐶1 − 𝛼𝐶1 )
𝑒𝑏𝐶1
× 𝑖+
− 𝑑1 𝛽1 𝐽1 (𝛽1 𝑟0 ) + 𝑑̂1 𝜇 𝛽̂1 𝐽1 (𝛽̂1 𝑟0 ) + 𝑒̂1 𝜇 𝛽̂1 𝑌1 (𝛽̂1 𝑟0 )
∗ ∗
2𝜋𝛾 2𝜋𝛾𝛼
= 0, × (𝑏𝑌0 (𝛽̂1 ) − 𝑙𝛽̂1 𝑌1 (𝛽̂1 ))
× 𝑒𝛽1 𝐽1 (𝛽1 𝑟0 ) 𝑖] / (𝑏𝐽0 (𝛽̂1 ) − 𝑙𝛽̂1 𝐽1 (𝛽̂1 ))

𝑑̂2 (𝑏𝐽0 (𝛽̂2 ) − 𝑙𝛽̂2 𝐽1 (𝛽̂2 )) + 𝑒̂2 (𝑏𝑌0 (𝛽̂2 ) − 𝑙𝛽̂2 𝑌1 (𝛽̂2 ))
× (𝛽1 𝐽1 (𝛽1 𝑟0 ) 𝑌0 (𝛽̂1 𝑟0 )
𝑏𝐶2
− 𝑖 = 0,
2𝜋𝜔𝑟 𝛾 −𝜇∗ 𝛽̂1 𝑌1 (𝛽̂1 𝑟0 ) 𝐽0 (𝛽1 𝑟0 ))
𝐶1 + (𝑏𝑌0 (𝛽̂1 ) − 𝑙𝛽̂1 𝑌1 (𝛽̂1 ))
𝑑2 𝐽0 (𝛽2 𝑟0 ) − 𝑑̂2 𝐽0 (𝛽̂2 𝑟0 ) − 𝑒̂2 𝑌0 (𝛽̂2 𝑟0 ) − 𝑖
2𝜋𝜔𝑟 𝛼
× (𝜇∗ 𝛽̂1 𝐽0 (𝛽1 𝑟0 ) 𝐽1 (𝛽̂1 𝑟0 )
𝐶2
+ 𝑖 = 0,
2𝜋𝜔𝑟 𝛾 −𝛽1 𝐽1 (𝛽1 𝑟0 ) 𝐽0 (𝛽̂1 𝑟0 )) ,
− 𝑑2 𝛽2 𝐽1 (𝛽2 𝑟0 ) + 𝑑̂2 𝜇∗ 𝛽̂2 𝐽1 (𝛽̂2 𝑟0 ) + 𝑒̂2 𝜇∗ 𝛽̂2 𝑌1 (𝛽̂2 𝑟0 )
𝑒̂1 = [ (𝜇∗ 𝛽̂1 𝐽0 (𝛽1 𝑟0 ) 𝐽1 (𝛽̂1 𝑟0 )
= 0.
(35) −𝛽1 𝐽0 (𝛽̂1 𝑟0 ) 𝐽1 (𝛽1 𝑟0 ))
𝑒𝑏𝐶1 𝑖 𝑒̂2 = [(𝜇∗ 𝛽̂2 𝐽0 (𝛽2 𝑟0 ) 𝐽1 (𝛽̂2 𝑟0 )

× − (𝑏𝐽0 (𝛽̂1 ) − 𝑙𝛽̂1 𝐽1 (𝛽̂1 ))
2𝜋𝛾
−𝛽2 𝐽0 (𝛽̂2 𝑟0 ) 𝐽1 (𝛽2 𝑟0 ))
× (𝛾𝐶1 − 𝛼𝐶1 )
𝑏𝐶1 𝑖
𝛽 𝑒𝑖 × − (𝑏𝐽0 (𝛽̂2 ) − 𝑙𝛽̂2 𝐽1 (𝛽̂2 ))
× 𝐽1 (𝛽1 𝑟0 ) 1 ] / (𝑏𝐽0 (𝛽̂1 ) − 𝑙𝛽̂1 𝐽1 (𝛽̂1 )) 2𝜋𝜔𝑟 𝛾
2𝜋𝛼𝛾
× (𝛾𝐶1 − 𝛼𝐶1 ) 𝐽1 (𝛽2 𝑟0 )
× (𝛽1 𝐽1 (𝛽1 𝑟0 ) 𝑌0 (𝛽̂1 𝑟0 )
𝛽2 𝑖
−𝜇∗ 𝛽̂1 𝑌1 (𝛽̂1 𝑟0 ) 𝐽0 (𝛽1 𝑟0 )) × ] / (𝑏𝐽0 (𝛽̂2 ) − 𝑙𝛽̂2 𝐽1 (𝛽̂2 ))
2𝜋𝜔𝑟 𝛼𝛾
+ (𝑏𝑌0 (𝛽̂1 ) − 𝑙𝛽̂1 𝑌1 (𝛽̂1 )) × (𝛽2 𝐽1 (𝛽2 𝑟0 ) 𝑌0 (𝛽̂2 𝑟0 )
× (𝜇∗ 𝛽̂1 𝐽0 (𝛽1 𝑟0 ) 𝐽1 (𝛽̂1 𝑟0 ) − 𝛽1 𝐽1 (𝛽1 𝑟0 ) 𝐽0 (𝛽̂1 𝑟0 )) , −𝜇∗ 𝛽̂2 𝑌1 (𝛽̂2 𝑟0 ) 𝐽0 (𝛽2 𝑟0 ))
+ (𝑏𝑌0 (𝛽̂2 ) − 𝑙𝛽̂2 𝑌1 (𝛽̂2 ))

𝑑2 = 𝜇∗ [ (𝐽1 (𝛽̂2 𝑟0 ) 𝑌0 (𝛽̂2 𝑟0 ) − 𝐽0 (𝛽̂2 𝑟0 ) 𝑌1 (𝛽̂2 𝑟0 ))
× (𝜇∗ 𝛽̂2 𝐽0 (𝛽2 𝑟0 ) 𝐽1 (𝛽̂2 𝑟0 )
𝑏𝛽̂ 𝐶 𝑖
× 2 1 + (𝛾𝐶2 − 𝛼𝐶2 ) −𝛽2 𝐽1 (𝛽2 𝑟0 ) 𝐽0 (𝛽̂2 𝑟0 )) .
2𝜋𝜔𝑟 𝛾
(36)
× [𝐽1 (𝛽̂2 𝑟0 ) (𝑏𝑌0 (𝛽̂2 ) − 𝑙𝛽̂2 𝑌1 (𝛽̂2 ))
−𝑌1 (𝛽̂2 𝑟0 ) (𝑏𝐽0 (𝛽̂2 ) − 𝑙𝛽̂2 𝐽1 (𝛽̂2 ))] To show the flow behavior and the effect of the slip
parameter, we investigate the velocity profiles in the arteries
𝛽̂2 𝑖 with different values of the slip parameter under various
× ] / (𝑏𝐽0 (𝛽̂2 ) − 𝑙𝛽̂2 𝐽1 (𝛽̂2 )) different conditions. In the first example of investigation, the
2𝜋𝜔𝑟 𝛾𝛼
radius of the artery is taken as 𝑟 = 𝑏 = 0.15 cm, and the other
× (𝛽2 𝐽1 (𝛽2 𝑟0 ) 𝑌0 (𝛽̂2 𝑟0 ) parameters are set to 𝐴 0 = 698.65 dyne/cm3 , 𝐴 𝑔 = 0.5𝑔,
𝑓𝑏 = 𝑓𝑝 = 1.2, 𝜙 = 0, 𝐶1 = 6.6, 𝐶2 = 4.64, 𝐴 1 = 1.2𝐴 0 , and
−𝜇∗ 𝛽̂2 𝑌1 (𝛽̂2 𝑟0 ) 𝐽0 (𝛽2 𝑟0 )) 𝜌1 /𝜌2 = 1. Figure 1 shows the 3-dimensional velocity profile
as a function of time and location and the 2-dimensional
+ (𝑏𝑌0 (𝛽̂2 ) − 𝑙𝛽̂2 𝑌1 (𝛽̂2 )) velocity profile as a function of time at three different radial
locations for two different slip parameters 𝑙 = 0 (no-slip)
× (𝜇∗ 𝛽̂2 𝐽0 (𝛽2 𝑟0 ) 𝐽1 (𝛽̂2 𝑟0 ) and 𝑙 = 2. The results show that boundary slip has a very
dramatical effect on the fluid flow in the artery. It affects
not only the magnitude of the flow velocity significantly, but
−𝛽2 𝐽1 (𝛽2 𝑟0 ) 𝐽0 (𝛽̂2 𝑟0 )) ,
also the flow pattern and velocity profile on the cross-section
of the artery. For the no-slip flow (𝑙𝑏 = 0), the pulsatile
𝑑̂2 = [(𝛽2 𝐽1 (𝛽2 𝑟0 ) 𝑌0 (𝛽̂2 𝑟0 )
flow nature gradually disappears toward the arterial wall,
while with boundary slip, the flow near the arterial wall also
−𝜇∗ 𝛽̂2 𝑌1 (𝛽̂2 𝑟0 ) 𝐽0 (𝛽2 𝑟0 ))
displays a pulsatile nature.
We then investigate whether the above observed flow phe-
𝑏𝐶1 (𝛾𝐶1 − 𝛼𝐶1 )
× 𝑖+ nomena associated with boundary slip are affected or not by
2𝜋𝜔𝑟 𝛾 2𝜋𝛾𝛼𝜔𝑟 the radius of the artery, and for this purpose, we consider the
fluid flow through an artery with a larger radius 𝑟 = 0.5 cm.
× (𝑏𝑌0 (𝛽̂2 ) − 𝑙𝛽̂2 𝑌1 (𝛽̂2 )) The constant pressure gradient is set to 𝐴 0 = 32 dyne/cm3 in
order to achieve a mean velocity magnitude approximately
×𝛽2 𝐽1 (𝛽2 𝑟0 ) 𝑖] / (𝑏𝐽0 (𝛽̂2 ) − 𝑙𝛽̂2 𝐽1 (𝛽̂2 )) equal to that in the smaller artery, while all other parameters
are set to the same values as those used for the smaller
× (𝛽2 𝐽1 (𝛽2 𝑟0 ) 𝑌0 (𝛽̂2 𝑟0 ) radius. Figure 2 shows the velocity profile in the artery for
two different slip parameter values including 𝑙𝑏 = 0 (no-slip)
−𝜇∗ 𝛽̂2 𝑌1 (𝛽̂2 𝑟0 ) 𝐽0 (𝛽2 𝑟0 )) and 𝑙𝑏 = 2. The 3-dimensional graph shows the variation
of the flow velocity with time and radial position, while the
+ (𝑏𝑌0 (𝛽̂2 ) − 𝑙𝛽̂2 𝑌1 (𝛽̂2 )) 2-dimensional graphs demonstrate the variation of the flow
× (𝜇∗ 𝛽̂2 𝐽0 (𝛽2 𝑟0 ) 𝐽1 (𝛽̂2 𝑟0 ) velocity with time at three different radial locations including
𝑟 = 0 (centre), 𝑟 = 0.6 (inner-outer layers interface),
−𝛽2 𝐽1 (𝛽2 𝑟0 ) 𝐽0 (𝛽̂2 𝑟0 )) , and 𝑟 = 1 (arterial wall). From Figures 1 and 2, it is clear
that the boundary slip related flow phenomena and behavior general case. Our analytical and numerical studies show that
observed for the smaller artery also appear in the artery with for the flow of fluids with the Fahraeus-Lindqvist effect,
a larger radius, and further, a more significant pulsatile nature boundary slip has a very significant influence on the magni-
of fluid flow is observed for the larger artery. tude of the mean flow velocity and on the flow pattern and
To further investigate the effect of the slip parameter on velocity profile on the cross-section. With boundary slip, the
the velocity profile near the artery wall, we show in Figure 3 boundary layer near the wall also displays significant pulsatile
the velocity of fluid on the artery wall for four different flow nature. The results also show that as the boundary slip
values of the slip parameter including 𝑙𝑏 = 0, 2, 4, 6, and 8. length increases, the convergence rate of numerical results to
The results clearly demonstrate that the slip parameter has a the exact solutions decreases and the time required to achieve
very significant effect on the near-wall velocity and that the the steady state pulsatile flow increases.
magnitude of the average wall velocity is proportional to the
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http://dx.doi.org/10.1155/2013/380484
Research Article
Analytical Solutions of Boundary Values Problem of
2D and 3D Poisson and Biharmonic Equations by Homotopy
Decomposition Method
Abdon Atangana1 and Adem KJlJçman2

1
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State,
Bloemfontein 9300, South Africa
2
Department of Mathematics and Institute for Mathematical Research, University of Putra Malaysia,
43400 Serdang, Selangor, Malaysia
Correspondence should be addressed to Adem Kılıçman; kilicman@yahoo.com
Received 13 June 2013; Accepted 18 August 2013
Copyright © 2013 A. Atangana and A. Kılıçman. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The homotopy decomposition method, a relatively new analytical method, is used to solve the 2D and 3D Poisson equations and
biharmonic equations. The method is chosen because it does not require the linearization or assumptions of weak nonlinearity, the
solutions are generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical
problems. The method does not require any corrected function or any Lagrange multiplier and it avoids repeated terms in the series
solutions compared to the existing decomposition method including the variational iteration method, the Adomian decomposition
method, and Homotopy perturbation method. The approximated solutions obtained converge to the exact solution as N tends to
infinity.
1. Introduction the biharmonic equation in three dimensions. Khattar et al.

[6] derived a fourth-order finite difference approximation
The numerical solution of Poisson equations and biharmonic based on arithmetic average discretization for the solution
equations is an important problem in numerical analysis. of three-dimensional nonlinear biharmonic partial differen-
A vast arrangement of investigating effort has been pub- tial equations on a 19-point compact stencil using coupled
lished on the development of numerical solution of Poisson approach. Altas et al. [7] used multigrid and precondi-
equations and biharmonic equations. The finite difference tioned Krylov iterative methods to solve three-dimensional
schemes of second and fourth order for the solution of nonlinear biharmonic partial differential equations. Jeon
Poisson’s equation in polar coordinates have been derived by [8] derived scalar boundary integral equation formulas for
Mittal and Gahlaut [1]. A numerical method to interpolate the both interior and exterior biharmonic equations with the
source terms of Poisson’s equation by using B-spline approx- Dirichlet boundary data. A spectral collocation method for
imation has been devised by Perrey-Debain and ter Morsche numerically solving two-dimensional biharmonic boundary-
[2]. Sutmann and Steffen [3] proposed compact approxima- value problems has been reported in [9]. An indirect radial-
tion schemes for the Laplace operator of fourth and sixth basis-function collocation method for numerically solving
order; the schemes are based on a Padé approximation of the biharmonic boundary-value problems has been reported in
Taylor expansion for the discretized Laplace operator. Ge [4] [10]. A high-order boundary integral equation method for the
used fourth-order compact difference discretization scheme solution of biharmonic equations has been presented in [11].
with unequal mesh sizes in different coordinate directions to A Galerkin boundary node method for solving biharmonic
solve a 3D Poisson equation on a cubic domain. Gumerov and problems was developed in [12]. An integral collocation
Duraiswami [5] developed a complete translation theory for approach based on Chebyshev polynomials for numerically
solving biharmonic equations for the case of irregularly Using the homotopy scheme the solution of the previous
shaped domains has been developed by Mai-Duy et al. integral equation is given in a series form as
[13]. A numerical method, based on neural-network-based ∞
functions, for solving partial differential equations has been 𝑈 (𝑥, 𝑡, 𝑝) = ∑ 𝑝𝑛 𝑈𝑛 (𝑥, 𝑡) ,
in [14]. Mai-Duy and Tanner [15] presented a collocation 𝑛=0 (6)
method based on a Cartesian grid and a 1D integrated radial 𝑈 (𝑥, 𝑡) = lim 𝑈 (𝑥, 𝑡, 𝑝)
basis function scheme for numerically solving partial differ- 𝑝→1
ential equations in rectangular domains and Haar wavelet and the nonlinear term can be decomposed as
presented in [16]. The aim of this paper is to solve these ∞
problems via the homotopy decomposition method. 𝑁𝑈 (𝑟, 𝑡) = ∑ 𝑝𝑛 H𝑛 (𝑈) , (7)
𝑛=1
2. Method where 𝑝 ∈ (0, 1] is an embedding parameter. H𝑛 (𝑈) is He’s
In this study we follow the method of [17–20]. In order to polynomials [21] that can be generated by
illustrate the basic idea of this method we consider a gen- 𝑛
1 𝜕𝑛 [
eral nonlinear nonhomogeneous partial differential equation H𝑛 (𝑈0 , . . . , 𝑈𝑛 ) = 𝑁 ( ∑ 𝑝𝑗 𝑈𝑗 (𝑥, 𝑡))] ,
with initial conditions of the following form 𝑛! 𝜕𝑝𝑛 𝑗=0 (8)
[ ]
𝜕𝑚 𝑈 (𝑥, 𝑡) 𝑛 = 0, 1, 2 . . . .
= 𝐿 (𝑈 (𝑥, 𝑡)) + 𝑁 (𝑈 (𝑥, 𝑡)) + 𝑓 (𝑥, 𝑡) ,
𝜕𝑡𝑚 (1) The homotopy decomposition method is obtained by the
𝑚 = 1, 2, 3, . . . , graceful coupling of decomposition method with He’s poly-
subject to the initial conditions nomials and is given by
∞
𝜕𝑖 𝑈 (𝑥, 0) 𝜕𝑚−1 𝑈 (𝑥, 0) ∑ 𝑝𝑛 𝑈𝑛 (𝑥, 𝑡)
= 𝑓𝑚 (𝑥) , = 0,
𝜕𝑡𝑖 𝜕𝑡𝑚−1 (2) 𝑛=0
𝑖 = 0, 1, 2, . . . , 𝑚 − 2, = 𝑇 (𝑥, 𝑡)
𝑡
1
where 𝑓 is a known function, 𝑁 is the general nonlinear dif- +𝑝 ∫ (𝑡 − 𝜏)𝑚−1
ferential operator, and 𝐿 represents a linear differential oper- (𝑚 − 1)! 0
ator. The method’s first step here is to apply the inverse ∞
operator 𝜕𝑚 /𝜕𝑡𝑚 of on both sides (1) to obtain × [𝑓 (𝑥, 𝜏) + 𝐿 ( ∑ 𝑝𝑛 𝑈𝑛 (𝑥, 𝜏))
𝑛=0
𝑚−1 𝑘
𝑡 𝑑𝑘 𝑢 (𝑥, 0)
𝑈 (𝑥, 𝑡) = ∑ ∞
𝑘=0
𝑘! 𝑑𝑡𝑘 + ∑ 𝑝𝑛 H𝑛 (𝑈)] 𝑑𝜏
𝑛=0
𝑡 𝑡1 𝑡𝑚−1 (3)
(9)
+ ∫ ∫ ⋅⋅⋅∫ 𝐿 (𝑈 (𝑥, 𝜏)) + 𝑁 (𝑈 (𝑥, 𝜏))
0 0 0 with
+ 𝑓 (𝑥, 𝜏) 𝑑𝜏 ⋅ ⋅ ⋅ 𝑑𝑡. 𝑚−1 𝑘
𝑡 𝑑𝑘 𝑢 (𝑥, 𝑡)
The multi-integrals in (3) can be transformed to 𝑇 (𝑥, 𝑡) = ∑ { | 𝑡 = 0} . (10)
𝑘=0
𝑘! 𝑑𝑡𝑘
𝑡 𝑡1 𝑡𝑚−1
∫ ∫ ⋅⋅⋅∫ 𝐿 (𝑈 (𝑥, 𝜏)) + 𝑁 (𝑈 (𝑥, 𝜏)) Comparing the terms of the same power of 𝑝 gives the solu-
0 0 0 tions of various orders. The initial guess of the approximation
+ 𝑓 (𝑥, 𝜏) 𝑑𝜏 ⋅ ⋅ ⋅ 𝑑𝑡1 is 𝑇(𝑥, 𝑡). Some further related results can be seen in [22–25].
𝑡
1
= ∫ (𝑡 − 𝜏)𝑚−1 𝐿 (𝑈 (𝑥, 𝜏)) + 𝑁 (𝑈 (𝑥, 𝜏)) Lemma 1 (see [17]). The complexity of the homotopy decom-
(𝑚 − 1)! 0 position method is of order 𝑂(𝑛).
+ 𝑓 (𝑥, 𝜏) 𝑑𝜏.
(4) Proof. The number of computations including product, addi-
So that (3) can be reformulated as tion, subtraction, and division are as follows.
In step 2
𝑈 (𝑥, 𝑡)
𝑈0 : 0 because it is obtained directly from the initial
𝑚−1 𝑘 𝑘
𝑡 𝑑 𝑢 (𝑥, 0) conditions
= ∑ { }
𝑘! 𝑑𝑡𝑘 𝑈1 : 3
𝑘=0
𝑡 ..
1 .
+ ∫ (𝑡 − 𝜏)𝑚−1 𝐿 (𝑈 (𝑥, 𝜏)) + 𝑁 (𝑈 (𝑥, 𝜏))
(𝑚 − 1)! 0 𝑈𝑛 : 3.
+ 𝑓 (𝑥, 𝜏) 𝑑𝜏. Now in step 4 the total number of computations is equal to
(5) ∑𝑛𝑗=0 𝑈𝑗 (𝑥, 𝑡) = 3𝑛 = 𝑂(𝑛).
3. Solutions of the Main Problems 𝑥

𝜕2 𝑢𝑛−1
𝑝𝑛 : 𝑢𝑛 (𝑥, 𝑦) = ∫ (𝑥 − 𝜏) [− ] 𝑑𝜏,
0 𝜕𝑦2
Problem 1. Consider the following equation
𝑢𝑛 (𝑥, 𝑦) = 0 along the boundaries.
𝜕2 𝑢 𝜕2 𝑢 (15)
+ = sin (𝜋𝑥) sin (𝜋𝑦) ;
𝜕𝑥2 𝜕𝑦2 The following solutions are obtained:
𝑢 (𝑥, 𝑦) = 0 along the boundaries, 0 ≤ 𝑥, 𝑦 ≤ 1; (11)
sin (𝜋𝑦)
sin (𝑦𝜋) 𝑢0 (𝑥, 𝑦) = − 𝑥,
𝑢𝑥 (0, 𝑦) = − . 2𝜋
2𝜋
𝑥 𝜋𝑥3 sin (𝜋𝜏) sin (𝜋𝑦)
𝑢1 (𝑥, 𝑦) = [ − ] sin (𝜋𝑦) − ,
The exact solution of the previous equation is given as 𝜋 2 × 3! 𝜋2
sin (𝑥𝜋) sin (𝜋𝑦) 𝑥 𝜋𝑥3 𝜋3 𝑥5 sin (𝜋𝜏) sin (𝜋𝑦)

𝑢2 (𝑥, 𝑦) = [− + − ]sin (𝜋𝑦)+ ,
𝑢 (𝑥, 𝑦) = . (12) 𝜋 6 240 𝜋2
−2𝜋2
𝑥 𝜋𝑥3 𝜋3 𝑥5 𝜋5 𝑥7
In the view of the homotopy decomposition method, (11) can 𝑢3 (𝑥, 𝑦) = [ − + − ] sin (𝜋𝑦)
𝜋 6 120 10080
be first transformed to
sin (𝜋𝜏) sin (𝜋𝑦)
sin (𝜋𝑦) − ,
𝑢 (𝑥, 𝑦) = 𝑢 (0, 𝑦) − 𝑥 𝜋2
2𝜋 𝑢4 (𝑥, 𝑦)
𝑥
+ ∫ (𝑥 − 𝜏) [sin (𝜋𝜏) sin (𝜋𝑦) − 𝑢𝑦𝑦 (𝜏, 𝑦)] , 𝑥 𝜋𝑥3 𝜋3 𝑥5 𝜋5 𝑥7 𝜋7 𝑥9
0 = [− + − + − ] sin (𝜋𝑦)
∞ 𝜋 6 120 5040 725760
𝑢 (𝑥, 𝑦, 𝑝) = ∑ 𝑝𝑛 𝑢𝑛 (𝑥, 𝑦) .
𝑛=0 sin (𝜋𝜏) sin (𝜋𝑦)
+ ,
(13) 𝜋2
𝑢5 (𝑥, 𝑦)
Following the decomposition techniques, we obtain the fol-
lowing equation 𝑥 𝜋𝑥3 𝜋3 𝑥5 𝜋5 𝑥7 𝜋7 𝑥9 𝜋9 𝑥11
=[ − + − + − ] sin (𝜋𝑦)
𝜋 6 120 5040 362880 79833600
∞
∑ 𝑝𝑛 𝑢𝑛 (𝑥, 𝑦) sin (𝜋𝜏) sin (𝜋𝑦)
𝑛=0 − ,
𝜋2
= 𝑇 (𝑥, 𝑦)
𝑥 (14) 𝑢6 (𝑥, 𝑦)
+ 𝑝 ∫ (𝑥 − 𝜏) [ sin (𝜋𝜏) sin (𝜋𝑦)
0 𝑥 𝜋𝑥3 𝜋3 𝑥5 𝜋5 𝑥7 𝜋7 𝑥9
= [− + − + −
𝜕2 ∞ 𝑛 𝜋 6 120 5040 362880
− [ ∑ 𝑝 𝑢𝑛 (𝑥, 𝑦)]] .
𝜕𝑦2 𝑛=0 𝜋9 𝑥11 𝜋11 𝑥13
+ − ] sin (𝜋𝑦)
39916800 12454041600
Comparing the terms of the same power of 𝑝 leads to
sin (𝜋𝜏) sin (𝜋𝑦)
+ ,
𝜋2
sin (𝜋𝑦)
𝑝0 : 𝑢0 (𝑥, 𝑦) = − 𝑥, 𝑢7 (𝑥, 𝑦)
2𝜋
𝑥
𝜕2 𝑥 𝜋𝑥3 𝜋3 𝑥5 𝜋5 𝑥7 𝜋7 𝑥9 𝜋9 𝑥11
𝑝1 : 𝑢1 (𝑥, 𝑦) = ∫ (𝑥 − 𝜏) [sin (𝜋𝜏) sin (𝜋𝑦) − [𝑢 ]] 𝑑𝜏, =[ − + − + −
0 𝜕𝑦2 0 𝜋 6 120 5040 362880 39916800
𝑢1 (𝑥, 𝑦) = 0 along the boundaries,
𝜋11 𝑥13 𝜋13 𝑥15
𝑥 2 + − ] sin (𝜋𝑦)
𝜕 6227020800 2615348736000
𝑝2 : 𝑢2 (𝑥, 𝑦) = ∫ (𝑥 − 𝜏) [− [𝑢 ]] 𝑑𝜏,
0 𝜕𝑦2 1 sin (𝜋𝜏) sin (𝜋𝑦)
− .
𝑥
𝜕2 𝜋2
𝑝3 : 𝑢3 (𝑥, 𝑦) = ∫ (𝑥 − 𝜏) [− 2 [𝑢2 ]] 𝑑𝜏,
0 𝜕𝑦 (16)
Table 1: Evaluation of numerical errors for 𝑁 = 4.

𝑥 𝑌 𝑢(𝑥, 𝑦) exact 𝑢(𝑥, 𝑦) 𝑁 = 4 Error
0.25 −0.0253303 −0.0253303 6.27007 ⋅ 10−11
0.5 −0.0358224 −0.0358224 8.86722 ⋅ 10−11
0.25
0.75 −0.0253303 −0.0253303 6.27007 ⋅ 10−11
0.95 −0.00560387 −0.00560387 1.38714 ⋅ 10−11
Exact solution
0.04
1.0
0.25 −0.0358224 −0.0358224 1.26904 ⋅ 10−11
0.02
0.5 −0.0506606 −0.0506604 1.79469 ⋅ 10−7
0.5
0.75 −0.0358224 −0.0358223 1.26904 ⋅ 10−11 0.00
0.0 0.5
0.95 −0.00792506 −0.00792506 2.80752 ⋅ 10−8 y
0.25 −0.0253303 −0.0253195 1.07646 ⋅ 10−5
0.5
0.5 −0.0358224 −0.0358072 1.52235 ⋅ 10−5 x
0.75
0.75 −0.0253303 −0.0253195 1.07646 ⋅ 10−5 0.0
1.0
0.95 −0.00560387 −0.00560148 2.38148 ⋅ 10−6
0.25 −0.00560387 −0.00546191 0.000141956 Figure 1: Exact solution.
0.5 −0.00792506 −0.00772431 000200756
0.95
0.75 −0.00560387 −0.00546191 0.000141956
0.95 −0.00123975 −0.00120835 3.14051 ⋅ 10−5
In the same manner one can obtain the rest of the compo-
nents. But for eight terms were computed and the asymptotic
0.04
Approximate
solution is given by 1.0

0
𝑢𝑁=8 (𝑥, 𝑦)
0.00
0.0 0.5
y
𝑥 𝜋𝑥3 𝜋3 𝑥5 𝜋5 𝑥7 𝜋7 𝑥9
=[ − + − +
2𝜋 2 × 3! 2 × 5! 2 × 7! 2 × 9! 0.5
(17) x
𝜋9 𝑥11 𝜋11 𝑥13 𝜋13 𝑥15 0.0
− + − ] sin (𝜋𝑦) 1.0
2 × 11! 2 × 13! 2 × 15!
Figure 2: Approximated solution for the 4 first terms.
1
− 2 sin (𝑥𝜋) sin (𝑦𝜋) .
𝜋
Problem 2. Consider 3D Poisson equation:
Therefore in general for any 𝑁 > 8 we have
𝜕2 𝑢 𝜕2 𝑢 𝜕2 𝑢
+ + = sin (𝜋𝑥) sin (𝜋𝑦) sin (𝜋𝑧) ,
1 𝑁 (−1)𝑛 (𝑥𝜋)2𝑛+1 𝜕𝑥2 𝜕𝑦2 𝜕𝑧2 (19)
𝑢𝑁=𝑛 (𝑥, 𝑦) = [ ∑ ] sin (𝜋𝑦)
2𝜋2 𝑛=0 (2𝑛 + 1)! 𝑢 (𝑥, 𝑦, 𝑧) = 0 along the boundaries, 0 ≤ 𝑥, 𝑦 ≤ 1.
1 Following the discussion presented earlier we obtain the fol-
− sin (𝑥𝜋) sin (𝑦𝜋) ,
𝜋2 lowing set of integral equations:
1 (18) sin (𝜋𝑦) sin (𝜋𝑧)
lim 𝑢𝑁 (𝑥, 𝑦) = sin (𝜋𝑥) sin (𝜋𝑦) 𝑝0 : 𝑢0 (𝑥, 𝑦) = − 𝑥,
𝑁→∞ 2𝜋2 3𝜋
1 𝑝1 : 𝑢1 (𝑥, 𝑦)
− sin (𝑥𝜋) sin (𝑦𝜋)
𝜋2
1
𝑥
𝜕2 𝑢0
= − 2 sin (𝑥𝜋) sin (𝑦𝜋) . = ∫ (𝑥 − 𝜏) [sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧) − ] 𝑑𝜏,
2𝜋 0 𝜕𝑦2
𝑥
𝜕2 𝑢𝑛−1
This is the exact solution of the problem. Figures 1 and 2 show 𝑝𝑛 : 𝑢𝑛 (𝑥, 𝑦) = ∫ (𝑥 − 𝜏) [− ] 𝑑𝜏,
the comparison of the exact solution and the approximated 0 𝜕𝑦2
one for 𝑁 = 4. The approximate solution and the exact
solution are compared in Figures 1 and 2, respectively. 𝑢𝑛 (𝑥, 𝑦) = 0 along the boundaries, 𝑛 ≥ 2.
The numerical errors for 𝑁 = 4 are evaluated in Table 1. (20)
The following solutions are obtained: 1 𝑥 𝜋𝑥3 𝜋3 𝑥5 𝜋5 𝑥7 𝜋7 𝑥9

𝑢(𝑥, 𝑦, 𝑧)𝑁=6 = [ − + − +
3𝜋2 3𝜋 18 360 15120 1088640
sin (𝜋𝑦) sin (𝜋𝑧)
𝑢0 (𝑥, 𝑦, 𝑧) = − 𝑥, 𝜋9 𝑥11
3𝜋 − ]sin (𝜋𝑦) sin (𝜋𝑧)
119750400
1 𝜋𝑥3 2 sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧)
𝑢1 (𝑥, 𝑦, 𝑧) = [ 𝑥 − ] sin (𝜋𝑦) sin (𝜋𝑧) − ,
𝜋 9 3𝜋2
sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧) 1 (𝜋𝑥)3 (𝜋𝑥)5 (𝜋𝑥)7
− , 𝑢(𝑥, 𝑦, 𝑧)𝑁=6 = 2 [𝜋𝑥 − + −
𝜋2 3𝜋 3! 5! 7!
(𝜋𝑥)9 (𝜋𝑥)3
2 𝜋𝑥3 𝜋3 𝑥5 + − ]sin(𝜋𝑦) sin(𝜋𝑧)
𝑢2 (𝑥, 𝑦, 𝑧) = [− 𝑥 + − ] sin (𝜋𝑦) sin (𝜋𝑧) 9! 11!
𝜋 9 90
2 sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧)
2 sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧) − .
+ , 3𝜋2
𝜋2 (22)
𝑢3 (𝑥, 𝑦, 𝑧) Therefore, for any 𝑛 ≥ 6, the partial sum is given as
4 2𝜋𝑥3 𝜋3 𝑥5 𝜋5 𝑥7 1 𝑁
(−1)𝑘 (𝜋𝑥)2𝑘+1
=[ 𝑥− + − ] sin (𝜋𝑦) sin (𝜋𝑧) 𝑢𝑁=𝑛 (𝑥, 𝑦, 𝑧) = [ ∑ ] sin (𝜋𝑦) sin (𝜋𝑧)
𝜋 3 30 1890 3𝜋2 𝑘=1 (2𝑘 + 1)!
4 sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧) 2 sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧)
− , − .
𝜋2 3𝜋2
(23)
𝑢4 (𝑥, 𝑦, 𝑧)
Thus
8 4𝜋𝑥3 𝜋3 𝑥5 𝜋5 𝑥7
= [− 𝑥 + − + ] sin (𝜋𝑦) sin (𝜋𝑧) 𝑢 (𝑥, 𝑦, 𝑧) = lim 𝑢𝑁=𝑛 (𝑥, 𝑦, 𝑧)
𝜋 3 15 630 𝑁→∞
8 sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧) sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧)
+ , =
𝜋2 3𝜋2
(24)
2 sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧)
16 8𝜋𝑥3 2𝜋3 𝑥5 𝜋5 𝑥7 −
𝑢5 (𝑥, 𝑦, 𝑧) = [ 𝑥 − + − 3𝜋2
𝜋 3 15 315
sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧)
7 9 7 11 =− .
𝜋𝑥 𝜋𝑥 3𝜋2
+ − ] sin (𝜋𝑦) sin (𝜋𝑧)
22680 3742200
And this is the exact solution to the problem. One can
evaluate error committed by choosing the 𝑁 first terms in
16 sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧) the series solutions, in the same manner as in Table 1. The
− .
𝜋2 accuracy of the results is estimated by error function
(21)
󵄨 󵄨
𝑅𝑁 (𝑥, 𝑦, 𝑧) = 󵄨󵄨󵄨𝑢𝑁 (𝑥, 𝑦, 𝑧) − 𝑢 (𝑥, 𝑦, 𝑧)󵄨󵄨󵄨 . (25)
In the same manner one can obtain the rest of the compo- Problem 3. Let us consider the following biharmonic equa-
nents. But for six terms were computed and the asymptotic tion
solution is given by
𝑑4 𝑢 (𝑥)
+ 4𝑢 (𝑥) = 0, (26)
𝑥 𝜋𝑥3 𝜋3 𝑥5 𝜋5 𝑥7 𝜋7 𝑥9 𝑑𝑥4
𝑢(𝑥, 𝑦, 𝑧)𝑁=6 = [ − + − +
3𝜋 18 360 15120 1088640 for which the exact solution is
9 11
𝜋𝑥 Exp [1 − 𝑥] cos [𝑥]
− ] sin (𝜋𝑦) sin (𝜋𝑧) 𝑢 (𝑥) = . (27)
119750400 cos [1]
2 sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧) The aim of this part is to compare the numerical results
− , obtained via HDM and the method used in [26].
𝜋2
Table 2: Comparison of the HDM and [1] results with the exaction Problem 4. We consider the 2D biharmonic equation
solution for 𝑁 = 6.
𝜕4 𝑢 𝜕4 𝑢 𝜕4 𝑢
𝑥 HDM Exact ADM Err for HDM Err for ADM + 2 + = sin (3𝜋𝑥) sin (3𝜋𝑦) , 0 ≤ 𝑥, 𝑦 ≤ 1,
−1.0 7.38906 7.38906 7.38906 6.78𝐸 − 16 8.88𝐸 − 16
𝜕𝑥4 𝜕𝑥2 𝜕𝑦2 𝜕𝑥4
(31)
−0.6 7.56598 7.56598 7.56598 4.76𝐸 − 12 7.96𝐸 − 12
−0.2 6.02244 6.02244 6.02244 0.015𝐸 − 11 1.46𝐸 − 11 subject to the initial conditions:
0.2 4.03696 4.03696 4.03696 0.017𝐸 − 11 1.80𝐸 − 11
0.6 2.27883 2.27883 2.27883 0.015𝐸 − 11 1.46𝐸 − 11 𝜕𝑢 (𝑥, 𝑦) sin (3𝜋𝑦)
| (𝑥 = 0) = , 𝜕𝑥,𝑥 𝑢 (0, 𝑦) = 0,
1.0 1.0 1.0 1.0 1.24𝐸 − 15 2.22𝐸 − 15 𝜕𝑥 108𝜋3
(32)
sin (3𝜋𝑦)
𝜕𝑥,𝑥,𝑥 𝑢 (0, 𝑦) = − .
Applying the steps involved in the HDM, we arrive at the 12𝜋
following: In the view of the homotopy decomposition method, the fol-
3
lowing integral equations are obtained:
𝑥
𝑢0 (𝑥) = 𝑒 Sec (1) (1 − 𝑥 + ),
3 sin (3𝜋𝑦) sin (3𝜋𝑦) 𝑥 3
𝑝0 : 𝑢0 (𝑥, 𝑦) = 3
𝑥− ,
108𝜋 12𝜋 3!
𝑥4 𝑥5 𝑥7
𝑢1 (𝑥) = −𝑒 Sec (1) [ − + ], 𝑝1 : 𝑢1 (𝑥, 𝑦)
4 20 420
𝑥8 𝑥9 𝑥11
𝑥
𝜕2 𝑢0 𝜕4 𝑢0
𝑢2 (𝑥) = −𝑒 Sec (1) [− + − ], = ∫ (𝑥 − 𝜏)[sin (𝜋𝜏) sin (𝜋𝑦) − − 2 ]𝑑𝜏,
1120 10080 554400 0 𝜕𝑦2 𝜕𝑥2 𝜕𝑦2
𝑥12 𝑥13 𝑥15

𝑥
𝜕2 𝑢𝑛−1 𝜕4 𝑢
𝑢3 (𝑥) = −𝑒 Sec (1) [ − + ], 𝑝𝑛 : 𝑢𝑛 (𝑥, 𝑦) = ∫ (𝑥 − 𝜏) [− 2
− 2 2 0 2 ] 𝑑𝜏,
2217600 64864800 1135134000 0 𝜕𝑦 𝜕𝑥 𝜕𝑦
𝑢4 (𝑥) 𝑢𝑛 (𝑥, 𝑦) = 0 along the boundaries, 𝑛 ≥ 2.

(33)
𝑥16 𝑥17
= −𝑒 Sec (1) [− + It is worth noting that if the zeroth component 𝑢0 (𝑥, 𝑦) is
16144128000 274450176000
defined, then the remaining components 𝑛 ≥ 1 can be com-
𝑥19 pletely determined such that each term is determined by
− ], using the previous terms, and the series solutions are thus
46930980096000
entirely determined. Finally, the solution 𝑢(𝑥, 𝑦) is approxi-
𝑢5 (𝑥) mated for 𝑛 = 4:
𝑥20 𝑥21 𝑢 (𝑥, 𝑦)

= −𝑒 Sec (1) [ −
312873200640000 6570337213440000 𝑥 𝑥3 𝜋𝑥5 3𝜋3 𝑥7 3𝜋5 𝑥9
= sin (3𝜋𝑦) [ 3
− + − +
𝑥23 108𝜋 72𝜋 160 2240 17920
+ ].
1662295315000320000 27𝜋7 𝑥11 81𝜋9 𝑥13 81𝜋11 𝑥15
(28) − + − ],
1971200 102502400 7175168000
In the same manner, one can obtain the remaining term by (34)
using the following recursive formula: 𝑢 (𝑥, 𝑦)
𝑥
𝑢𝑛+1 (𝑥) = − ∫ (𝑥 − 𝑡)3 𝑢𝑛 (𝑡) 𝑑𝑡. (29) sin (3𝜋𝑦) (3𝜋𝑥)3 (3𝜋𝑥)5 (3𝜋𝑥)7 (3𝜋𝑥)9
0
= [3𝜋𝑥 − + − +
324𝜋4 3! 5! 7! 9!
In this paper we consider only the first six terms of the series
(3𝜋𝑥)11 (3𝜋𝑥)13 (3𝜋𝑥)15
solution as follows: − + − ].
11! 13! 15!
5 (35)
𝑢𝑁=6 = ∑ 𝑢𝑛 (𝑥) . (30)
𝑛=0 Therefore for any 𝑁 ≥ 4 we have the following:
To access the accuracy of the method used in paper, we com- sin (3𝜋𝑦) 𝑁 (3𝜋𝑥)2𝑛+1
pare in Table 2 the numerical results of the above equation, 𝑢𝑁 (𝑥, 𝑦) = ∑ . (36)
the solution obtained in [26] with the exact solution. 324𝜋4 𝑛=0 (2𝑛 + 1)!
Proof. Use the step of the homotopy decomposition method.
Problem 5. We consider the 3D biharmonic equation:
𝜕4 𝑢 𝜕4 𝑢 𝜕4 𝑢 𝜕4 𝑢 𝜕4 𝑢 𝜕4 𝑢
0.00002 + 2 + 2 + 2 + +
Approximate
1.0 𝜕𝑥4 𝜕𝑥2 𝜕𝑦2 𝜕𝑥2 𝜕𝑧2 𝜕𝑧2 𝜕𝑦2 𝜕𝑦4 𝜕𝑧4
0
−0.00002 = sin (𝜋𝑥) sin (𝜋𝑦) sin (𝜋𝑧) , (41)
0.0 0.5 y 0 ≤ 𝑥, 𝑦, 𝑧 ≤ 1,
0.5 𝑢 = 0, 𝑢𝑥,𝑥 = 𝑢𝑦,𝑦 = 𝑢𝑧,𝑧 = 0.
x
1.0 0.0
In the view of the homotopy decomposition method, the fol-
lowing integral equations are obtained:
Figure 3: Analytical solution.
sin (3𝜋𝑦) sin (3𝜋𝑦) 𝑥 3
𝑝0 : 𝑢0 (𝑥, 𝑦) = 3
𝑥− ,
108𝜋 12𝜋 3!
𝑝1 : 𝑢1 (𝑥, 𝑦)
𝑥
= ∫ (𝑥 − 𝜏) [ sin (𝜋𝜏) sin (𝜋𝑦) sin (𝜋𝑧)
0.00003 0
Approximate
0.00002 1.0
𝜕4 𝑢0 𝜕4 𝑢0
0.00001 −2 − 2
𝜕𝑥2 𝜕𝑦2 𝜕𝑥2 𝜕𝑧2
0
0.0 0.5 y
𝜕4 𝑢0 𝜕4 𝑢0 𝜕4 𝑢0
−2 − − ] 𝑑𝜏,
0.5 𝜕𝑧2 𝜕𝑦2 𝜕𝑦4 𝜕𝑧4
x
1.0 0.0 𝑝𝑛 : 𝑢𝑛 (𝑥, 𝑦)
Figure 4: Absolute value of the solution.

𝑥
𝜕4 𝑢𝑛−1 𝜕4 𝑢
= ∫ (𝑥 − 𝜏) [−2 2 2
− 2 2 𝑛−12
0 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑧
Thus 𝜕4 𝑢𝑛−1 𝜕4 𝑢𝑛−1 𝜕4 𝑢𝑛−1

−2 − − ] 𝑑𝜏,
sin (3𝜋𝑦) sin (3𝜋𝑥) 𝜕𝑧2 𝜕𝑦2 𝜕𝑦4 𝜕𝑧4
lim 𝑢𝑁 (𝑥, 𝑦) = . (37)
𝑁→∞ 324𝜋4 𝑢𝑛 (𝑥, 𝑦) = 0 along the boundaries, 𝑛 ≥ 2.
The exact solution of (31) is given by (42)
sin (3𝜋𝑦) sin (3𝜋𝑥) Solving the previous integral equations, the series solutions
= 𝑢 (𝑥, 𝑦) . (38)
324𝜋4 for the first 𝑁 terms are given as
Figures 3 and 4 are the graphical representation of the previ-
ous solution. We have plotted the solution for (31) in Figure 3 sin (𝑧𝜋) sin (𝜋𝑦) 𝑁 (𝜋𝑥)2𝑛+1
𝑢𝑁 (𝑥, 𝑦, 𝑧) = ∑ . (43)
and showed absolute value of the solution in Figure 4. 9𝜋4 𝑛=0 (2𝑛 + 1)!
Therefore taking the limit at 𝑁 tending to infinity we obtained

Theorem 2. Let 𝑚 be a nonzero natural number and let
(x, y) ∈ [0, 1] × [0, 1]; then two dimensional biharmonic equa- sin (𝑥𝜋) sin (𝑧𝜋) sin (𝜋𝑦)
tion of form 𝑢 (𝑥, 𝑦, 𝑧) = lim 𝑢𝑁 (𝑥, 𝑦, 𝑧) = .
𝑁→∞ 9𝜋4
4 4 4 (44)
𝜕𝑢 𝜕𝑢 𝜕𝑢
+ 2 2 2 + 4 = sin (𝑚𝜋𝑥) sin (𝑚𝜋𝑦) (39)
𝜕𝑥4 𝜕𝑥 𝜕𝑦 𝜕𝑦
4. Conclusion
with 𝑢(𝑥, 𝑦) = 0 along the boundaries has an exact solution as
follows In this paper the recent homotopy decomposition [18–21] is
used to solve the 2D and 3D Poisson equations and bihar-
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𝑢 (𝑥, 𝑦) = . (40)
4𝑚4 𝜋4 require the linearization or assumptions of weak nonlinearity,
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http://dx.doi.org/10.1155/2013/921879
Research Article
Pattern Dynamics in a Spatial Predator-Prey
System with Allee Effect
Gui-Quan Sun,1,2,3,4 Li Li,5 Zhen Jin,1,2 Zi-Ke Zhang,3 and Tao Zhou6
1
Complex Sciences Center, Shanxi University, Taiyuan, Shan’xi 030006, China
2
School of Mathematical Sciences, Shanxi University, Taiyuan, Shan’xi 030006, China
3
Institute of Information Economy, Hangzhou Normal University, Hangzhou 310036, China
4
Department of Mathematics, North University of China, Taiyuan, Shan’xi 030051, China
5
Department of Mathematics, Taiyuan Institute of Technology, Taiyuan, Shan’xi 030008, China
6
Web Sciences Center, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China
Correspondence should be addressed to Gui-Quan Sun; gquansun@126.com
Copyright © 2013 Gui-Quan Sun et al. This is an open access article distributed under the Creative Commons Attribution License,
We investigate the spatial dynamics of a predator-prey system with Allee effect. By using bifurcation analysis, the exact Turing
domain is found in the parameters space. Furthermore, we obtain the amplitude equations and determine the stability of different
patterns. In Turing space, it is found that predator-prey systems with Allee effect have rich dynamics. Our results indicate that
predator mortality plays an important role in the pattern formation of populations. More specifically, as predator mortality rate
increases, coexistence of spotted and stripe patterns, stripe patterns, spotted patterns, and spiral wave emerge successively. The
results enrich the finding in the spatial predator-prey systems well.
1. Introduction However, these previous works did not take into account the
effect of space.
The Allee effect, named after the ecologist Warder Clyde There are also some works done on spatial predator-
Allee, has been recognized as an important phenomenon of prey systems with Allee effect [14–16]. Petrovskii et al. found
positive density dependence in low-density population [1–5]. that the deterministic system with Allee effect can induce
Allee effect can occur whenever fitness of an individual in a patch invasion [14]. Morozov et al. found that the temporal
small or sparse population decreases as the population size or population oscillations can exhibit chaotic dynamics even
density also declines [6, 7]. Since the outstanding work of when the distribution of the species in the space was regular
Allee [1], the Allee effect has been regarded as one of the cen- [15]. Moreover, they found that the chaos accompanied with
tral and highly important issues in the population and com- patch invasion even though the environments were het-
munity ecology. And its critical importance has widely been erogeneous [16]. However, their results were obtained by
realized in the conservation biology that Allee effect is most choosing particular initial conditions. Then, it is natural to
likely to increase the extinction risk of low-density popula- ask what kind of patterns can be obtained in predator-prey
tions. As a result, studies on Allee effect have received more systems with Allee effect by using other initial conditions. To
and more attention from both mathematicians and ecologists. understand that mechanism well, we will investigate a
Long time series of the density of both prey and predator predator-prey system with Allee effect.
is needed, so it is difficult to analyse their dynamics. As a Because of the insightful work of many scientists over
result, it may provide useful information by constructing recent years, we can make research on pattern selection by
mathematical models to investigate the dynamical behaviors using the standard multiple scale analysis [17, 18], in which
of predator-prey systems. There have been a large group of the control parameters and the derivatives are expanded in
papers on predator-prey systems with Allee effect [8–13]. terms of a small enough parameter. In the neighborhood of
the bifurcation points (Hopf and Turing bifurcation points), we obtain the following equations:
the critical amplitudes follow the normal forms, and thus
their general forms can be obtained from the methods of
𝜕𝑢
symmetry-breaking bifurcations. = 𝛾𝑢 (𝑢 − 𝛽) (1 − 𝑢) − 𝑢V + Δ𝑢, (4a)
The paper is organized as follows. In Section 2, we present 𝜕𝑡
a predator-prey system with Allee effect and give Turing 𝜕V
region in parameters space. In Section 3, by using multiple = 𝑢V − 𝛿V2 + 𝜀ΔV, (4b)
𝜕𝑡
scale analysis, we obtain amplitude equations. In Section 4,
we show the spatial patterns by a series of numerical simu- where
lations. Finally, conclusions and discussions are presented in
𝐻0 4𝜔𝐾
Section 5. 𝛽= , 𝛾= 2
,
𝐾 𝐴𝜅(𝐾 − 𝐻0 )
(5)
2. A Predator-Prey System with Allee Effect 𝑀 𝐷
𝛿= , 𝜀 = 2.
𝑎 𝐷1
We consider the following model of two-dimensional spa-
tiotemporal system [14–16, 19]: First of all, we need to investigate the dynamics of nonspatial
model of systems (4a) and (4b)
𝜕𝐻
= 𝐹 (𝐻) − 𝑓 (𝐻, 𝑃) + 𝐷1 Δ𝐻, (1a) 𝑑𝑢
𝜕𝑇 = 𝛾𝑢 (𝑢 − 𝛽) (1 − 𝑢) − 𝑢V, (6a)
𝑑𝑡
𝜕𝑃
= 𝜅𝑓 (𝐻, 𝑃) − 𝐷 (𝑃) + 𝐷2 Δ𝑃, (1b) 𝑑V
𝜕𝑇 = 𝑢V − 𝛿V2 . (6b)
𝑑𝑡
where 𝐻 = 𝐻(𝑋, 𝑌, 𝑇) and 𝑃 = 𝑃(𝑋, 𝑌, 𝑇) are densities of Systems (6a) and (6b) have three boundary equilibrium
prey and predator, respectively, at time 𝑇 and position (𝑋, 𝑌). named 𝐸0 = (0, 0), 𝐸1 = (1, 0), and 𝐸2 = (𝛽, 0) and two
The function 𝐹(𝐻) represents the intrinsic prey growth, interior equilibriums named 𝐸3 and 𝐸4 , where
𝑓(𝐻, 𝑃) = 𝑓(𝐻)𝑃 represents predation term, 𝜅 is the food
utilization coefficient, 𝐷1 and 𝐷2 are diffusion coefficients,
and 𝐷(𝑃) describes predator mortality. 𝛾𝛿 + 𝛾𝛽𝛿 − 1 + √𝑄 𝛾𝛿 + 𝛾𝛽𝛿 − 1 + √𝑄
𝐸3 = ( , ), (7a)
It is assumed that the predation term is a bilinear form 2𝛾𝛿 2𝛾𝛿2
of prey and predator density and predator mortality is a
nonlinear function of predator density. As a result, we choose 𝛾𝛿 + 𝛾𝛽𝛿 − 1 − √𝑄 𝛾𝛿 + 𝛾𝛽𝛿 − 1 − √𝑄
𝐸4 = ( , ), (7b)
𝑓(𝐻, 𝑃) = 𝐻𝑃 and 𝐷(𝑃) = 𝑀𝑃2 [20]. 2𝛾𝛿 2𝛾𝛿2
When the prey population obeys Allee dynamics, its
growth rate can be parameterized as follows [14, 15, 21]: where 𝑄 = (𝛾𝛿)2 − 2(𝛾𝛿)2 𝛽 − 2𝛾𝛽𝛿 + 1.
From a biological point of view, we are concerned with the
dynamics of 𝐸3 and 𝐸4 . The Jacobian matrix corresponding
4𝜔
𝐹 (𝐻) = 2
𝐻 (𝐻 − 𝐻0 ) (𝐾 − 𝐻) , (2) to the equilibrium point is that
(𝐾 − 𝐻0 )
𝑎 𝑎
𝐽 = ( 11 12 ) , (8)
𝑎21 𝑎22
where 𝐾 is the prey-carrying capacity, 𝜔 is the maximum
per capita growth rate, and 𝐻0 quantifies the intensity of the where
Allee effect. If 0 < 𝐻0 < 𝐾, 𝐹(𝐻) is a strong Allee effect; if
2
−𝐾 < 𝐻0 < 0, 𝐹(𝐻) is a weak Allee effect; if 𝐻0 ≤ −𝐾, the 𝑎11 = 2𝛾𝑢∗ − 𝛾𝛽 − 3𝛾(𝑢∗ ) + 2𝛾𝛽𝑢∗ ,
Allee effect is absent.
In order to minimize the number of parameters involved 𝑎12 = −𝑢∗ ,
in the model system, it is extremely useful to write the (9)
system in a nondimensionalized form. Although there is no 𝑎21 = V∗ − 𝛿,
unique method of doing this, it is often a good idea to relate
𝑎22 = 𝑢∗ .
the variables to some key relevant parameters. Introducing
dimensionless variables Diffusion-driven instability requires the stable, homoge-
neous steady state is driven unstable by the interaction of the
𝐻 𝑃 dynamics and diffusion of the species; and therefore
𝑢= , V= , 𝑡 = 𝑎𝑇,
𝐾 𝜅𝐾
(3) 𝑎11 + 𝑎22 < 0,
𝑎 𝑎 (10)
𝑋 = 𝑋√ , 𝑌 = 𝑌√ ,
𝐷1 𝐷1 𝑎11 𝑎22 − 𝑎12 𝑎21 > 0.
It is found from direct calculations that 𝐸3 is unstable and

𝐸4 is stable. Denote 𝐸4 = (𝑢∗ , V∗ ). 2.2
Following the standard linear analysis of the reaction-
diffusion equation [22], we consider a perturbation near the 2
steady state:
𝑢 (𝑟,⃗𝑡) = 𝑢∗ + 𝑢 (𝑟, 𝑡) , 1.8

(11)
V (𝑟,⃗𝑡) = V∗ + V (𝑟, 𝑡) , 1.6
𝛿
where 𝑢(𝑟, 𝑡) ≪ 𝑢∗ , V(𝑟, 𝑡) ≪ V∗ , and 𝑟 = (𝑋, 𝑌). Assume 1.4
that
𝑢 (𝑟, 𝑡) 𝛼 1.2
( ) = ( 1 ) 𝑒𝜆𝑡 𝑒𝑖(𝜅𝑋 𝑋+𝜅𝑌 𝑌) , (12)
V (𝑟, 𝑡) 𝛼2
1
where 𝜆 is the growth rate of perturbation in time 𝑡, 𝛼1 and
T
𝛼2 represent the amplitudes, and 𝜅𝑋 and 𝜅𝑌 are the wave 0.8
number of the solutions.
The characteristic equation of the systems (4a) and (4b) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
is 𝛽
Turing line
𝑢
(𝐴 − 𝜆𝐼) ( ) = 0, (13) Hopf line
V
Figure 1: Bifurcation diagram for the systems (4a) and (4b). The
where green one is the Hopf bifurcation critical line and the red one, Turing
bifurcation critical line. The figure shows the Turing space which is
2
𝑎11 − (𝜅𝑋 + 𝜅𝑌2 ) 𝑎12 marked by 𝑇. Parameters values: 𝛾 = 1.5 and 𝜀 = 0.15.
𝐴=( ). (14)
2
𝑎21 𝑎22 − 𝜀 (𝜅𝑋 + 𝜅𝑌2 )
respect to the homogeneous perturbations, but they lose
As a result, we have characteristic polynomial: their stability with respect to the perturbations of specific
wave numbers 𝜅. In this region, stationary patterns can be
𝜆2 − 𝑡𝑟𝜅 𝜆 + Δ 𝜅 = 0, (15) observed. To see the effect of parameter 𝛿 well, we plot
Δ in Figure 2 the dispersion relation corresponding to several
𝑡𝑟𝜅 = 𝑎11 + 𝑎22 − 𝜅2 (1 + 𝜀) = 𝑡𝑟𝐽 − 𝜅2 (1 + 𝜀) , values of 𝛿 while keeping the other parameters fixed. We see
that the available Turing modes shift to higher wave numbers
Δ 𝜅 = 𝑎11 𝑎22 − 𝑎12 𝑎21 − 𝜅2 (𝑎11 𝜀 + 𝑎22 ) + 𝜅4 𝜀 (16) when 𝛽 decreases.
Δ
= Δ 𝐽 − 𝜅2 (𝑎11 𝜀 + 𝑎22 ) + 𝜅4 𝜀,
3. Spatial Dynamics of Systems (4a) and (4b)
2 2
where 𝜅 = + 𝜅𝑋 𝜅𝑌2 . In the following, we use multiple scale analysis to determine
The roots of (15) can be obtained by the following form: the amplitude equations when |𝜅| = 𝜅𝑇 . Denote 𝛿 as the
1 controlled parameters. When the controlled parameter is
𝜆𝜅 = (𝑡𝑟𝜅 ± √𝑡𝑟𝜅2 − 4Δ 𝜅 ) . (17) larger than the critical value of Turing point, the solutions of
2 systems (4a) and (4b) can be expanded as
When Im(𝜆 𝜅 ) ≠0 and Re(𝜆 𝜅 ) = 0, Hopf bifurcation 𝑁
will emerge. Then, we have that the critical value of Hopf 𝑐 = 𝑐0 + ∑ (𝐴 𝑖 exp (𝑖𝜅𝑖 𝑟)⃗ + (𝐴𝑖 exp (−𝑖𝜅𝑖 𝑟))
⃗, (19)
bifurcation parameter-𝛿 equals 𝑖=1
𝛾 (𝛾 + 𝛽 − 1) with |𝜅| = 𝜅𝑇 . 𝐴 𝑗 and the conjugate 𝐴𝑗 are the amplitudes

𝛿𝐻 = . (18) associated with the modes 𝜅𝑗 and −𝜅𝑗 .
𝛾2 𝛽2+ 𝛾2 − 2𝛾2 𝛽 − 1
Close to onset 𝛽 = 𝛽𝑇 , one has that
When 𝜅2 = (𝜅𝑇 )2 = √Δ 𝐽 /𝜀 and Im(𝜆 𝜅 ) = 0, Re(𝜆 𝜅 ) = 0, 𝜕𝐴 𝑖
= 𝑠𝑖 𝐴 𝑖 + 𝐹𝑖 (𝐴 𝑖 , 𝐴 𝑗 , . . .) . (20)
Turing bifurcation will occur. Denote 𝛿𝑇 as the critical value 𝜕𝑡
of 𝛿 as Turing instability occurs. Since the expression is Based on the center manifold near the Turing bifurcation
complicated, we omit it here. point, it can be concluded that amplitude 𝐴 𝑗 satisfies
In Figure 1, we show the two critical lines in the parameter
space spanned by 𝛽 and 𝛿. The equilibria that can be found 𝜕𝐴 𝑖
= 𝐹𝑖 (𝐴 𝑖 , 𝐴𝑖 , 𝐴 𝑗 , 𝐴𝑗 , . . .) . (21)
in the region, marked by 𝑇 (Turing space), are stable with 𝜕𝑡
Due to spatial translational symmetry, we have the follow-

0.05 (e) ing equation:
(d) 𝜅
0.2 0.4 0.6 0.8 1 1.2 1.4 𝜕𝐴 𝑘
𝜏 exp (𝑖𝜅𝑘 𝑟0 )
0
(c) 𝜕𝑡
(b) = 𝜇𝐴 𝑘 exp (𝑖𝜅𝑘 𝑟0 ) + ∑ℎ𝑙𝑚 𝐴 𝑙 𝐴 𝑚 exp [𝑖 (𝜅𝑙 + 𝜅𝑚 ) 𝑟0 ]
−0.05 𝑙𝑚
(a)
Re (𝜆)
+ ∑ 𝑔𝑙𝑚𝑛 𝐴 𝑙 𝐴 𝑚 𝐴 𝑛 exp [𝑖 (𝜅𝑙 + 𝜅𝑚 + 𝜅𝑛 ) 𝑟0 ] .

𝑙𝑚𝑛
−0.1
(26)
Comparing (25) with (26), one can find that the two
−0.15 equations hold only if 𝜅𝑘 = 𝜅𝑙 + ⋅ ⋅ ⋅ + 𝜅𝑚 . From the center
manifold theory, we know that amplitude equations do not
include the amplitude with unstable mode. As a result, we
−0.2
have the following equations:
Figure 2: Dispersion relation for different 𝛿. Parameters values: 𝛽 = 𝜕𝐴 1

𝜏0 = 𝜇𝐴 1 + ℎ𝐴2 𝐴3
0.02, 𝛾 = 1.5, and 𝜀 = 0.15. (a) 𝛿 = 1.08; (b) 𝛿 = 1.04; (c) 𝛿 = 1; (d) 𝑑𝑡
𝛿 = 0.96; and (e) 𝛿 = 0.92.
󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2
− (𝑔1 󵄨󵄨󵄨𝐴 1 󵄨󵄨󵄨 + 𝑔2 (󵄨󵄨󵄨𝐴 2 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝐴 3 󵄨󵄨󵄨 )) 𝐴 1 ,
𝜕𝐴 2
In order to obtain the amplitude equations, we first need 𝜏0 = 𝜇𝐴 2 + ℎ𝐴1 𝐴3
𝑑𝑡 (27)
to investigate the linearized form of systems (4a) and (4b) at
󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2
the equilibrium point 𝐸4 . By setting 𝑢 = 𝑢∗ +𝑥 and V = V∗ +𝑦, − (𝑔1 󵄨󵄨󵄨𝐴 2 󵄨󵄨󵄨 + 𝑔2 (󵄨󵄨󵄨𝐴 1 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝐴 3 󵄨󵄨󵄨 )) 𝐴 2 ,
we have the following equations:
𝜕𝐴 3
𝜏0 = 𝜇𝐴 3 + ℎ𝐴1 𝐴2
𝑑𝑡
𝜕𝑥 2 2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2
= [2𝛾𝑢∗ − 3𝛾(𝑢∗ ) + 2𝛾(𝑢∗ ) 𝛽 − 𝛾𝛽 − V∗ ] 𝑥 − (𝑔1 󵄨󵄨󵄨𝐴 3 󵄨󵄨󵄨 + 𝑔2 (󵄨󵄨󵄨𝐴 1 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝐴 2 󵄨󵄨󵄨 )) 𝐴 3 ,
𝜕𝑡 (22a)
+ (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) 𝑥2 − 𝛾𝑥3 − 𝑥𝑦 + Δ𝑥, where 𝜇 = (𝛿𝑇 − 𝛿)/𝛿𝑇 and 𝜏0 is a typical relaxation time.
In the following part, we will give the expressions of 𝜏0 , ℎ,
𝜕𝑦 𝑔1 , and 𝑔2 . Let
= V∗ 𝑥 + 𝑢∗ 𝑦 + 𝑥𝑦 − 2𝛿V∗ 𝑦 − 𝛿𝑦2 + 𝜀Δ𝑦. (22b)
𝜕𝑡
𝑥
𝑋 = ( ),
Close to onset 𝛿 = 𝛿𝑇 , the solutions of systems (4a) and 𝑦
(4b) can be expanded as series form: (28)
𝑁
𝑁 = ( 1) .
3
𝑁2
𝑈 = 𝑈𝑠 + ∑ 𝑈0 [𝐴 𝑗 exp (𝑖𝜅𝑗 𝑟)⃗ + 𝐴𝑗 exp (−𝑖𝜅𝑗 𝑟)]
⃗. (23)
𝑗=1
Then systems (4a) and (4b) can be written as:
𝜕𝑋
System (19) can be expanded as = 𝐿𝑋 + 𝑁, (29)
𝜕𝑡
3 where
𝑈∗ = ∑𝑈0 [𝐴 𝑗 exp (𝑖𝜅𝑗 𝑟)⃗ + 𝐴𝑗 exp (−𝑖𝜅𝑗 𝑟)]
⃗, (24) 2 2
2𝛾𝑢∗ − 3𝛾(𝑢∗ ) + 2𝛾𝛽(𝑢∗ ) − 𝛾𝛽 − V∗ + Δ 0
𝑗=1
𝐿=( ),
∗
V 𝑢 − 2𝛿V∗ + 𝜀Δ
∗
∗
where 𝑈0 = ((𝑎11 ∗
𝜀 + 𝑎11 ∗
)/(2𝑎21 ), 1)𝑇 is the eigenvector of the
(𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) − 𝛾𝑥3 − 𝑥𝑦
linearized operator. 𝑁=( ).
From the standard multiple scale analysis, up to the third 𝑥𝑦 − 𝛿𝑦2
order in the perturbations, the spatiotemporal evolution of (30)
the amplitudes can be described as
We need to investigate the dynamical behavior when 𝛿 is
close to 𝛿𝑇 , and thus we expand 𝛿 as:
𝜕𝐴 𝑘
𝜏 = 𝜇𝐴 𝑘 + ∑ℎ𝑙𝑚 𝐴 𝑙 𝐴 𝑚 + ∑ 𝑔𝑙𝑚𝑛 𝐴 𝑙 𝐴 𝑚 𝐴 𝑛 . (25)
𝜕𝑡 𝑙𝑚 𝑙𝑚𝑛 𝛿𝑇 − 𝛿 = 𝜖𝛿1 + 𝜖2 𝛿2 + 𝜖3 𝛿3 + 𝑂 (𝜖4 ) , (31)
where 𝜖 is a small enough parameter. We expand 𝑋 and 𝑁 as We first consider the case of the first order of 𝜀. Since 𝐿 𝑇 is
the series form of 𝜖: the linear operator of the system close to the onset, (𝑥1 , 𝑦1 )𝑇
𝑥 𝑥 𝑥 𝑥 is the linear combination of the eigenvectors that corresponds
𝑋 = ( ) = 𝜖 ( 1 ) + 𝜖2 ( 2 ) + 𝜖3 ( 3 ) + ⋅ ⋅ ⋅ , to the eigenvalue zero. Since that
𝑦 𝑦1 𝑦2 𝑦3
𝑁=(
(𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) (𝑥12 𝜖2 + 2𝑥1 𝑥2 𝜖3 ) − 𝛾𝑥13 𝜖3 − 𝑥1 𝑦1 𝜖2 − (𝑥2 𝑦1 + 𝑥1 𝑦2 ) 𝜖3 + 𝑜 (𝜖4 )
). 3 𝐴𝑥𝑖
𝑥
𝑥1 𝑦1 𝜖2 + (𝑥2 𝑦1 + 𝑥1 𝑦2 ) 𝜖3 + 𝑜 (𝜖4 ) ( ) = ∑ ( ) exp (𝑖𝜅𝑖 𝑟)⃗+ c.c., (41)
𝑦 𝑦
(32) 𝑖=1 𝐴 𝑖
Linear operator 𝐿 can be expanded as

we have that
𝐿 = 𝐿 𝑇 + (𝛿𝑇 − 𝛿) 𝑀, (33)
where ∗ ∗
(𝑎11 + Δ) 𝑥1 + 𝑎12 𝑦1 = 0, (42a)
𝑎∗ + Δ ∗
𝑎12 𝑏 𝑏
𝐿 𝑇 = ( 11 ∗ ∗ ), 𝑀 = ( 11 12 ) . (34) ∗ ∗
𝑎21 𝑎22 + 𝜀Δ 𝑏21 𝑏22 𝑎21 𝑥1 + (𝑎22 + 𝜀Δ) 𝑦1 = 0. (42b)
Let ∗ ∗ ∗ ∗ 2 ∗
2
As 𝜀𝑎12 = ((𝑎22 − 𝜀𝑎11 )/2𝑎21 ) , we can obtain that 𝑥1 = (𝑎22 −
𝑇0 = 𝑡, 𝑇1 = 𝜖𝑡, 𝑇2 = 𝜖 𝑡, (35) ∗ ∗
𝜀𝑎11 )/(2𝑎21 ) by assuming 𝑦1 = 1.
∗ ∗ ∗
Let 𝑅 = (𝑎11 𝜀 − 𝑎22 )/2𝑎21 then
and 𝑇𝑖 is a dependent variable. For the derivation of time, we
have that
𝑥 𝑅
𝜕 𝜕 𝜕 𝜕 ( 1 ) = ( ) (𝑊1 exp (𝑖𝜅1 𝑟)⃗+ 𝑊2 exp (𝑖𝜅2 𝑟)⃗
= +𝜖 + 𝜖2 + 𝑜 (𝜖3 ) . (36) 𝑦1 1 (43)
𝜕𝑡 𝜕𝑇0 𝜕𝑇1 𝜕𝑇2
⃗ + c.c.,
+𝑊3 exp (𝑖𝜅3 𝑟))
The solutions of systems (4a) and (4b) have the following
form:
where |𝜅𝑗 | = 𝜅𝑇∗ and 𝑊𝑗 is the amplitude of the mode
3 𝐴𝑥𝑖 exp(𝑖𝜅𝑗 𝑟).
𝑥
𝑋 = ( ) = ∑( ) exp (𝑖𝜅𝑖 𝑟)⃗+ ⋅ ⋅ ⋅ . (37) Now, we consider the case of the second order of 𝜀. Note
𝑦 𝑦
𝑖=1 𝐴𝑖 that
This expression implies that the bases of the solutions have
𝑥 𝜕 𝑥1 𝑏 𝑥 +𝑏 𝑦
nothing to do with time and the amplitude 𝐴 is a variable that 𝐿 𝑇 ( 2) = ( ) − 𝛿𝑇 ( 11 1 12 1 )
changes slowly. As a result, one has the following equation: 𝑦2 𝑦
𝜕𝑇1 1 𝑏21 𝑥1 + 𝑏22 𝑦1
𝜕𝐴 𝜕𝐴 𝜕𝐴 (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) 𝑥12 − 𝑥1 𝑦1

=𝜖 + 𝜖2 + 𝑜 (𝜖3 ) . (38) −(
𝑥1 𝑦1
) (44)
𝜕𝑡 𝜕𝑇1 𝜕𝑇2
Substituting the above equations into (29) and expanding 𝐹
(29) according to different orders of 𝜖, we can obtain three = ( 𝑥) .
𝐹𝑦
equations as follows:
𝑥 According to the Fredholm solubility condition, the
𝜖 : 𝐿 𝑇 ( 1 ) = 0; vector function of the right hand of the above equation must
𝑦1
be orthogonal with the zero eigenvectors of operator L+𝑐 . And
𝑥 𝜕 𝑥1 𝑥 the zero eigenvectors of operator L+𝑐 are
𝜖2 : 𝐿 𝑇 ( 2 ) = ( ) − 𝛿1 𝑀 ( 1 )
𝑦2 𝜕𝑇1 𝑦1 𝑦1
1
(𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) 𝑥12 − 𝑥1 𝑦1
−( ); (39) ( 1 ) exp (𝑖𝜅𝑗 𝑟)⃗ + c.c (𝑗 = 1, 2, 3) . (45)
𝑥1 𝑦1 − 𝑅
𝜀
𝑥 𝜕 𝑥2 𝜕 𝑥1 𝑥
𝜖3 : 𝐿 𝑇 ( 3 ) = ( )+ ( ) − 𝛿1 𝑀 ( 2 )
𝑦3 𝜕𝑇1 𝑦2 𝜕𝑇2 𝑦1 𝑦2 It can be found from the orthogonality condition that
𝑥 1
− 𝛿2 𝑀 ( 1 ) − 𝐸, 𝐹𝑥𝑖
𝑦1
( 1 ) ( ) = 0, (46)
where − 𝑅 𝐹𝑦𝑖
𝜀
2𝑥 𝑥 (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) − 𝛾𝑥13 − (𝑥2 𝑦1 + 𝑥1 𝑦2 )
𝐸=( 1 2 ).
𝑥2 𝑦1 + 𝑥1 𝑦2 where 𝐹𝑥𝑖 and 𝐹𝑦𝑖 represent the coefficients corresponding to
(40) exp(𝑖𝜅𝑗 𝑟) in 𝐹𝑥 and 𝐹𝑦 .
⃗ one has
By investigating exp(𝑖𝜅1 𝑟), 𝑋
( 𝑗𝑗 )
𝑌𝑗𝑗
𝜕𝑊1 ∗
(𝑎22 − 4𝜀𝜅𝑇2 ) [−𝑅2 (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) + 𝑅] + 𝑅𝑎12
∗
𝐹𝑥1 𝑅 𝑏11 𝑅𝑊1 + 𝑏12 𝑊1
𝜕𝑇1 ∗ − 4𝜅2 ) (𝑎∗ − 4𝜀𝜅2 ) − 𝑎∗ 𝑎∗
(𝑎11
( )=( ) − 𝛿1 ( ) 𝑇 22 𝑇 12 21
𝜕𝑊1 =( )
𝐹𝑦1 𝑏21 𝑅𝑊1 + 𝑏22 𝑊1 ∗
𝑎21 [𝑅2 (𝛽𝛾 − 3𝛾𝑢∗ +𝛾) − 2𝑅] − 𝑅 (𝑎11
∗
− 4𝜀𝜅𝑇2 )
𝜕𝑇1
∗ − 4𝜅2 ) (𝑎∗ − 4𝜀𝜅2 ) − 𝑎∗ 𝑎∗
(𝑎11 𝑇 22 𝑇 12 21
2 ∗
2𝑅 (𝛽𝛾 − 3𝛾𝑢 + 𝛾) 𝑊2 𝑊3 + 2𝑅𝑊2 𝑊3
−( ). × 𝑊𝑗2 ,
2𝑅𝑊2 𝑊3
(47) 𝑋
( 𝑗𝑘 )
𝑌𝑗𝑘
∗
(𝑎22 − 3𝜀𝜅𝑇2 ) [−2𝑅2 (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) + 2𝑅] + 2𝑅𝑎12
∗
It can be obtained from the orthogonality condition that
∗ − 3𝜅2 ) (𝑎∗ − 3𝜀𝜅2 ) − 𝑎∗ 𝑎∗
(𝑎11 𝑇 22 𝑇 12 21
=( ∗ )
𝑎21 [2𝑅2 (𝛽𝛾 − 3𝛾𝑢∗ +𝛾) − 2𝑅] − 2𝑅 (𝑎11
∗
− 3𝜅𝑇2 )
𝜀 − 1 𝜕𝑊1 𝑅 ∗ − 3𝜅2 ) (𝑎∗ − 3𝜀𝜅2 ) − 𝑎∗ 𝑎∗
𝑅 = 𝛿 (𝑅𝑏11 + 𝑏22 − (𝑅𝑏21 + 𝑏22 ) 𝑊1 ) (𝑎11 𝑇 22 𝑇 12 21
𝜀 𝜕𝑇1 𝜀
× 𝑊𝑗 𝑊𝑘 ,
1 1
+ 2𝑅2 (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾 + − ) 𝑊2 𝑊3 . (51)
𝑅 𝜀
(48)
where 𝜅𝑇2 = √(𝑎11∗ 𝑎∗ − 𝑎∗ 𝑎∗ )/𝜀.
22 12 21
By using the same methods, one has For the third order of 𝜀, we have that
𝑥 𝜕 𝑥2 𝜕 𝑥1
3
𝑋 𝐿 𝑇 ( 3) = ( )+ ( )
𝑥 𝑋 𝑦3 𝜕𝑇1 𝑦2 𝜕𝑇2 𝑦1
( 2 ) = ( 0 ) + ∑ ( 𝑗 ) exp (𝑖𝜅𝑗 𝑟)⃗
𝑦2 𝑌0 𝑌𝑗 (52)
𝑗=1
𝑥 𝑥
(49) − 𝛿1 𝑀 ( 2 ) − 𝛿2 𝑀 ( 1 ) − 𝑆,
3 𝑦2 𝑦1
𝑋
+ ∑ ( 𝑗𝑗 ) exp (2𝑖𝜅𝑗 𝑟)⃗ + 𝑄 + c.c.,
𝑌𝑗𝑗
𝑗=1
where
where 2𝑥 𝑥 (𝛽𝛾 − 3𝛾𝑢∗ ) − 𝛾𝑥13 − (𝑥2 𝑦1 + 𝑥1 𝑦2 )

𝑆=( 1 2 ). (53)
𝑥2 𝑦1 + 𝑥1 𝑦2
𝑋 𝑋
𝑄 = ( 12 ) exp (𝑖 (𝜅1 − 𝜅2 ) 𝑟)⃗+ ( 23 ) exp (𝑖 (𝜅2 − 𝜅3 ) 𝑟)⃗ Using the Fredholm solubility condition, we can obtain
𝑌12 𝑌23
𝑋 𝜀 − 1 𝜕𝑊1 𝜀 − 1 𝜕𝑌1
+ ( 31 ) exp (𝑖 (𝜅3 − 𝜅1 ) 𝑟)⃗. 𝑅 + 𝑅
𝑌31 𝜀 𝜕𝑇2 𝜀 𝜕𝑇1
(50)
1
= 𝛿2 [𝑅𝑏11 + 𝑏12 − 𝑅 (𝑅𝑏21 + 𝑏22 )] 𝑊1 (54)
𝜀
By solving the sets of the linear equations about exp(0),
⃗ exp(2𝑖𝜅𝑗 𝑟),
exp(𝑖𝜅𝑗 𝑟), ⃗ we obtain that
⃗ and exp(𝑖(𝜅𝑗 − 𝜅𝑘 )𝑟), 1
+ 𝛿1 [𝑅𝑏11 + 𝑏12 − (𝑅𝑏21 + 𝑏22 )] 𝑌1 + 𝑍,
𝜀
𝑋 where
( 0)
𝑌0
1
∗
𝑎22 [−2𝑅2 (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) + 2𝑅] + 2𝑅𝑎12
∗ 𝑍 = [2𝑅2 (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) − 2𝑅 − 2 ]
𝜀
∗ 𝑎∗ − 𝑎∗ 𝑎∗
𝑎11 22 12 21
=( ∗
) × [𝑊1 𝑌0 + 𝑊 − 2𝑌12 + 𝑊3 𝑌13 + 𝑊1 𝑌11
𝑎21 [2𝑅2 (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) − 2𝑅] + 2𝑅𝑎11
∗
∗ 𝑎∗ − 𝑎∗ 𝑎∗
𝑎11 22 12 21
+ 𝑊2 𝑌3 + 𝑊3 𝑌2 ]
󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2
× (󵄨󵄨󵄨𝑊1 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑊2 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑊3 󵄨󵄨󵄨 ) , − (𝐺1 󵄨󵄨󵄨𝑊1 󵄨󵄨󵄨 + 𝐺2 󵄨󵄨󵄨𝑊2 󵄨󵄨󵄨 + 𝐺3 󵄨󵄨󵄨𝑊3 󵄨󵄨󵄨 ) 𝑊1 ,
Table 1: Coefficients for different parameter sets.
𝛽 𝛿 ℎ 𝑔1 𝑔2 𝜇1 𝜇2 𝜇3 𝜇4
0.02 0.92 −19.08604 7599.215 6906.578 0.0042531 0 5.770186 0.046076
0.02 0.96 2.1329690 −740.11 −1429.72 −0.000315 0 −0.00708 −0.00611
0.02 1 8.4304106 −207.521 −474.186 −0.015371 0 −0.20741 −0.01250
0.02 1.12 11.304093 −99.3194 −193.856 −0.0655924 0 −1.42005 −0.04391
(a) (b) (c)
(d) (e) (f)
Figure 3: Spatial pattern of prey population at different time. Parameters set: 𝛾 = 1.5, 𝜀 = 0.15, and 𝛿 = 0.92. (a) 𝑡 = 0; (b) 𝑡 = 100; (c)
𝑡 = 200; (d) 𝑡 = 500; (e) 𝑡 = 1000; and (f) 𝑡 = 2000.
1 The other two equations can be obtained through the

𝐺1 = ( 𝑅 − 1) [𝑅 (𝑦11 + 𝑦0 ) + 𝑥11 + 𝑥0 ]
𝜀 transformation of the subscript of 𝐴. By calculations, we
obtain the expressions of the coefficients of 𝜏0 , ℎ, 𝑔1 , and
− 2𝑅 (𝑥11 + 𝑥0 ) (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) + 3𝛾𝑅3 ,
(55) 𝑔2 as follows:
1
𝐺2 = ( 𝑅 − 1) [𝑅 (𝑦12 + 𝑦0 ) + 𝑥12 + 𝑥0 ]
𝜀
𝜀−1
− 2𝑅 (𝑥12 + 𝑥0 ) (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) + 6𝛾𝑅3 . 𝜏0 = 𝑅 ,
𝛿𝑇 [𝑅𝑏11 + 𝑏12 − (𝑅/𝜀) (𝑅𝑏21 + 𝑏22 )]
By using the same methods, we can obtain the other two
equations. The amplitude 𝐴 𝑖 can be expanded as [2𝑅2 (𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) − 2𝑅 − 2 (𝑅2 /𝜀)]
ℎ= ,
2 3
𝛿𝑇 [𝑅𝑏11 + 𝑏12 − (𝑅/𝜀) (𝑅𝑏21 + 𝑏22 )]
𝐴 𝑖 = 𝜖𝑊𝑖 + 𝜖 𝑉𝑖 + 𝑜 (𝜖 ) . (56) (58)
𝐺1
As a result, we have 𝑔1 = ,
𝛿𝑇 [𝑅𝑏11 + 𝑏12 − (𝑅/𝜀) (𝑅𝑏21 + 𝑏22 )]
𝜕𝐴 1 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2
𝜏0 = 𝜇𝐴 1 + ℎ𝐴2 𝐴3 − (𝑔1 󵄨󵄨󵄨𝐴 1 󵄨󵄨󵄨 + 𝑔2 󵄨󵄨󵄨𝐴 2 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝐴 3 󵄨󵄨󵄨 ) 𝐴 1 . 𝐺2
𝜕𝑡 𝑔2 = ,
(57) 𝛿𝑇 [𝑅𝑏11 + 𝑏12 − (𝑅/𝜀) (𝑅𝑏21 + 𝑏22 )]
(a) (b) (c)
(d) (e) (f)
Figure 4: Spatial pattern of prey population at different time. Parameters set: 𝛾 = 1.5, 𝜀 = 0.15, and 𝛿 = 0.96. (a) 𝑡 = 0; (b) 𝑡 = 50; (c) 𝑡 = 100;
(d) 𝑡 = 200; (e) 𝑡 = 500; and (f) 𝑡 = 1000.
where 𝐺1 = ((𝑅/𝜀) − 1)[𝑅(𝑦0 + 𝑦11 ) + 𝑥0 + 𝑥11 ] − 2𝑅(𝑥0 + (2) Stripe patterns (𝑆), given by
𝑥11 )(𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) + 3𝛾𝑅3 and 𝐺2 = ((𝑅/𝜀) − 1)[𝑅(𝑦0 + 𝑦12 ) +
𝑥0 + 𝑥12 ] − 2𝑅(𝑥0 + 𝑥12 )(𝛽𝛾 − 3𝛾𝑢∗ + 𝛾) + 6𝛾𝑅3 . 𝜇
𝜌1 = √ ≠0, 𝜌2 = 𝜌3 = 0, (61)
By using substitutions, we have 𝑔1
𝜕𝜑 𝜌2 𝜌2 + 𝜌12 𝜌32 + 𝜌22 𝜌32
𝜏0 = −ℎ 1 2 sin 𝜑, are stable for 𝜇 > 𝜇3 = ℎ2 𝑔1 /(𝑔2 − 𝑔1 )2 , and unstable
𝑑𝑡 𝜌1 𝜌2 𝜌3
for 𝜇 < 𝜇3 .
𝜕𝜌1
𝜏0 = 𝜇𝜌1 + ℎ𝜌2 𝜌3 cos 𝜑 − 𝑔1 𝜌13 − 𝑔2 (𝜌22 𝜌32 ) 𝜌1 , (3) Hexagon patterns (𝐻0 , 𝐻𝜋 ) are given by
𝑑𝑡 (59)
𝜕𝜌2 |ℎ| ± √ℎ2 + 4 (𝑔1 + 2𝑔2 𝜇)
𝜏0 = 𝜇𝜌2 + ℎ𝜌1 𝜌3 cos 𝜑 − 𝑔1 𝜌23 − 𝑔2 (𝜌12 𝜌32 ) 𝜌2 , 𝜌1 = 𝜌2 = 𝜌3 = , (62)
𝑑𝑡
2 (𝑔1 + 2𝑔2 )
𝜕𝜌3
𝜏0 = 𝜇𝜌3 + ℎ𝜌1 𝜌2 cos 𝜑 − 𝑔1 𝜌33 − 𝑔2 (𝜌12 𝜌22 ) 𝜌3 ,
𝑑𝑡 with 𝜑 = 0 or 𝜋, and exist when
where 𝜑 = 𝜑1 + 𝜑2 + 𝜑3 . In order to see the relationships
between different parameters, we give the values of coeffi- −ℎ2
𝜇 > 𝜇1 = . (63)
cients for different parameter sets in Table 1. 4 (𝑔1 + 2𝑔2 )
The dynamical systems (4a) and (4b) possess five kinds of
solutions [23] as follows.
The solution 𝜌+ = |ℎ| + √ℎ2 + 4(𝑔1 + 2𝑔2 𝜇)/2(𝑔1 +
(1) The stationary state (𝑂), given by 2𝑔2 ) is stable only for
𝜌1 = 𝜌2 = 𝜌3 = 0, (60) 2𝑔1 + 𝑔2
𝜇 < 𝜇4 = 2
ℎ2 , (64)
is stable for 𝜇 < 𝜇2 = 0 and unstable for 𝜇 > 𝜇2 . (𝑔2 − 𝑔1 )
(a) (b) (c)
(d) (e) (f)
Figure 5: Spatial pattern of prey population at different time. Parameters set: 𝛾 = 1.5, 𝜀 = 0.15, and 𝛿 = 1. (a) 𝑡 = 0; (b) 𝑡 = 150; (c) 𝑡 = 300;
(d) 𝑡 = 500; (e) 𝑡 = 600; and (f) 𝑡 = 1000.
and 𝜌− = (|ℎ| − √ℎ2 + 4(𝑔1 + 2𝑔2 𝜇))/2(𝑔1 + 2𝑔2 ) is By setting 𝛾 = 1.5, 𝜀 = 0.15, and 𝛿 = 0.96, one can obtain
always unstable. that 𝜇 > 𝜇4 . In Figure 4, we show the spatial pattern of prey
population when 𝑡 equals 0, 50, 100, 200, 500, and 1000. At
(4) The mixed states are given by the initial time, the prey population shows patched invasion.
As time increases, stripe pattern appears and the structure
|ℎ| 𝜇 − 𝑔1 𝜌12
𝜌1 = , 𝜌2 = 𝜌3 = √ , (65) does not change a lot. While keeping other parameters fixed
𝑔2 − 𝑔1 𝑔1 + 𝑔2 and increasing 𝛿, we find that stripe pattern will occupy the
whole space. However, some stripe patterns connect with
with 𝑔2 > 𝑔1 . They exist when 𝜇 > 𝜇3 and are always each other and cause the emergence of spotted patterns which
unstable. are shown in Figure 5.
Figure 6 shows the evolution of the spatial pattern of prey
4. Spatial Pattern of Systems (4a) and (4b) population at 𝑡 = 0, 100, 300, 500, 1000, and 2000 iterations,
with small random perturbation of the stationary solution
In this section, we perform extensive numerical simulations of the spatially homogeneous systems (4a) and (4b). The
of the spatially extended systems (4a) and (4b) in two- corresponding parameters values are 𝛾 = 1.5, 𝜀 = 0.15,
dimensional spaces. All our numerical simulations employ and 𝛿 = 1.04. By the amplitude equations, we can conclude
the zero-flux boundary conditions with a system size of 200 × that there are spotted patterns of prey population for this
200. The space step is Δ𝐻 = 1, and the time step is Δ𝑡 = parameter set. In this case, one can see that for the systems
0.00001. (4a) and (4b), the random initial distribution leads to the
In Figure 3, we show the spatial pattern of prey population formation of an irregular transient pattern in the domain.
at different time. In the parameter set, 𝛾 = 1.5, 𝜀 = 0.15, After these forms, it grows slightly and spotted patterns
and 𝛿 = 0.92, we find that 𝜇 ∈ (𝜇3 , 𝜇4 ), which means that emerge. When the time is large enough, the spotted patterns
there is coexistence of spotted and stripe patterns. As shown prevail over the two-dimensional space. As time further
in this figure, our theoretical results are consistent with the increases, the pattern structures of the prey population do not
numerical results. undergo any further changes.
(a) (b) (c)
(d) (e) (f)
Figure 6: Spatial pattern of prey population at different time. Parameters set: 𝛾 = 1.5, 𝜀 = 0.15, and 𝛿 = 1.04. (a) 𝑡 = 0; (b) 𝑡 = 100; (c)
𝑡 = 300; (d) 𝑡 = 500; (e) 𝑡 = 1000; and (f) 𝑡 = 2000.
5. Conclusion and Discussion bifurcation, and spiral waves occupy the whole domain
instead of stationary patterns, which is shown in Figure 7. The
Allee effect has been paid much attention due to its strong stability of spiral wave can be done by using the spectrum
potential impact on population dynamics [24]. In this paper, theory analysis [25, 26]. In the further study, we will use the
we investigated the pattern dynamics of a spatial predator- spectrum theory to show the stability of spiral wave.
prey systems with Allee effect. Based on the bifurcation In [15], they found that a spatial predator-prey model with
analysis, exact Turing pattern region is obtained. By using Allee effect and linear death rate could increase the system’s
amplitude equations, the Turing pattern selection of the complexity and enhance chaos in population dynamics.
predator-prey system is well presented. It is found that the However, in this paper, we showed that a spatial population
predator-prey systems with Allee effect have rich spatial model with Allee effect and nonlinear death rate can induce
dynamics by performing a series of numerical simulations. stationary patterns, which is different from the previous
It should be noted that our results were obtained under results.
the assumption that predation is modeled by the bilinear From a biological point of view, our results show that
function of the prey and predator densities. However, this predator mortality plays an important role in the spatial
function has limitations to describe many realistic phenom- invasions of populations. More specifically, low predator
ena in the biology. By numerical simulations, we find that mortality will induce stationary patterns (cf. Figures 3–
the system exhibits similar behaviour when the functional 6), and high predator mortality corresponds to travelling
patterns (cf. Figure 7). When the populations exhibit wave
response is of other types, such as Holling-II and Holling-III
distribution in space, the dynamics of populations may
forms.
be accompanied with chaotic properties [27, 28]. If the
To compare the spatial dynamics for different parame- chaotic behavior occurs, it may lead to the extinction of the
ters, we give the spatial patterns of population 𝑢 when the population, or the population may be out of control [29, 30].
parameter values are out of the domain of Turing space. In that case, we need to find out the best way to control the
For this parameter set, systems (4a) and (4b) have Hopf chaos or change the chaotic behavior.
(a) (b) (c)
(d) (e) (f)
Figure 7: Spatial pattern of prey population at different time. Parameters set: 𝛾 = 1.5, 𝜀 = 0.15, and 𝛿 = 1.2. (a) 𝑡 = 0; (b) 𝑡 = 100; (c) 𝑡 = 200;
(d) 𝑡 = 300; (e) 𝑡 = 400; and (f) 𝑡 = 500.
Acknowledgments [7] P. A. Stephens and W. J. Sutherland, “Consequences of the

Allee effect for behaviour, ecology and conservation,” Trends in
This research was partially supported by the National Natural Ecology and Evolution, vol. 14, no. 10, pp. 401–405, 1999.
Science Foundation of China under Grant nos. 11301490, [8] J. Cushing and J. T. Hudson, “Evolutionary dynamics and strong
11301491, 11331009, 11147015, 11171314, 11305043, and 11105024; allee effects,” Journal of Biological Dynamics, vol. 6, pp. 941–958,
Natural Science Foundation of Shan’xi Province Grant nos. 2012.
2012021002-1 and 2012011002-2, the Opening Foundation [9] S. N. Elaydi and R. J. Sacker, “Population models with Allee
of Institute of Information Economy, Hangzhou Normal effect: a new model,” Journal of Biological Dynamics, vol. 4, no.
University, Grant no. PD12001003002003; and the specialized 4, pp. 397–408, 2010.
research fund for the doctoral program of higher education [10] J. Shi and R. Shivaji, “Persistence in reaction diffusion models
(preferential development) Grant no. 20121420130001. with weak Allee effect,” Journal of Mathematical Biology, vol. 52,
no. 6, pp. 807–829, 2006.
[11] J. Wang, J. Shi, and J. Wei, “Predator-prey system with strong
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http://dx.doi.org/10.1155/2013/203875
Research Article
The Analytical Solution of Some Fractional Ordinary
Differential Equations by the Sumudu Transform Method
Hasan Bulut,1 Haci Mehmet Baskonus,2 and Fethi Bin Muhammad Belgacem3
1
Department of Mathematics, Firat University, Elazig, Turkey
2
Department of Computer Engineering, Tunceli University, Tunceli, Turkey
3
Department of Mathematics, Faculty of Basic Education, PAAET, Shamiya, Kuwait
Correspondence should be addressed to Hasan Bulut; hbulut@firat.edu.tr
Received 16 March 2013; Accepted 31 July 2013
Copyright © 2013 Hasan Bulut et al. This is an open access article distributed under the Creative Commons Attribution License,
We introduce the rudiments of fractional calculus and the consequent applications of the Sumudu transform on fractional
derivatives. Once this connection is firmly established in the general setting, we turn to the application of the Sumudu transform
method (STM) to some interesting nonhomogeneous fractional ordinary differential equations (FODEs). Finally, we use the
solutions to form two-dimensional (2D) graphs, by using the symbolic algebra package Mathematica Program 7.
1. Introduction also used to solve many ordinary differential equations with

integer order [23–29]. The application of STM turns out to
The Sumudu transform was first defined in its current be pragmatic in getting analytical solution of the fractional
shape by Watugala as early as 1993, which he used to solve ordinary differential equations fast. Notably, implementa-
engineering control problems. Although he might have had tions of difference methods such as in the differential trans-
ideas for it sooner than that (1989) as some conference form method (DTM), the Adomian decomposition method
proceedings showed, he used it to control engineering prob- (ADM) [30–33], the variational iteration method (VIM)
lems [1, 2]. Later, Watugala extended in 2002 the Sumudu [34–40] empowered us to achieve approximate solutions of
transform to two variables [3]. The first applications to various ordinary differential equations. STM [41–44] which
differential equations and inversion formulae were done by is newly submitted to the literature is a suitable technique for
Weerakoon in two papers in 1994 and 1998 [4, 5]. The solving various kinds of ordinary differential equations with
Sumudu transform was also first defended by Weerakoon fractional order (FODEs). In this sense, it is estimated that
against Deakin’s definition who claimed that there is no this novel approach that is used to solve homogeneous and
difference between the Sumudu and the Laplace and who nonhomogeneous problems will be particularly valuable as a
reminded Weerakoon that the Sumudu transform is really the tool for scientists and applied mathematicians.
Carson or the S-multiplied transform disguised [6, 7]. The
applications followed in three consecutive papers by Asiru
dealing with the convolution-type integral equations and the 2. Fundamental Properties of Fractional
discrete dynamic systems [8, 9]. At this point, Belgacem et Calculus and STM
al. using previous references and connections to the Laplace
transform extended the theory and the applications of the 2.1. Fundamental Facts of the Fractional Calculus. Firstly, we
Sumudu transform in [10–17] to various applications. In mention some of the fundamental properties of the fractional
the meantime, subsequent to exchanges between Belgacem calculus. Fractional derivatives (and integrals as well) defi-
and other scholars, the following papers sprang up in the nitions may differ, but the most widely used definitions are
last decade [18–22]. Moreover, the Sumudu transform was those of Abel-Riemann (A-R). Following the nomenclature
in [45], a derivative of fractional order in the A-R sense is Theorem 1. If 𝐹(𝑢)is the Sumudu transform of 𝑓(𝑡), one knows
defined by that the Sumudu transform of the derivatives with integer order
is given as follows [46–49]:
𝐷𝛼 [𝑓 (𝑡)]
𝑑𝑓 (𝑡) 1
1 𝑑 𝑡 𝑓 (𝜏) 𝑆[ ] = [𝐹 (𝑢) − 𝑓 (0)] . (9)
{
{ ∫ 𝑑𝜏, 𝑚 − 1 < 𝛼 ≤ 𝑚, 𝑑𝑡 𝑢
{ Γ [𝑚 − 𝛼] 𝑑𝑡𝑚 0 (𝑡 − 𝜏)𝛼−𝑚+1
={ 𝑚 Proof. Let us take the Sumudu transform [46–49] of 𝑓󸀠 (𝑡) =
{𝑑
{
{ 𝑑𝑡𝑚 𝑓 (𝑡) , 𝛼 = 𝑚, 𝑑𝑓(𝑡)/𝑑𝑡 as follows:
𝑑𝑓 (𝑡) ∞ 𝑑𝑓 𝑝 𝑑𝑓
(1) (𝑢𝑡) −𝑡 (𝑢𝑡) −𝑡
𝑆[ ]=∫ 𝑒 𝑑𝑡 = lim ∫ 𝑒 𝑑𝑡
where 𝑚 ∈ Z+ and 𝛼 ∈ 𝑅+ . 𝐷𝛼 is a derivative operator here, 𝑑𝑡 0 𝑑𝑡 𝑝→∞ 0 𝑑𝑡
and 󵄨󵄨𝑝 1 𝑝
1
−𝛼 1 𝑡 = lim [ 𝑒−(𝑡/𝑢) 𝑓 (𝑡)󵄨󵄨󵄨󵄨 + 2 ∫ 𝑒−(𝑡/𝑢) 𝑓 (𝑡) 𝑑𝑡]
𝐷 [𝑓 (𝑡)] = ∫ (𝑡 − 𝜏)𝛼−1 𝑓 (𝜏) 𝑑𝜏, 0 < 𝛼 ≤ 1. (2) 𝑝→∞ 𝑢 󵄨0 𝑢 0
Γ [𝛼] 0
1 󵄨󵄨𝑝
On the other hand, according to A-R, an integral of = lim [ 𝑒−(𝑡/𝑢) 𝑓 (𝑡)󵄨󵄨󵄨󵄨
fractional order is defined by implementing the integration 𝑝→∞ 𝑢 󵄨0
operator 𝐽𝛼 in the following manner:
1 1 𝑝
1 𝑡 + ( ∫ 𝑒−(𝑡/𝑢) 𝑓 (𝑡) 𝑑𝑡)]
𝐽𝛼 [𝑓 (𝑡)] = ∫ (𝑡 − 𝜏)𝛼−1 𝑓 (𝜏) 𝑑𝜏, 𝑡 > 0, 𝛼 > 0. (3) 𝑢 𝑢 0
Γ [𝛼] 0
When it comes to some of the fundamental properties 1 1 1 𝑝
= lim [− 𝑓 (0) + ( ∫ 𝑒−(𝑡/𝑢) 𝑓 (𝑡) 𝑑𝑡)]
of fractional integration and fractional differentiation, these 𝑝→∞ 𝑢 𝑢 𝑢 0
have been introduced to the literature by Podlubny [46].
Among these, we mention 1 1
= − 𝑓 (0) + 𝐹 (𝑢) .
Γ [1 + 𝑛] 𝑛+𝛼 𝑢 𝑢
𝐽𝛼 [𝑡𝑛 ] = 𝑡 , (10)
Γ [1 + 𝑛 + 𝛼]
(4)
𝛼 𝑛 Γ [1 + 𝑛] 𝑛−𝛼
𝐷 [𝑡 ] = 𝑡 . Equation (10) gives us the proof of Theorem 1. When we
Γ [1 + 𝑛 − 𝛼] continue in the same manner, we get the Sumudu transform
Another main definition of the fractional derivative is that of the second-order derivative as follows [46–49]:
of Caputo [46, 47] who defined it by
𝑑2 𝑓 (𝑡) 1 𝑑𝑓 (𝑡) 󵄨󵄨󵄨󵄨
𝑆[ ] = [𝐹 (𝑢) − 𝑓 (0) − 𝑢 󵄨 ]. (11)
𝑑𝑡 󵄨󵄨󵄨𝑡=0
𝐶 𝛼
𝐷 [𝑓 (𝑡)] 𝑑𝑡2 𝑢2
{ 1 𝑡 𝑓(𝑚) (𝜏) If we go on the same way, we get the Sumudu transform

{
{ ∫ 𝑑𝜏, 𝑚 − 1 < 𝛼 < 𝑚,
{Γ − 𝛼] 0 (𝑡 − 𝜏)𝛼−𝑚+1 of the 𝑛-order derivative as follows:
= { [𝑚
{
{ 𝑑𝑚
{ 𝑑𝑛 𝑓 (𝑡) 𝑛−1
𝑑𝑛 𝑓 (𝑡) 󵄨󵄨󵄨󵄨
𝑓 (𝑡) , 𝛼 = 𝑚. 𝑆[ ] = 𝑢−𝑛 [𝐹 (𝑢) − ∑ 𝑢𝑘 󵄨 ]. (12)
{ 𝑑𝑡𝑚 𝑑𝑡𝑛
𝑘=0
𝑑𝑡𝑛 󵄨󵄨󵄨𝑡=0
(5)
Theorem 2. If 𝐹(𝑢) is the Sumudu transform of 𝑓(𝑡), one can
A fundamental feature of the Caputo fractional derivative
take into consideration the Sumudu transform of the Riemann-
is that [17]
Liouville fractional derivative as follows [17]:
∞
𝑡𝑘
𝐽𝛼 [𝐶𝐷𝛼 𝑓 (𝑡)] = 𝑓 (𝑡) − ∑ 𝑓(𝑘) (0+ ) . (6) 𝑛
𝑘=0
𝑘! 𝑆 [𝐷𝛼 𝑓 (𝑡)] = 𝑢−𝛼 [𝐹 (𝑢) − ∑ 𝑢𝛼−𝑘 [𝐷𝛼−𝑘 (𝑓 (𝑡))]𝑡=0 ] ,
𝑘=1 (13)
2.2. Fundamental Facts of the Sumudu Transform Method.
The Sumudu transform is defined in [1, 2] as follows. Over − 1 < 𝑛 − 1 ≤ 𝛼 < 𝑛.
the set of functions
Proof. Let us take the Laplace transform of 𝑓󸀠 (𝑡) = 𝑑𝑓(𝑡)/𝑑𝑡
𝐴 = {𝑓 (𝑡) | ∃𝑀, 𝜏1 , 𝜏2 > 0, as follows:
(7) 𝑛−1
󵄨󵄨 󵄨
󵄨󵄨𝑓 (𝑡)󵄨󵄨󵄨 < 𝑀𝑒 𝑖 , if 𝑡 ∈ (−1) × [0, ∞)} ,
|𝑡|/𝜏 𝑗
𝐿 [𝐷𝛼 𝑓 (𝑡)] = 𝑠𝛼 𝐹 (𝑠) − ∑ 𝑠𝑘 [𝐷𝛼−𝑘−1 (𝑓 (𝑡))]𝑡=0
𝑘=0
the Sumudu transform of 𝑓(𝑡) is defined as (14)
𝑛
∞ 𝛼 𝑘−1 𝛼−𝑘
𝐹 (𝑢) = 𝑆 [𝑓 (𝑡)] = ∫ 𝑓 (𝑢𝑡) 𝑒−𝑡 𝑑𝑡, 𝑢 ∈ (−𝜏1 , 𝜏2 ) . (8) = 𝑠 𝐹 (𝑠) − ∑ 𝑠 [𝐷 (𝑓 (𝑡))]𝑡=0 .
0 𝑘=0
Therefore, when we substitute 1/𝑢 for 𝑠, we get the where 𝑔(𝑢) is defined by ∑𝑛𝑘=1 𝑢𝛼−𝑘 [𝐷𝛼−𝑘 (𝑈(𝑡))]𝑡=0 −
Sumudu transform of fractional order of 𝑓(𝑡) as follows: 𝑢𝛼−1 (𝜕𝑈(𝑡)/𝜕𝑡)|𝑡=0 . When we take the inverse Sumudu
𝑛 transform of (18) by using the inverse transform table in
𝑆 [𝐷𝛼 𝑓 (𝑡)] = 𝑢−𝛼 [𝐹 (𝑢) − ∑ 𝑢𝛼−𝑘 [𝐷𝛼−𝑘 (𝑓 (𝑡))]𝑡=0 ] . (15) [11, 17], we get the solution of (16) by using STM as follows:
𝑘=1 1
𝑈 (𝑡) = 𝑆−1 [
1 − 𝑢 − 𝑢𝛼−1 − 𝑢𝛼
𝛼−2
Now, we will introduce the improvement form of STM × [𝑔 (𝑢) − 𝑢𝛼−1 𝑈 (0) − 𝑢𝛼−2 𝑈 (0) + 𝑐𝑢𝛼 ] ] .
for solving FODEs. We take into consideration a general
linear ordinary differential equation with fractional order as (19)
follows:
𝜕𝛼 𝑈 (𝑡) 𝜕2 𝑈 (𝑡) 𝜕𝑈 (𝑡) 3. Applications of STM to Nonhomogeneous
= + + 𝑈 (𝑡) + 𝑐, (16)
𝜕𝑡𝛼 𝜕𝑡2 𝜕𝑡 Fractional Ordinary Differential Equations
being subject to the initial condition In this section, we have applied STM to the nonhomogeneous
𝑈 (0) = 𝑓 (0) . (17) fractional ordinary differential equations as follows.
Then, we will obtain the analytical solutions of some of Example 3. Firstly, we consider the nonhomogeneous frac-
the fractional ordinary differential equations by using STM. tional ordinary differential equation as follows [50]:
When we take the Sumudu transform of (16) under the terms
2 1
of (12) and (15), we obtain the Sumudu transform of (16) as 𝐷𝛼 [𝑈 (𝑡)] = −𝑈 (𝑡) + 𝑡2−𝛼 − 𝑡1−𝛼
follows: Γ [3 − 𝛼] Γ [2 − 𝛼] (20)
𝜕𝛼 𝑈 (𝑡) 𝜕2 𝑈 (𝑡) 𝜕𝑈 (𝑡) + 𝑡2 − 𝑡, 𝑡 > 0, 0 < 𝛼 ≤ 1,
𝑆[ ] = 𝑆 [ ] + 𝑆[ ] + 𝑆 [𝑈 (𝑡)] + 𝑆 [𝑐] ,
𝜕𝑡𝛼 𝜕𝑡2 𝜕𝑡 With the initial condition being
𝑛
𝑢−𝛼 [𝐹 (𝑢) − ∑ 𝑢𝛼−𝑘 [𝐷𝛼−𝑘 (𝑈 (𝑡))]𝑡=0 ] 𝑈 (0) = 0. (21)
𝑘=1 In order to solve (20) by using STM, when we take the
1 𝜕𝑓 (𝑡) 󵄨󵄨󵄨󵄨 Sumudu transform of both sides of (20), we get the Sumudu
= 2 [𝐹 (𝑢) − 𝑓 (0) − 𝑢 󵄨 ]
𝑢 𝜕𝑡 󵄨󵄨󵄨𝑡=0 transform of (20) as follows:
1 𝑆 [𝐷𝛼 𝑈 (𝑡)] + 𝑆 [𝑈 (𝑡)]
+ [𝐹 (𝑢) − 𝑓 (0)] + 𝐹 (𝑢) + 𝑐,
𝑢 2 1
𝑛 = 𝑆[ 𝑡2−𝛼 − 𝑡1−𝛼 + 𝑡2 − 𝑡] ,
𝐹 (𝑢) − ∑ 𝑢 𝛼−𝑘
[𝐷 𝛼−𝑘
(𝑈 (𝑡))]𝑡=0 Γ [3 − 𝛼] Γ [2 − 𝛼]
𝑘=1 2
𝑆 [𝐷𝛼 𝑈 (𝑡)] + 𝐹 (𝑢) = 𝑆 [ 𝑡2−𝛼 ]
𝜕𝑈 (𝑡) 󵄨󵄨󵄨󵄨 Γ [3 − 𝛼]
= 𝑢𝛼−2 [𝐹 (𝑢) − 𝑓 (0) − 𝑢 󵄨 ]
𝜕𝑡 󵄨󵄨󵄨𝑡=0 1
− 𝑆[ 𝑡1−𝛼 ] + 𝑆 [𝑡2 ] − 𝑆 [𝑡] ,
+ 𝑢𝛼−1 [𝐹 (𝑢) − 𝑓 (0)] Γ [2 − 𝛼]
+ 𝑢𝛼 𝐹 (𝑢) + 𝑐𝑢𝛼 , 󵄨
𝐹 (𝑢) 𝐷𝛼−1 [𝑈 (𝑡)] 󵄨󵄨󵄨 2
− 󵄨󵄨 + 𝐹 (𝑢) = 𝑆 [𝑡2−𝛼 ]
𝐹 (𝑢) = 𝑢𝛼−2 𝐹 (𝑢) − 𝑢𝛼−2 𝑈 (0) 𝑢𝛼 𝑢 󵄨󵄨󵄨𝑡=0 Γ [3 − 𝛼]
𝑛
𝜕𝑈 (𝑡) 󵄨󵄨󵄨󵄨 −
1
𝑆 [𝑡1−𝛼 ]
+ ∑ 𝑢𝛼−𝑘 [𝐷𝛼−𝑘 (𝑈 (𝑡))]𝑡=0 − 𝑢𝛼−1 󵄨 Γ [2 − 𝛼]
𝑘=1
𝜕𝑡 󵄨󵄨󵄨𝑡=0
𝛼−1 + 𝑆 [𝑡2 ] − 𝑆 [𝑡] ,
+𝑢 𝐹 (𝑢) − 𝑢𝛼−1 𝑓 (0)
+ 𝑢𝛼 𝐹 (𝑢) + 𝑐𝑢𝛼 , 𝐹 (𝑢) 2
+ 𝐹 (𝑢) = 𝑢2−𝛼 Γ [3 − 𝛼]
𝑢𝛼 Γ [3 − 𝛼]
𝐹 (𝑢) − 𝑢𝛼−2 𝐹 (𝑢) − 𝑢𝛼−1 𝐹 (𝑢) − 𝑢𝛼 𝐹 (𝑢)
1
= −𝑢𝛼−2 𝑓 (0) − 𝑢1−𝛼 Γ [2 − 𝛼] + 2𝑢2 − 𝑢,
Γ [2 − 𝛼]
𝑛
𝜕𝑈 (𝑡) 󵄨󵄨󵄨󵄨 1
+ ∑ 𝑢𝛼−𝑘 [𝐷𝛼−𝑘 (𝑈 (𝑡))]𝑡=0 − 𝑢𝛼−1 󵄨 (1 + 𝛼 ) 𝐹 (𝑢) = 2𝑢2−𝛼 − 𝑢1−𝛼 + 2𝑢2 − 𝑢,
𝑘=1
𝜕𝑡 󵄨󵄨󵄨𝑡=0 𝑢
𝛼−1 (1 + 𝑢𝛼 ) 𝐹 (𝑢) = 2𝑢2 − 𝑢 + 2𝑢2+𝛼 − 𝑢1+𝛼 ,
−𝑢 𝑈 (0) + 𝑐𝑢𝛼 ,
1 (1 + 𝑢𝛼 ) 𝐹 (𝑢) = 𝑢 (2𝑢 − 1) + 𝑢𝛼 𝑢 (2𝑢 − 1) ,
𝐹 (𝑢) =
1 − 𝑢 − 𝑢𝛼−1 − 𝑢𝛼
𝛼−2
𝐹 (𝑢) = (2𝑢 − 1) 𝑢,
𝛼−1 𝛼−2 𝛼
× [𝑔 (𝑢) − 𝑢 𝑈 (0) − 𝑢 𝑈 (0) + 𝑐𝑢 ] , 𝐹 (𝑢) = 2𝑢2 − 𝑢.
(18) (22)
When we take the inverse Sumudu transform of (22) by 6

using the inverse transform table in [11], we get the analytical
5
solution of (20) by STM as follows:
4
𝑈 (𝑡) = 𝑡2 − 𝑡. (23)
3
u
Remark 4. If we take the corresponding values for some
parameters into consideration, then the solution of (20) is in 2
full agreement with the solution of (30) mentioned in [50].
To our knowledge, the analytical solution of FODEs that we 1
find in this paper has been newly submitted to the literature.
0.5 1.0 1.5 2.0 2.5 3.0
Example 5. Secondly, we consider the nonhomogeneous t
fractional ordinary differential equation as follows [51]:
Analytical solution by STM
Γ [3] 1.5
𝐷0.5 𝑈 (𝑡) + 𝑈 (𝑡) = 𝑡2 + 𝑡 , 𝑡 > 0, (24)
Γ [2.5] Figure 1: The 2D surfaces of the obtained solution by means of STM
for (23) when 0 < 𝑡 < 3.
With the initial condition being
𝑈 (0) = 0. (25)
In order to solve (24) by using STM, when we take the 8
Sumudu transform of both sides of (24), we get the Sumudu
transform of (24) as follows: 6
Γ [3]
𝑆 [𝐷0.5 𝑈 (𝑡)] + 𝑆 [𝑈 (𝑡)] = 𝑆 [𝑡2 ] + 𝑆 [𝑡1.5 ] ,
u
Γ [2.5] 4
𝑆 [𝐷0.5 𝑈 (𝑡)] + 𝑆 [𝑈 (𝑡)] = 𝑆 [𝑡2 ] + 1.50451𝑆 [𝑡1.5 ] , 2

󵄨
𝐹 (𝑢) 𝐷𝛼−1 [𝑈 (𝑡)] 󵄨󵄨󵄨
− 󵄨󵄨 + 𝐹 (𝑢)
𝑢0.5 𝑢 󵄨󵄨 0.5 1.0 1.5 2.0 2.5 3.0
󵄨𝑡=0
t
= 2𝑢2 + 2𝑢1.5
Analytical solution by STM
𝐹 (𝑢) (26)
󳨐⇒ 0.5 + 𝐹 (𝑢) = 2𝑢2 + 2𝑢1.5 , Figure 2: The 2D surfaces of the obtained solution by means of STM
𝑢 for (27) when 0 < 𝑡 < 3.
1 + 𝑢0.5
( ) 𝐹 (𝑢) = 2𝑢2 + 2𝑢1.5
𝑢0.5
fractional ordinary differential equations have been solved
0.5 2 1.5 by using the Sumudu transform after giving the related
󳨐⇒ (1 + 𝑢 ) 𝐹 (𝑢) = 2𝑢 + 2𝑢 ,
formulae for the fractional integrals, the derivatives, and the
2𝑢2.5 2𝑢2 2𝑢2 (1 + 𝑢0.5 ) Sumudu transform of FODEs. The Sumudu technique can be
𝐹 (𝑢) = + = = 2𝑢2 . used to solve many types such as initial-value problems and
1 + 𝑢0.5 1 + 𝑢0.5 1 + 𝑢0.5 boundary-value problems in applied sciences, engineering
When we take the inverse Sumudu transform of (26) by fields, aerospace sciences, and mathematical physics. The
using the inverse transform table in [48], we get the analytical Sumudu transform method has been used for The discrete
solution of (24) by using STM as follows: fractional calculus in [43]. This technique has been inves-
tigated in terms of the double Sumudu transform in [44].
𝑈 (𝑡) = 𝑡2 . (27)
Consequently, this new approach has been implemented
Remark 6. The solution (27) obtained by using the Sumudu with success on interesting fractional ordinary differential
transform method for (24) has been checked by the Mathe- equations. As such and pragmatically so, it enriches the
matica Program 7. To our knowledge, the analytical solution library of integral transform approaches. Without a doubt,
that we find in this paper has been newly submitted to the and based on our findings such as Figures 1 and 2, the STM
literature. technique remains direct, robust and valuable tool for solving
same fractional differential equations.
4. Conclusion and Future Work
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http://dx.doi.org/10.1155/2013/414353
Research Article
Improved (𝐺󸀠/𝐺)-Expansion Method for the Space and Time
Fractional Foam Drainage and KdV Equations
Ali Akgül,1,2 Adem KJlJçman,3 and Mustafa Inc4

1
Department of Mathematics, Education Faculty, Dicle University, 21280 Diyarbakır, Turkey
2
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA
3
Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia
4
Department of Mathematics, Science Faculty, Fırat University, 23119 Elazığ, Turkey
Correspondence should be addressed to Mustafa Inc; minc@firat.edu.tr
Received 10 June 2013; Accepted 17 July 2013
Copyright © 2013 Ali Akgül et al. This is an open access article distributed under the Creative Commons Attribution License, which
The fractional complex transformation is used to transform nonlinear partial differential equations to nonlinear ordinary
differential equations. The improved (𝐺󸀠 /𝐺)-expansion method is suggested to solve the space and time fractional foam drainage
and KdV equations. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional
differential equations.
1. Introduction the fractional subequation method. Guo et al. [19] presented

the improved subequation method to solve the space-time
The soliton solutions of nonlinear evolution equations have fractional Whitham-Broer-Kaup and the generalized Hirota-
made a major impact in the flesh. These solitons appear in
Satsuma coupled KdV equations. Tang et al. [20] used the
various areas of physical and biological sciences. They show
generalized fractional subequation method to obtain exact
up in nonlinear optics, plasma physics, fluid dynamics, bio-
solutions of the space-time fractional Gardner equation with
chemistry, and mathematical chemistry. Fractional partial
variable coefficients. Lu [21] investigated the exact solutions
differential equations (FPDEs) have received considerable
interest in recent years and have been extensively investi- of the nonlinear fractional Klein-Gordon equation, the gen-
gated. These equations were applied for many real problems eralized time fractional Hirota-Satsuma coupled KdV system,
which are modeled in various areas, for example, in mathe- and the nonlinear fractional Sharma-Tasso-Olver equation.
matical physics [1], fluid and continuum mechanics [2], visco- Bin [22] solved the time-space fractional generalized Hirota-
plastic and viscoelastic flow [3], biology, chemistry, acoustics, Satsuma coupled KdV equations and the time fractional fifth-
and psychology [4, 5]. Some FPDEs do not have exact solu- order Sawada-Kotera equation by using the (𝐺󸀠 /𝐺)-expan-
tions, so approximation and numerical techniques must be sion method. Omran and Gepreel [23] used the improved
used. There are several approximation and numerical meth- (𝐺󸀠 /𝐺)-expansion method to calculate the exact solutions to
ods. The most commonly used ones are the homotopy pertur- the time-space fractional foam drainage and KdV equations.
bation method [6, 7], the Adomian decomposition method In this paper, we will apply the improved (𝐺󸀠 /𝐺)-expansion
[8, 9], the variational iteration method [10–12], the homotopy method to obtain the exact solutions for the time-space frac-
analysis method [13, 14], the generalized differential trans- tional foam drainage and KdV equations with the modified
form method [15], the finite difference method [16], and the Riemann-Liouville derivative defined by Jumarie [24–27]:
finite element method [17]. In recent years, some authors have
2
got exact solutions of FPDEs by using analytical methods. S. 𝜕𝛼 𝑢 1 𝜕2𝛽 𝑢 𝜕𝛽 𝑢 𝜕𝛽 𝑢
Zhang and H.-Q. Zhang [18] proposed to solve the nonlinear 𝛼
= 𝑢 2𝛽 + 2𝑢2 𝛽 + ( 𝛽 ) ,
𝜕𝑡 2 𝜕𝑥 𝜕𝑥 𝜕𝑥 (1)
time fractional biological population model and (4 + 1)-
dimensional space-time fractional Fokas equation by using 𝑡 > 0, 𝛼 > 0, 𝛽 ≤ 1,
𝜕𝛼 𝑢 𝜕𝛽 𝑢 𝜕3𝛽 𝑢 Theorem 4. Assume that the continuous function 𝑓(𝑥) has a

+ 𝛼𝑢 + = 0,
𝜕𝑡𝛼 𝜕𝑥𝛽 𝜕𝑥3𝛽 (2) fractional derivative of order 𝛼; then
𝑡 > 0, 𝛼 > 0, 𝛽 ≤ 1, 𝑑𝛼 𝛼
𝐼 𝑓 (𝑥) = 𝑓 (𝑥) ,
where 𝛼 is arbitrary constant. This paper is organized 𝑑𝑥𝛼
(7)
as follows. In Section 2, we introduce some basic defini- 𝛼 𝑑𝛼
𝐼 𝑓 (𝑥) = 𝑓 (𝑥) − 𝑓 (0) , 0 < 𝛼 ≤ 1,
tions of Jumarie’s modified Riemann-Liouville derivative. In 𝑑𝑥𝛼
Section 3, the main steps of the improved (𝐺󸀠 /𝐺)-expansion hold.
method are given. In Section 4, we construct the exact
solutions of (1) and (2) by the proposed method. Some
conclusions are given in Section 5. 3. Description of the Improved (𝐺󸀠 /𝐺)-
Expansion Method
2. Preliminaries In this section, we give the description of the improved
(𝐺󸀠 /𝐺)-expansion method for solving the nonlinear FPDEs
There are several definitions for fractional differential equa- as
tions. These definitions include Riemann-Liouville, Weyl,
Grünwald-Letnikov, Riesz, and Jumarie fractional deriva- 𝐹 (𝑢, 𝐷𝑡𝛼 𝑢, 𝐷𝑥𝛽 𝑢, 𝐷𝑦𝛾 𝑢, 𝐷𝑧𝛿 𝑢, 𝐷𝑡𝛼 𝐷𝑡𝛼 𝑢,
tives. The Riemann-Liouville fractional derivative of a con-
stant is not zero. So the fractional derivative is only defined for 𝐷𝑡𝛼 𝐷𝑥𝛽 𝑢, 𝐷𝑥𝛽 𝐷𝑥𝛽 𝑢, . . .) = 0, (8)
differentiable function. In order to deal with nondifferen- 0 < 𝛼, 𝛽, 𝛾, 𝛿 ≤ 1,
tiable functions, Jumarie [24–27] presented a modification of
the Riemann-Liouville definition which appears to provide a where 𝑢 is an unknown function and 𝐹 is a polynomial of 𝑢
framework for a fractional calculus. This modification was and its partial fractional derivatives, in which the highest
successfully applied in the probability calculus, fractional order derivatives and nonlinear terms are involved. We offer
Laplace problem, exact solutions of the nonlinear fractional an improved (𝐺󸀠 /𝐺)-expansion method [32]. The essential
differential equations, and many other types of linear and steps of this method are described as follows.
nonlinear fractional differential equations [28–30].
Step 1. Li and He [33] and He and Li [34] presented a frac-
Definition 1. The Riemann-Liouville fractional integral is tional complex transform to transform fractional differential
equations into ordinary differential equations. So, all ana-
defined [31] as
lytical methods devoted to advanced calculus can be easily
𝛼
0 𝐼𝑥 𝑓 (𝑥) = 𝐼𝛼 𝑓 (𝑥) dedicated to fractional calculus. The traveling wave variable
𝑥
1 is given as
= ∫ 𝑓 (𝜉) (𝑥 − 𝜉)𝛼−1 𝑑𝜉, 𝛼 > 0, (3)
Γ (𝛼) 0 𝑢 (𝑥, 𝑦, 𝑧, 𝑡) = 𝑢 (𝜉) ,
0
𝐼 𝑓 (𝑥) = 𝑓 (𝑥) . 𝐾𝑥𝛽 𝑁𝑦𝛾 𝐿𝑡𝛼 (9)
𝜉= + + ,
Definition 2. Jumarie [24–27] defined the fractional deriva- Γ (𝛽 + 1) Γ (𝛾 + 1) Γ (𝛼 + 1)
tive in the limit form by where 𝐾, 𝑁, and 𝐿 are nonzero arbitrary constants. So, (9) is
𝛼 reduced to (10):
Δ [𝑓 (𝑥) − 𝑓 (0)]
𝑓(𝛼) (𝑥) = lim , (4)
ℎ→0 ℎ𝛼 𝑝 (𝑢, 𝑢󸀠 , 𝑢󸀠󸀠 , 𝑢󸀠󸀠󸀠 , . . .) = 0, (10)
where 𝑓(𝑥) should be a continuous (but not necessarily dif- where 𝑢 = 𝑢(𝜉).
ferentiable) function and ℎ > 0 denotes a constant discreti-
zation span. So, the modified form of the Riemann-Liouville Step 2. Suppose that (10) has the solution (11):
derivative is defined as 𝑛
1 𝑑𝑛 𝑥 𝑢 (𝜉) = ∑𝑎𝑖 𝐹𝑖 (𝜉) , (11)

𝛼
0 𝐷𝑥 𝑓 (𝑥) = ∫ (𝑥 − 𝜉)(𝑛−𝛼) [𝑓 (𝜉) − 𝑓 (0)] , 𝑖=0
Γ (𝑛 − 𝛼) 𝑑𝑥𝑛 0
(5) where 𝑎𝑖 (𝑖 = 0, 1, . . . , 𝑛) are real constants to be determined,
the balancing number 𝑛 is a positive integer which can be
where 𝑥 ∈ [0, 1], 𝑛 − 1 ≤ 𝛼 < 𝑛 and 𝑛 ≥ 1. determined by balancing the highest derivative terms with
the highest power nonlinear terms in (10). More precisely, we
Lemma 3. The integral with respect to (𝑑𝑥)𝛼 is defined by Jum- define the degree of 𝑢(𝜉) as 𝐷[𝑢(𝜉)] = 𝑚, which gives rise to
arie [24, 25] as follows: the degrees of other expressions, as follows:
𝑥 𝑥
∫ 𝑓 (𝜉) (𝑑𝜉)𝛼 = 𝛼 ∫ (𝑥 − 𝜉) 𝑓 (𝜉) 𝑑𝜉, 0 < 𝛼 ≤ 1, 𝑑𝑞 𝑢
𝐷[ ] = 𝑚 + 𝑞,
0 0 𝑑𝜉𝑞
𝛼 𝑢(𝑥) 𝑠 (12)
𝑑 𝛼 󸀠 𝛼 (6) 𝑑𝑞 𝑢
𝑝
∫ 𝑓 (𝜉) (𝑑𝜉) = Γ (𝛼 + 1) 𝑓 [𝑢 (𝜉)] [𝑢 (𝜉)] , 𝐷 [𝑢 ( 𝑞 ) ] = 𝑚𝑝 + 𝑠 (𝑞 + 𝑚) .
𝑑𝑥𝛼 0 𝑑𝜉
0 < 𝛼 ≤ 1. Therefore, we can obtain the value of 𝑚 in (11).
Step 3. 𝐹(𝜉) is 4.1. The Time and Space-Fractional Nonlinear Foam Drainage
Equation. We apply the improved (𝐺󸀠 /𝐺)-expansion method
𝐺󸀠 (𝜉) to construct the exact solutions for the time-space fractional
𝐹 (𝜉) = , (13)
𝐺 (𝜉) nonlinear foam drainage equation in this subsection. Foams
are of great importance in many technological processes and
where 𝐺(𝜉) expresses the solution of the following auxiliary applications. Their properties are subject of intensive studies
ordinary differential equation from practical and scientific points of view [27, 35–37]. Liquid
2 foam is an example of soft matter with a very well-defined
𝐺 (𝜉) 𝐺󸀠󸀠 (𝜉) = 𝐴𝐺2 (𝜉) + 𝐵𝐺 (𝜉) 𝐺󸀠 (𝜉) + 𝐶[𝐺 (𝜉)] , (14)
structure, described by Joseph Plateau in the 19th century.
where the prime denotes derivative with respect to 𝜉. 𝐴, 𝐵, Foams are common in foods and personal care products
and 𝐶 are real parameters. such as lotions and creams. They have important applications
in food and chemical industries, mineral processing, fire
Step 4. Substituting (13) into (10), using (14), collecting all fighting, and structural material sciences [27, 35–37]. This
terms with the same order of (𝐺󸀠 (𝜉)/𝐺(𝜉)) together, and then equation is numerically and analytically taken into account by
equating each coefficient of the resulting polynomial to zero, different authors [38–40]. The space-time fractional nonlin-
we obtain a set of algebraic equations for 𝑎𝑖 (𝑖 = 0, 1, . . . , 𝑛), ear foam drainage equation is solved analytically only by
𝐴, 𝐵, 𝐶, 𝐾, 𝑁, and 𝐿. Omran and Gepreel [23]. We can see the fractional complex
transform as
Step 5. Using the general solutions of (14), with the aid of
Mathematica, we have the following four solutions of (13). 𝑢 (𝑥, 𝑡) = 𝑢 (𝜉) ,
𝐾𝑥𝛽 𝐿𝑡𝛼 (20)

Case 1. If 𝐵 ≠0 and Δ = 𝐵2 + 4𝐴 − 4𝐴𝐶 ≥ 0, then 𝜉= + ,
Γ (𝛽 + 1) Γ (𝛼 + 1)
𝐵 𝐵√Δ
𝐹 (𝜉) = + where 𝐾 and 𝐿 are constants. So, (20) reduces to (21):
2 (1 − 𝐶) 2 (1 − 𝐶)
(15) 1 2
𝑐1 exp ((√Δ/2) 𝜉) + 𝑐2 exp ((−√Δ/2) 𝜉) −𝐿𝑢󸀠 + 𝐾2 𝑢𝑢󸀠󸀠 + 2𝐾𝑢2 𝑢󸀠 + 𝐾2 (𝑢󸀠 ) = 0. (21)
× . 2
𝑐1 exp ((√Δ/2) 𝜉) − 𝑐2 exp ((−√Δ/2) 𝜉)
Balancing the highest order nonlinear term and the high-
Case 2. If 𝐵 ≠0 and Δ = 𝐵2 + 4𝐴 − 4𝐴𝐶 < 0, then est order linear term, we get 𝑛 = 1. Thus, we obtain
𝐵 𝐵√−Δ 𝐺󸀠 (𝜉)
𝐹 (𝜉) = + 𝑢 (𝜉) = 𝑎0 + 𝑎1 𝐹 (𝜉) , 𝐹 (𝜉) = , (22)
2 (1 − 𝐶) 2 (1 − 𝐶) 𝐺 (𝜉)
(16) where 𝑎0 and 𝑎1 will be determined constants. Substituting
𝑖𝑐1 cos ((√−Δ/2) 𝜉) − 𝑐2 sin ((√−Δ/2) 𝜉)
× . (22) into (21), using (14), collecting all the terms of powers of
𝑖𝑐1 sin ((√−Δ/2) 𝜉) + 𝑐2 cos ((√−Δ/2) 𝜉) (𝐺󸀠 /𝐺), and setting each coefficient to zero, we have the fol-
lowing system of algebraic equations:
Case 3. If 𝐵 = 0 and Δ = 𝐴(𝐶 − 1) ≥ 0, then
0
𝐺󸀠 1
√Δ 𝑐1 cos (√Δ𝜉) + 𝑐2 sin (√Δ𝜉) ( ) :2𝐴𝑎02 𝑎1 𝐾 + 𝐴2 𝑎12 𝐾2 + 𝐴𝑎0 𝑎1 𝐵𝐾2 − 𝐴𝑎1 𝐿 = 0,
𝐹 (𝜉) = . (17) 𝐺 2
(1 − 𝐶) 𝑐1 sin (√Δ𝜉) − 𝑐2 cos (√Δ𝜉)
1
𝐺󸀠 5
Case 4. If 𝐵 = 0 and Δ = 𝐴(𝐶 − 1) < 0, then ( ) :2𝑎02 𝑎1 𝐵𝐾 − 𝐴𝑎0 𝑎1 𝐾2 + 𝐴𝑎12 𝐵𝐾2
𝐺 2
√−Δ 𝑖𝑐1 cosh (√−Δ𝜉) − 𝑐2 sinh (√−Δ𝜉) 1
𝐹 (𝜉) = , (18) + 𝑎0 𝑎1 𝐵2 𝐾2 + 𝐴𝑎0 𝑎1 𝐶𝐾2 − 𝑎1 𝐵𝐿 = 0,
(1 − 𝐶) 𝑖𝑐1 sinh (√−Δ𝜉) − 𝑐2 cosh (√−Δ𝜉) 2
2
where 𝐺󸀠
( ) : − 2𝑎02 𝑎1 𝐾 + 2𝐴𝑎13 𝐾 + 4𝑎0 𝑎12 𝐵𝐾
𝐾𝑥𝛽 𝑁𝑦𝛾 𝐿𝑡𝛼 𝐺
𝜉= + + (19)
Γ (𝛽 + 1) Γ (𝛾 + 1) Γ (𝛼 + 1) 3
+ 2𝑎02 𝑎1 𝐶𝐾 − 3𝐴𝑎12 𝐾2 + 𝑎0 𝑎1 𝐵𝐾2
and 𝐴, 𝐵, 𝐶, 𝑐1 , and 𝑐2 are real parameters. 2
3 3
+ 𝑎12 𝐵2 𝐾2 + 3𝐴𝑎12 𝐶𝐾2 + 𝑎0 𝑎1 𝐵𝐶𝐾2
4. Applications 2 2
3 3
We use the improved (𝐺󸀠 /𝐺)-expansion method on the time- + 𝑎12 𝐵2 𝐾2 + 3𝐴𝑎12 𝐶𝐾2 + 𝑎0 𝑎1 𝐵𝐶𝐾2
2 2
space fractional nonlinear foam drainage equation and the
time-space fractional nonlinear KdV equation in this section. + 𝑎1 𝐿 − 𝑎1 𝐶𝐿 = 0,
3
𝐺󸀠 where
( ) : − 4𝑎0 𝑎12 𝐾 + 2𝑎13 𝐵𝐾 + 4𝑎0 𝑎12 𝐶𝐾
𝐺
𝐾𝑥𝛽 1
7 𝜉= + (4𝑎02 𝐾 + 𝑎0 𝐵𝐾2 + 2𝐴𝐾3 − 2𝐴𝐶𝐾3 )
+ 𝑎0 𝑎1 𝐾2 − 𝑎12 𝐵𝐾2 + 2𝑎0 𝑎1 𝐶𝐾2 Γ (𝛽 + 1) 2
2 (28)
𝑡𝛼
7 × .
+ 𝑎12 𝐵𝐶𝐾2 + 𝑎0 𝑎1 𝐶2 𝐾2 = 0, Γ (𝛼 + 1)
2
4
𝐺󸀠 (3) If we choose 𝐵 = 0 and Δ 1 = 𝐴(𝐶 − 1) ≥ 0, then we
( ) : − 2𝑎13 𝐾 + 2𝑎13 𝐶𝐾 + 2𝑎12 𝐾2 get another triangular function solution
𝐺
− 4𝑎12 𝐶𝐾2 + 2𝑎12 𝐶2 𝐾2 = 0. 𝑐1 cos (√Δ 1 𝜉) + 𝑐2 sin (√Δ 1 𝜉)

(23) 𝑢 (𝑥, 𝑡) = −𝐾√Δ 1 , (29)
𝑐1 sin (√Δ 1 𝜉) − 𝑐2 cos (√Δ 1 𝜉)
Solving the set of the above algebraic equations, we get the fol-
lowing result: where 𝜉 = 𝐾𝑥𝛽 /Γ(𝛽 + 1) − Δ 1 (𝐾3 𝑡𝛼 /Γ(𝛼 + 1)).
(4) If we choose 𝐵 = 0 and Δ 1 = 𝐴(𝐶 − 1) < 0, then we
8𝐴𝐵𝐾 (𝐶 − 1)
𝑎0 = , 𝑎1 = 𝐾 (𝐶 − 1) , obtain the hyperbolic function solution
𝐵2 + 6𝐴 − 6𝐴𝐶
1 (24) 𝑖𝑐1 cosh (√−Δ 1 𝜉) − 𝑐2 sinh (√−Δ 1 𝜉)
𝐿= (4𝑎02 𝐾 + 𝑎0 𝐵𝐾2 + 2𝐴𝐾3 − 2𝐴𝐶𝐾3 ) , 𝑢 (𝑥, 𝑡) = −𝐾√Δ 1 ,
2 𝑖𝑐1 sinh (√−Δ 1 𝜉) − 𝑐2 cosh (√−Δ 1 𝜉)
𝐾𝐵 (𝐶 − 1) ≠0. (30)
Substituting this value in (22) and by Cases 1–4, we obtain where 𝜉 = 𝐾𝑥𝛽 /Γ(𝛽 + 1) − Δ 1 (𝐾3 𝑡𝛼 /Γ(𝛼 + 1)).
the following exponential, hyperbolic and triangular function
solutions of (1). If we take 𝑐1 = −𝑐2 and 𝑐1 = 𝑐2 in (25), respectively, then
we get
(1) If we choose 𝐵 ≠0 and Δ = 𝐵2 + 4𝐴 − 4𝐴𝐶 ≥ 0, then
the exponential function solutions can be found as 16𝐴𝐵𝐾 (𝐶 − 1) − 𝐾𝐵 [Δ + 2𝐴 (𝐶 − 1)]
𝑢 (𝑥, 𝑡) =
2Δ + 2𝐴 (𝐶 − 1)
16𝐴𝐵𝐾 (𝐶 − 1) − 𝐾𝐵 [Δ + 2𝐴 (𝐶 − 1)] 𝐾𝐵√Δ
𝑢 (𝑥, 𝑡) = − 𝐾𝐵√Δ √Δ
2Δ + 2𝐴 (𝐶 − 1) 2 − tanh ( 𝜉) ,
2 2
𝑐1 exp ((√Δ/2) 𝜉) + 𝑐2 exp ((−√Δ/2) 𝜉) (31)
× , 16𝐴𝐵𝐾 (𝐶 − 1) − 𝐾𝐵 [Δ + 2𝐴 (𝐶 − 1)]
𝑢 (𝑥, 𝑡) =
𝑐1 exp ((√Δ/2) 𝜉) − 𝑐2 exp ((−√Δ/2) 𝜉) 2Δ + 2𝐴 (𝐶 − 1)
(25)
𝐾𝐵√Δ √Δ
− coth ( 𝜉) .
2 2
where
4.2. The Nonlinear Space-Time Fractional KdV Equation. The
𝐾𝑥𝛽 1
𝜉= + (4𝑎02 𝐾 + 𝑎0 𝐵𝐾2 + 2𝐴𝐾3 − 2𝐴𝐶𝐾3 ) KdV equation is the most popular soliton equation, and it
Γ (𝛽 + 1) 2 has been largely investigated. In addition, the space and time
(26) fractional KdV equations with initial conditions were widely
𝑡𝛼 worked by [27, 38, 39]. Integrating (2) with respect to 𝑢 and
× .
Γ (𝛼 + 1) ignoring the integral constants leads to
1 2 1 1 2
(2) If we choose 𝐵 ≠0 and Δ = 𝐵2 + 4𝐴 − 4𝐴𝐶 < 0, then 𝐿𝑢 + 𝑎𝐾𝑢3 + 𝐾3 (𝑢󸀠 ) = 0. (32)
2 6 2
the triangular function solution will be
Considering the homogeneous balance between the high-
16𝐴𝐵𝐾 (𝐶 − 1) − 𝐾𝐵 [Δ + 2𝐴 (𝐶 − 1)] 𝐾𝐵√−Δ est order derivatives and the nonlinear term in (32), we get
𝑢 (𝑥, 𝑡) = − 𝑛 = 2. So, we can suppose that (32) has the following ansatz:
2Δ + 2𝐴 (𝐶 − 1) 2
𝑖𝑐1 cos ((√−Δ/2) 𝜉) − 𝑐2 sin ((√−Δ/2) 𝜉) 𝑢 (𝜉) = 𝑎0 + 𝑎1 𝐹 (𝜉) + 𝑎2 𝐹2 (𝜉) , (33)

× ,
𝑖𝑐1 sin ((√−Δ/2) 𝜉) + 𝑐2 cos ((√−Δ/2) 𝜉)
where 𝑎0 , 𝑎1 , 𝑎2 , 𝐿, and 𝐾 are arbitrary constants to be
(27) determined later. Substituting (33) and (14), along with (13),
into (32) and using Mathematica yields a system of Equations Substituting (35) into (33) and according to (15)–(18), we
of (𝐺󸀠 /𝐺): obtain the following exponential function solutions, hyper-
0 bolic function solutions, and triangular function solutions of
𝐺󸀠 1
( ) : 𝑎𝑎03 𝐾 + 𝐴2 𝑎12 𝐾3 + 𝑎02 𝐿 = 0, (2), respectively.
𝐺 3
1 (1) If we choose 𝐵 ≠0 and Δ = 𝐵2 + 4𝐴 − 4𝐴𝐶 ≥ 0, then
𝐺󸀠 1
( ) : 𝑎𝑎02 𝑎1 𝐾 + 2𝐴2 𝑎1 𝑎2 𝐾3 the exponential function solution can be found as
𝐺 2
+ 𝐴𝑎12 𝐵𝐾3 + 𝑎0 𝑎1 𝐿 = 0, 3𝐾2 (Δ − 2𝐶) 6𝐶𝐾2 √Δ
𝑢 (𝑥, 𝑡) = −
2
𝑎 (𝐶 − 1) 𝑎 (𝐶 − 1)
𝐺󸀠 1 1
( ) : 𝑎𝑎0 𝑎12 𝐾 + 𝑎𝑎02 𝑎2 𝐾 − 𝐴𝑎12 𝐾3 𝑐1 exp (√Δ𝜉/2) + 𝑐2 exp ((−√Δ/2) 𝜉)
𝐺 2 2
×
1 𝑐1 exp ((√Δ/2) 𝜉) − 𝑐2 exp ((−√Δ/2) 𝜉)
+ 2𝐴2 𝑎22 𝐾3 + 4𝐴𝑎1 𝑎2 𝐵𝐾3 +
2𝑎12 𝐵2 𝐾3
6𝐶𝐾2 Δ
1 −
+ 𝐴𝑎12 𝐶𝐾3 + 𝑎12 𝐿 + 𝑎0 𝑎2 𝐿 = 0, 𝑎 (𝐶 − 1)
2
2
3 𝑐1 exp ((√Δ/2) 𝜉) + 𝑐2 exp (− (√Δ/2) 𝜉)
𝐺󸀠 1 ×[ ],
( ) : 𝑎𝑎13 𝐾 + 𝑎𝑎0 𝑎1 𝑎2 𝐾 − 4𝐴𝑎1 𝑎2 𝐾3
𝐺 6 𝑐1 exp ((√Δ/2) 𝜉) − 𝑐2 exp (− (√Δ/2) 𝜉)
(36)
− 𝑎12 𝐵𝐾3 + 4𝐴𝑎22 𝐵𝐾3 + 2𝑎1 𝑎2 𝐵2 𝐾3
+ 4𝐴𝑎1 𝑎2 𝐶𝐾3 + 𝑎12 𝐵𝐶𝐾3 + 𝑎1 𝑎2 𝐿 = 0,

where 𝜉 = 𝐾𝑥𝛽 /Γ(𝛽 + 1) − 𝐾3 Δ(𝑡𝛼 /Γ(𝛼 + 1)).
4
𝐺󸀠 1 1 1
( ) : 𝑎12 𝑎2 𝐾 + 𝑎𝑎0 𝑎22 𝐾 + 𝑎12 𝐾3 (2) If we choose 𝐵 ≠0 and Δ = 𝐵2 + 4𝐴 − 4𝐴𝐶 < 0, then
𝐺 2 2 2 the triangular function solution will be
− 4𝐴𝑎22 𝐾3 − 4𝑎1 𝑎2 𝐵𝐾3 + 2𝑎22 𝐵2 𝐾3
3𝐾2 (Δ − 2𝐶) 6𝐶𝐾2 √Δ
1 𝑢 (𝑥, 𝑡) = +
− 𝑎12 𝐶𝐾3 + 4𝑎1 𝑎2 𝐵𝐶𝐾3 + 𝑎12 𝐶2 𝐾3 𝑎 (𝐶 − 1) 𝑎 (𝐶 − 1)
2
1 𝑖𝑐1 cos (√−Δ𝜉/2) − 𝑐2 sin ((√−Δ/2) 𝜉)
+ 𝑎22 𝐿 = 0, ×
2 𝑖𝑐1 sin ((√−Δ/2) 𝜉) + 𝑐2 cos ((√−Δ/2) 𝜉)
5
𝐺󸀠 1 6𝐶𝐾2 Δ
( ) : 𝑎𝑎1 𝑎22 𝐾 + 2𝑎1 𝑎2 𝐾3 − 4𝑎22 𝐵𝐾3 +
𝐺 2 𝑎 (𝐶 − 1)
− 4𝑎1 𝑎2 𝐶𝐾3 + 4𝑎22 𝐵𝐶𝐾3 + 2𝑎1 𝑎2 𝐶2 𝐾3 = 0, 𝑖𝑐1 cos ((√−Δ/2) 𝜉) − 𝑐2 sin ((√−Δ/2) 𝜉)
2
×[ ],
𝐺󸀠 1
6 𝑖𝑐1 sin ((√−Δ/2) 𝜉) + 𝑐2 cos ((√−Δ/2) 𝜉)
( ) : 𝑎1 𝑎22 𝐾 + 2𝑎22 𝐾3 − 4𝑎22 𝐶𝐾3 (37)
𝐺 6
+ 2𝑎22 𝐶2 𝐾3 = 0.
(34) where 𝜉 = 𝐾𝑥𝛽 /Γ(𝛽 + 1) − 𝐾3 Δ(𝑡𝛼 /Γ(𝛼 + 1)).
Solving the set of the above algebraic equations by use of (3) If we choose 𝐵 = 0 and Δ 1 = 𝐴(𝐶 − 1) ≥ 0, then the
Mathematica, we get the following results: triangular function solution is given as
12𝐴𝐾2
𝑎 ≠0, 𝑎0 = − (𝐶 − 1) , 12𝐾2 Δ 1 12𝐾2 Δ 1
𝑎 𝑢 (𝑥, 𝑡) = − −
𝑎 𝑎 (𝐶 − 1)
12𝐵𝐾2 (38)
𝑎1 = (𝐶 − 1) , 2
𝑎 (35) 𝑐1 cos (√Δ 1 𝜉) + 𝑐2 sin (√Δ 1 𝜉)
×[ ],
12𝐾2 𝑐1 sin (√Δ 1 𝜉) − 𝑐2 cos (√Δ 1 𝜉)
𝑎2 = − (𝐶 − 1) ,
𝑎
𝐿 = −𝐾3 (𝐵2 − 4𝐴𝐶 + 4𝐴) . where 𝜉 = 𝐾𝑥𝛽 /Γ(𝛽 + 1) + 4𝐾3 Δ 1 (𝑡𝛼 /Γ(𝛼 + 1)).
(4) If we choose 𝐵 = 0 and Δ 1 = 𝐴(𝐶 − 1) < 0, then the 5. Conclusion

hyperbolic function solution is given as
In this paper, we introduced an improved (𝐺󸀠 /𝐺)-expansion
method and carried it out to obtain new travelling wave solu-
12𝐾2 Δ 1 12𝐾2 Δ 1
𝑢 (𝑥, 𝑡) = − + tions of the space-time fractional foam drainage equation and
𝑎 𝑎 (𝐶 − 1) the space-time fractional KdV equation. This method gives
2
(39) new exact solutions for nonlinear FPDEs. These solutions
𝑖𝑐1 cosh (√−Δ 1 𝜉) − 𝑐2 sinh (√−Δ 1 𝜉) include the hyperbolic function solution, the exponential
×[ ],
𝑖𝑐1 sinh (√−Δ 1 𝜉) − 𝑐2 cosh (√−Δ 1 𝜉) function solution, the triangular function solution, and the
trigonometric function solution. These solutions are useful to
understand the mechanisms of the complicated nonlinear
where 𝜉 = 𝐾𝑥𝛽 /Γ(𝛽 + 1) − 4𝐾3 Δ 1 (𝑡𝛼 /Γ(𝛼 + 1)).
physical phenomena.
Equation (36) can be rewritten at 𝑐1 = −𝑐2 ; so we get
the other hyperbolic function solution of (2):
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http://dx.doi.org/10.1155/2013/932085
Research Article
The Solution to the BCS Gap Equation for Superconductivity
and Its Temperature Dependence
Shuji Watanabe
Division of Mathematical Sciences, Graduate School of Engineering, Gunma University, 4-2 Aramaki-machi, Maebashi 371-8510, Japan
Correspondence should be addressed to Shuji Watanabe; shuwatanabe@gunma-u.ac.jp
Copyright © 2013 Shuji Watanabe. This is an open access article distributed under the Creative Commons Attribution License,
From the viewpoint of operator theory, we deal with the temperature dependence of the solution to the BCS gap equation for
superconductivity. When the potential is a positive constant, the BCS gap equation reduces to the simple gap equation. We first
show that there is a unique nonnegative solution to the simple gap equation, that it is continuous and strictly decreasing, and that it
is of class 𝐶2 with respect to the temperature. We next deal with the case where the potential is not a constant but a function. When
the potential is not a constant, we give another proof of the existence and uniqueness of the solution to the BCS gap equation, and
show how the solution varies with the temperature. We finally show that the solution to the BCS gap equation is indeed continuous
with respect to both the temperature and the energy under a certain condition when the potential is not a constant.
1. Introduction where 𝑈(⋅, ⋅) > 0 is the potential multiplied by the density of

states per unit energy at the Fermi surface and is a function
We use the unit 𝑘𝐵 = 1, where 𝑘𝐵 stands for the Boltzmann of 𝑥 and 𝜉. In (1) we introduce 𝜀 > 0, which is small enough
constant. Let 𝜔𝐷 > 0 and 𝑘 ∈ R3 stand for the Debye fre- and fixed (0 < 𝜀 ≪ ℎ𝜔𝐷). In the original BCS model, the
quency and the wave vector of an electron, respectively. Let integration interval is [0, ℎ𝜔𝐷]; it is not [𝜀, ℎ𝜔𝐷]. However,
ℎ > 0 be Planck’s constant, and set ℎ = ℎ/(2𝜋). Let 𝑚 > 0 and we introduce very small 𝜀 > 0 for the following mathematical
𝜇 > 0 stand for the electron mass and the chemical potential, reasons. In order to show the continuity of the solution to
respectively. We denote by 𝑇(≥ 0) the absolute temperature, the BCS gap equation with respect to the temperature and in
and by 𝑥 the kinetic energy of an electron minus the chemical order to show that the transition to a superconducting state
potential; that is, 𝑥 = ℎ2 |𝑘|2 /(2𝑚)−𝜇. Note that 0 < ℎ𝜔𝐷 ≪ 𝜇. is a second-order phase transition, we make the form of the
In the BCS model [1, 2] of superconductivity, the solution BCS gap equation somewhat easier to handle. So we choose
to the BCS gap equation (1) is called the gap function. The the closed interval [𝜀, ℎ𝜔𝐷] as the integration interval in (1).
gap function corresponds to the energy gap between the The integral with respect to 𝜉 in (1) is sometimes replaced
superconducting ground state and the superconducting first by the integral over R3 with respect to the wave vector 𝑘.
excited state. Accordingly, the value of the gap function (the Odeh [3] and Billard and Fano [4] established the existence
solution) is nonnegative. We regard the gap function as a and uniqueness of the positive solution to the BCS gap equa-
function of both 𝑇 and 𝑥 and denote it by 𝑢; that is, 𝑢 : tion in the case 𝑇 = 0. For 𝑇 ≥ 0, Vansevenant [5] determined
(𝑇, 𝑥) 󳨃
→ 𝑢(𝑇, 𝑥) (≥ 0). The BCS gap equation is the following the transition temperature (the critical temperature) and
nonlinear integral equation (0 < 𝜀 ≤ 𝑥 ≤ ℎ𝜔𝐷): showed that there is a unique positive solution to the BCS gap
ℎ𝜔𝐷
𝑈 (𝑥, 𝜉) 𝑢 (𝑇, 𝜉) equation. Recently, Frank et al. [6] gave a rigorous analysis
𝑢 (𝑇, 𝑥) = ∫ of the asymptotic behavior of the transition temperature at
𝜀 √𝜉2 + 𝑢(𝑇, 𝜉)2 weak coupling. Hainzl et al. [7] proved that the existence of a
(1)
positive solution to the BCS gap equation is equivalent to the
√𝜉2 + 𝑢(𝑇, 𝜉)2
× tanh 𝑑𝜉, existence of a negative eigenvalue of a certain linear operator
2𝑇 to show the existence of a transition temperature. Moreover,
Hainzl and Seiringer [8] derived upper and lower bounds on Proposition 2 (see [10, Proposition 2.2]). Let Δ be as in (5).
the transition temperature and the energy gap for the BCS gap Then there is a unique nonnegative solution Δ 1 : [0, 𝜏1 ] →
equation. [0, ∞) to the simple gap equation (3) such that the solution
Since the existence and uniqueness of the solution were Δ 1 is continuous and strictly decreasing on the closed interval
established for each fixed 𝑇 in the previous literature, the tem- [0, 𝜏1 ]:
perature dependence of the solution is not covered. It is well
known that studying the temperature dependence of the solu- Δ 1 (0) = Δ > Δ 1 (𝑇1 ) > Δ 1 (𝑇2 )
tion to the BCS gap equation is very important in condensed (6)
matter physics. This is because, by dealing with the ther- > Δ 1 (𝜏1 ) = 0, 0 < 𝑇1 < 𝑇2 < 𝜏1 .
modynamical potential, this study leads to a mathematical
proof of the statement that the transition to a superconduct- Moreover, the solution Δ 1 is of class 𝐶2 on the interval [0, 𝜏1 )
ing state is a second-order phase transition. So, in condensed and satisfies
matter physics, it is highly desirable to study the temperature
dependence of the solution to the BCS gap equation. Δ󸀠1 (0) = Δ󸀠󸀠1 (0) = 0, lim Δ󸀠1 (𝑇) = −∞. (7)
𝑇↑𝜏1
When the potential 𝑈(⋅, ⋅) in (1) is a positive constant, the
BCS gap equation reduces to the simple gap equation (3). In Proof. Setting 𝑌 = Δ 1 (𝑇)2 turns (3) into
this case, one assumes in the BCS model that there is a unique
nonnegative solution to the simple gap equation (3) and that ℎ𝜔𝐷
1 √ 𝜉2 + 𝑌
the solution is of class 𝐶2 with respect to the temperature 1 = 𝑈1 ∫ tanh 𝑑𝜉. (8)
𝜀 √ 𝜉2 + 𝑌 2𝑇
𝑇 (see e.g., [1] and [9, (11.45), page 392]). In this paper,
applying the implicit function theorem, we first show that this Note that the right side is a function of the two variables 𝑇 and
assumption of the BCS model indeed holds true; we show that 𝑌. We see that there is a unique function 𝑇 󳨃 → 𝑌 defined by (8)
there is a unique nonnegative solution to the simple gap equa- implicitly, that the function 𝑇 󳨃→ 𝑌 is continuous and strictly
tion (3) and that the solution is of class 𝐶2 with respect to the decreasing on [0, 𝜏1 ], and that 𝑌 = 0 at 𝑇 = 𝜏1 . We moreover
temperature 𝑇. We next deal with the case where the potential see that the function 𝑇 󳨃 → 𝑌 is of class 𝐶2 on the closed
is not a constant but a function. In order to show how the interval [0, 𝜏1 ].
solution varies with the temperature, we then give another
proof of the existence and uniqueness of the solution to the Remark 3. We set Δ 1 (𝑇) = 0 for 𝑇 > 𝜏1 .
BCS gap equation (1) when the potential is not a constant.
More precisely, we show that the solution belongs to the Remark 4. In Proposition 2, Δ 1 (𝑇) is nothing but √𝑓(𝑇) in
subset 𝑉𝑇 (see (12)). Note that the subset 𝑉𝑇 depends on 𝑇. [10, Proposition 2.2].
We finally show that the solution to the BCS gap equation (1)
is indeed continuous with respect to both 𝑇 and 𝑥 when 𝑇 We introduce another positive constant 𝑈2 > 0. Let 0 <
satisfies (20) when the potential is not a constant. 𝑈1 < 𝑈2 . We assume the following condition on 𝑈(⋅, ⋅):
Let
2
𝑈1 ≤ 𝑈 (𝑥, 𝜉)
𝑈 (𝑥, 𝜉) = 𝑈1 at all (𝑥, 𝜉) ∈ [𝜀, ℎ𝜔𝐷] , (2)
2 2
where 𝑈1 > 0 is a constant. Then the gap function depends ≤ 𝑈2 at all (𝑥, 𝜉) ∈ [𝜀, ℎ𝜔𝐷] , 𝑈 (⋅, ⋅) ∈ 𝐶 ([𝜀, ℎ𝜔𝐷] ) .
on the temperature 𝑇 only. So we denote the gap function by (9)
Δ 1 in this case; that is, Δ 1 : 𝑇 󳨃
→ Δ 1 (𝑇). Then (1) leads to the
simple gap equation When 𝑈(𝑥, 𝜉) = 𝑈2 at all (𝑥, 𝜉) ∈ [𝜀, ℎ𝜔𝐷]2 , an argument
similar to that in Proposition 2 gives that there is a unique
ℎ𝜔𝐷
1 √𝜉2 + Δ 1 (𝑇)2 nonnegative solution Δ 2 : [0, 𝜏2 ] → [0, ∞) to the simple gap
1 = 𝑈1 ∫ tanh 𝑑𝜉. (3) equation
𝜀 √𝜉2 + Δ 1 (𝑇)2 2𝑇
ℎ𝜔𝐷
1
The following is the definition of the temperature 𝜏1 > 0. 1 = 𝑈2 ∫
𝜀 √𝜉2 + Δ 2 (𝑇)2
Definition 1 (see [1]). Consider (10)
√ 𝜉2 2
ℎ𝜔𝐷
1 𝜉 + Δ 2 (𝑇)
1 = 𝑈1 ∫ tanh 𝑑𝜉. (4) × tanh 𝑑𝜉, 0 ≤ 𝑇 ≤ 𝜏2 .
𝜀 𝜉 2𝜏1 2𝑇
Here, 𝜏2 > 0 is defined by
2. The Simple Gap Equation
ℎ𝜔𝐷
1 𝜉
Set 1 = 𝑈2 ∫ tanh 𝑑𝜉. (11)
𝜀 𝜉 2𝜏2
√(ℎ𝜔𝐷 − 𝜀𝑒1/𝑈1 ) (ℎ𝜔𝐷 − 𝜀𝑒−1/𝑈1 )
Δ= . (5) We again set Δ 2 (𝑇) = 0 for 𝑇 > 𝜏2 . A straightforward calcu-
sinh (1/𝑈1 ) lation gives the following.
x fixed x fixed
Δ 2 (0) Δ 2 (0)
Δ 1 (0) Δ 1 (0) Δ 2 (T)

Δ 2 (T)
Δ 1 (T) Δ 1 (T)
0 𝜏1 𝜏2 0 𝜏1 𝜏2
Temperature Temperature
Figure 1: The graphs of the functions Δ 1 and Δ 2 . Figure 2: For each 𝑇, the solution 𝑢0 (𝑇, 𝑥) lies between Δ 1 (𝑇) and
Δ 2 (𝑇).
nonnegative solution 𝑢0 (𝑇, ⋅) ∈ 𝑉𝑇 to the BCS gap equation

Lemma 5 ([11, Lemma 1.5]). (a) The inequality 𝜏1 < 𝜏2 holds. (1) (𝑥 ∈ [𝜀, ℎ𝜔𝐷]):
(b) If 0 ≤ 𝑇 < 𝜏2 , then Δ 1 (𝑇) < Δ 2 (𝑇). If 𝑇 ≥ 𝜏2 , then
Δ 1 (𝑇) = Δ 2 (𝑇) = 0. ℎ𝜔𝐷
𝑈 (𝑥, 𝜉) 𝑢0 (𝑇, 𝜉)
𝑢0 (𝑇, 𝑥) = ∫
Note that Proposition 2 and Lemma 5 point out how Δ 1
𝜀 √𝜉2 + 𝑢0 (𝑇, 𝜉)2
and Δ 2 depend on the temperature and how Δ 1 and Δ 2 vary (13)
with the temperature; see Figure 1. √𝜉2 + 𝑢0 (𝑇, 𝜉)2
× tanh 𝑑𝜉.
Remark 6. On the basis of Proposition 2, the present author 2𝑇
[10, Theorem 2.3] proved that the transition to a supercon- Consequently, the solution is continuous with respect to 𝑥 and
ducting state is a second-order phase transition under the varies with the temperature as follows:
restriction (2).
Δ 1 (𝑇) ≤ 𝑢0 (𝑇, 𝑥)
(14)
≤ Δ 2 (𝑇) at (𝑇, 𝑥) ∈ [0, 𝜏2 ] × [𝜀, ℎ𝜔𝐷] .
3. The BCS Gap Equation
Proof. We define a nonlinear integral operator 𝐴 on 𝑉𝑇 by
Let 0 ≤ 𝑇 ≤ 𝜏2 and fix 𝑇, where 𝜏2 is that in (11). We consider ℎ𝜔𝐷
the Banach space 𝐶[𝜀, ℎ𝜔𝐷] consisting of continuous func- 𝑈 (𝑥, 𝜉) 𝑢 (𝑇, 𝜉)
𝐴𝑢 (𝑇, 𝑥) = ∫
tions of 𝑥 only and deal with the following subset 𝑉𝑇 : 𝜀 √𝜉2 + 𝑢(𝑇, 𝜉)2
(15)
√𝜉2 + 𝑢(𝑇, 𝜉) 2
𝑉𝑇 = {𝑢 (𝑇, ⋅) ∈ 𝐶 [𝜀, ℎ𝜔𝐷] : Δ 1 (𝑇) × tanh 𝑑𝜉,
(12) 2𝑇
≤ 𝑢 (𝑇, 𝑥) ≤ Δ 2 (𝑇) at 𝑥 ∈ [𝜀, ℎ𝜔𝐷]} . where 𝑢(𝑇, ⋅) ∈ 𝑉𝑇 . Clearly, 𝑉𝑇 is a bounded, closed, and con-
vex subset of the Banach space 𝐶[𝜀, ℎ𝜔𝐷]. A straightforward
calculation gives that the operator 𝐴 : 𝑉𝑇 → 𝑉𝑇 is compact.
Remark 7. The subset 𝑉𝑇 depends on 𝑇. So we denote each Therefore, the Schauder fixed point theorem applies, and
element of 𝑉𝑇 by 𝑢(𝑇, ⋅); see Figure 1. hence the operator 𝐴 : 𝑉𝑇 → 𝑉𝑇 has at least one fixed point
𝑢0 (𝑇, ⋅) ∈ 𝑉𝑇 . Moreover, we can show the uniqueness of the
As it is mentioned in the introduction, the existence and fixed point; see Figure 2.
uniqueness of the solution to the BCS gap equation were
established for each fixed 𝑇 in the previous literature, and the The existence of the transition temperature 𝑇𝑐 is pointed
temperature dependence of the solution is not covered. We out in the previous papers [5–8]. In our case, it is defined as
therefore give another proof of the existence and uniqueness follows.
of the solution to the BCS gap equation (1) so as to show how
the solution varies with the temperature. More precisely, we Definition 9. Let 𝑢0 (𝑇, ⋅) ∈ 𝑉𝑇 be as in Theorem 8. The tran-
show that the solution belongs to 𝑉𝑇 . sition temperature 𝑇𝑐 stemming from the BCS gap equation
(1) is defined by
Theorem 8 (see [11, Theorem 2.2]). Assume condition (9)
on 𝑈(⋅, ⋅). Let 𝑇 ∈ [0, 𝜏2 ] be fixed. Then there is a unique 𝑇𝑐 = inf {𝑇 > 0 : 𝑢0 (𝑇, 𝑥) = 0 at all 𝑥 ∈ [𝜀, ℎ𝜔𝐷]} . (16)
Remark 10. Combining Definition 9 with Theorem 8 implies x fixed

that 𝜏1 ≤ 𝑇𝑐 ≤ 𝜏2 . For 𝑇 > 𝑇𝑐 , we set 𝑢0 (𝑇, 𝑥) = 0 at all
𝑥 ∈ [𝜀, ℎ𝜔𝐷]. Δ 2 (0)
4. Continuity of the Solution with respect to Δ 1 (0)

Δ 2 (T)
the Temperature
Δ 1 (T)
Let 𝑈0 > 0 be a constant satisfying 𝑈0 < 𝑈1 < 𝑈2 . An
argument similar to that in Proposition 2 gives that there is
a unique nonnegative solution Δ 0 : [0, 𝜏0 ] → [0, ∞) to the
simple gap equation
ℎ𝜔𝐷 0 T1 𝜏1 𝜏2
1
1 = 𝑈0 ∫ Temperature
𝜀 √𝜉2 + Δ 0 (𝑇)2
(17) Figure 3: The solution 𝑢0 is continuous on [0, 𝑇1 ] × [𝜀, ℎ𝜔𝐷 ].
√𝜉2 + Δ 0 (𝑇)2
× tanh 𝑑𝜉, 0 ≤ 𝑇 ≤ 𝜏0 .
2𝑇 Theorem 14 (see [12, Theorem 1.2]). Assume (9). Let 𝑢0 be as
Here, 𝜏0 > 0 is defined by in Theorem 8 and 𝑉 as in (21). Then 𝑢0 ∈ 𝑉. Consequently, the
gap function 𝑢0 is continuous on [0, 𝑇1 ] × [𝜀, ℎ𝜔𝐷].
ℎ𝜔𝐷
1 𝜉
1 = 𝑈0 ∫ tanh 𝑑𝜉. (18) Proof. We define a nonlinear integral operator 𝐵 on 𝑉 by
𝜀 𝜉 2𝜏0
We set Δ 0 (𝑇) = 0 for 𝑇 > 𝜏0 . A straightforward calculation ℎ𝜔𝐷
𝑈 (𝑥, 𝜉) 𝑢 (𝑇, 𝜉)
gives the following. 𝐵𝑢 (𝑇, 𝑥) = ∫
𝜀 √𝜉2 + 𝑢(𝑇, 𝜉)2
Lemma 11. (a) 𝜏0 < 𝜏1 < 𝜏2 . (22)
(b) If 0 ≤ 𝑇 < 𝜏0 , then 0 < Δ 0 (𝑇) < Δ 1 (𝑇) < Δ 2 (𝑇). √𝜉2 + 𝑢(𝑇, 𝜉)2
(c) If 𝜏0 ≤ 𝑇 < 𝜏1 , then 0 = Δ 0 (𝑇) < Δ 1 (𝑇) < Δ 2 (𝑇). × tanh 𝑑𝜉,
2𝑇
(d) If 𝜏1 ≤ 𝑇 < 𝜏2 , then 0 = Δ 0 (𝑇) = Δ 1 (𝑇) < Δ 2 (𝑇).
(e) If 𝜏2 ≤ 𝑇, then 0 = Δ 0 (𝑇) = Δ 1 (𝑇) = Δ 2 (𝑇). where 𝑢 ∈ 𝑉.
Clearly, 𝑉 is a closed subset of the Banach space
Remark 12. Let the functions Δ 𝑘 (𝑘 = 0, 1, 2) be as above. For 𝐶([0, 𝑇1 ] × [𝜀, ℎ𝜔𝐷]). A straightforward calculation gives that
each Δ 𝑘 , there is the inverse Δ−1
𝑘 : [0, Δ 𝑘 (0)] → [0, 𝜏𝑘 ]. Here, the operator 𝐵 : 𝑉 → 𝑉 is contractive as long as (20) holds
true. Therefore, the Banach fixed-point theorem applies, and
√(ℎ𝜔𝐷 − 𝜀𝑒1/𝑈𝑘 ) (ℎ𝜔𝐷 − 𝜀𝑒−1/𝑈𝑘 ) hence the operator 𝐵 : 𝑉 → 𝑉 has a unique fixed point
Δ 𝑘 (0) = , (19) 𝑢0 ∈ 𝑉. The solution 𝑢0 ∈ 𝑉 to the BCS gap equation is thus
sinh (1/𝑈𝑘 )
continuous both with respect to the temperature and with
and Δ 0 (0) < Δ 1 (0) < Δ 2 (0). respect to the energy 𝑥; see Figure 3.
We introduce another temperature. Let 𝑇1 satisfy 0 <

Acknowledgment
𝑇1 < Δ−1
0 (Δ 0 (0)/2) and
Shuji Watanabe is supported in part by the JSPS Grant-in-Aid
Δ 0 (0) Δ (0)
tanh −1 0 for Scientific Research (C) 24540112.
4Δ−1
2 (Δ 0 (𝑇1 )) 4Δ 2 (Δ 0 (𝑇1 ))
(20)
1 4ℎ2 𝜔𝐷2
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http://dx.doi.org/10.1155/2013/490689
Research Article
Numerical Solution for IVP in Volterra Type Linear
Integrodifferential Equations System
F. Ghomanjani,1 A. KJlJçman,2 and S. Effati1

1
Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
2
Department of Mathematics, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
Correspondence should be addressed to A. Kılıçman; kilicman@yahoo.com
Received 23 May 2013; Accepted 9 July 2013
Copyright © 2013 F. Ghomanjani et al. This is an open access article distributed under the Creative Commons Attribution License,
A method is proposed to determine the numerical solution of system of linear Volterra integrodifferential equations (IDEs) by using
Bezier curves. The Bezier curves are chosen as piecewise polynomials of degree n, and Bezier curves are determined on [𝑡0 , 𝑡𝑓 ] by
𝑛 + 1 control points. The efficiency and applicability of the presented method are illustrated by some numerical examples.
1. Introduction IDEs (see [13, 14]). Rashidinia and Tahmasebi developed and
modified Taylor series method (TSM) introduced in [15] to
Integrodifferential equations (IDEs) have been found to solve the system of linear Volterra IDEs.
describe various kinds of phenomena, such as glass forming In the present work, we suggest a technique similar to the
process, dropwise condensation, nanohydrodynamics, and one which was used in [16] for solving the system of linear
wind ripple in the desert (see [1, 2]). Volterra IDEs in the following form:
There are several numerical and analytical methods for
solving IDEs. Some different methods are presented to 𝑛 𝛼𝑚𝑖 𝑛 𝑡 𝛼𝑚𝑖
solve integral and IDEs in [3, 4]. Maleknejad et al. [5] (𝑗) (𝑗)
∑ ∑ 𝑝𝑚𝑖𝑗 (𝑡) 𝑦𝑖 (𝑡) + ∑ ∫ (𝑘𝑚𝑖 (𝑡, 𝑥) ∑ 𝑦𝑖 (𝑥)) 𝑑𝑥
used rationalized Haar functions method to solve the linear 𝑖=1 𝑗=0 𝑖=1 𝑡0 𝑗=0 (1)
IDEs system. Linear IDEs system has been solved by using
Galerkin methods with the hybrid Legendre and block- = 𝑓𝑚 (𝑡) , 𝑚 = 1, 2, . . . , 𝑛, 𝑡0 ≤ 𝑡 ≤ 𝑡𝑓 ,
Pulse functions on interval [0, 1) (see [6]). Yusufoğlu [7]
presented an application of He’s homotopy perturbation with the initial conditions
(HPM) method to solve the IDEs system. He’s variational
iteration method has been used for solving two systems of 𝑦𝑖(0) (𝑡0 ) = 𝑐𝑖0 ,
(𝛼𝑚𝑖 −1)
𝑦𝑖(1) (𝑡0 ) = 𝑐𝑖1 , . . . , 𝑦𝑖 (𝑡0 ) = 𝑐𝑖(𝛼𝑚𝑖 −1) ,
Volterra integrodifferential equations (see [8]). Arikoglu and
(2)
Ozkol [9] presented differential transform method (DTM) for
integrodifferential and integral equation systems. He’s homo- (𝑗)
topy perturbation (HPM) method was proposed for system of where 𝑦𝑖 (𝑡) stands for 𝑗th-order derivative of 𝑦𝑖 (𝑡). 𝑓𝑚 (𝑡),
integrodifferential equations (see [10]). A numerical method 𝑘𝑚𝑖 (𝑡, 𝑥), and 𝑝𝑚𝑖𝑗 (𝑡) are known functions (𝑚, 𝑖 = 1, 2, . . .,
based on interpolation of unknown functions at distinct 𝑛; 𝑗 = 0, 1, . . . , 𝛼𝑚𝑖 ), and 𝑡0 , 𝑡𝑓 , and 𝑐𝑖𝑗 (𝑖 = 1, 2, . . . , 𝑛; 𝑗 =
interpolation points has been introduced for solving linear 0, 1, . . . , 𝛼𝑚𝑖 − 1) are appropriate constants.
IDEs system with initial values (see [11]). Recently, Biazar The current paper is organized as follows. In Section 2,
introduced a new modification of homotopy perturbation function approximation will be introduced. Numerical exam-
method (NHPM) to obtain the solution of linear IDEs system ples will be stated in Section 3. Finally, Section 4 will give a
(see [12]). Taylor expansion method has been used for solving conclusion briefly.
2.6 0.8
2.4 0.7
2.2 0.6
2 0.5
1.8 0.4
1.6 0.3
1.4 0.2
1.2 0.1
1 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
t t
Approximate y1
Approximate y2
Exact y1
Exact y2
Figure 1: The graph of approximated 𝑦1 (𝑡) for Example 1. Figure 2: The graph of approximated 𝑦2 (𝑡) for Example 1.
2. Function Approximation where ‖ ⋅ ‖ is the Euclidean norm. Our aim is to solve the
following problem over the interval [𝑡0 , 𝑡𝑓 ]:
Our strategy is to use Bezier curves to approximate the
solutions 𝑦𝑖 (𝑡), for 1 ≤ 𝑖 ≤ 𝑛, which are given below. min 𝑅 (𝑡)
Define the Bezier polynomials of degree 𝑁 that approximate,
respectively, the actions of 𝑦𝑖 (𝑡) over the interval [𝑡0 , 𝑡𝑓 ] as s.t. 𝑦𝑖(0) (𝑡0 ) = 𝑐𝑖0 , (7)
follows:
(𝛼𝑚𝑖 −1)
𝑁
𝑡 − 𝑡0 𝑦𝑖(1) (𝑡0 ) = 𝑐𝑖1 , . . . , 𝑦𝑖 (𝑡0 ) = 𝑐𝑖(𝛼𝑚 −1) .
𝑦𝑖 (𝑡) = ∑𝑎𝑟𝑖 𝐵𝑟,𝑁 ( ), (3) 𝑖
𝑟=0 ℎ The mathematical programming problem (7) can be

solved by many subroutine algorithms, and we used Maple
where ℎ = 𝑡𝑓 − 𝑡0 and 𝑎𝑟 is the control point of Bezier curve, 12 to solve this optimization problem.
and
Remark 1. In Chapter 1 of [19], it was proved that 𝑁 satisfies
𝑡 − 𝑡0 𝑁 1 𝑁−𝑟 𝑟
𝐵𝑟,𝑁 ( ) = ( ) 𝑁 (𝑡𝑓 − 𝑡) (𝑡 − 𝑡0 ) (4) 𝑆
ℎ 𝑟 ℎ 𝑁> , (8)
𝛿2 𝜖
is the Bernstein polynomial of degree 𝑁 over the interval where 𝑆 = ‖𝑦𝑖 (𝑡)‖, and because of this reason that 𝑦𝑖 (𝑡) is
[𝑡0 , 𝑡𝑓 ] (see [17]). By substituting (3) in (2), 𝑅𝑚 (𝑡) can be uniformly continuous on [𝑡0 , 𝑡𝑓 ], we have 𝑠, 𝑡 ∈ [𝑡0 , 𝑡𝑓 ] that
defined for 𝑡 ∈ [𝑡0 , 𝑡𝑓 ] as |𝑡 − 𝑠| < 𝛿 and −(𝜖/2) < 𝑦𝑖 (𝑡) − 𝑦𝑖 (𝑠) < 𝜖/2, for more details
see [19].
𝑛 𝛼𝑚𝑖
(𝑗)
𝑅𝑚 (𝑡) = ∑ ∑ 𝑝𝑚𝑖𝑗 (𝑡) 𝑦𝑖 (𝑡)
𝑖=1 𝑗=0 3. Applications and Numerical Results
𝑛 𝑡 𝛼𝑚𝑖 Consider the following examples which can be solved by
(𝑗)
(5)
+ ∑ ∫ (𝑘𝑚𝑖 (𝑡, 𝑥) ∑𝑦𝑖 (𝑥)) 𝑑𝑥 − 𝑓𝑚 (𝑡) , using the presented method.
𝑖=1 0 𝑗=0
Example 1. Consider a system of third-order linear Volterra
IDEs on the interval [0, 1] (see [4]):
𝑚 = 1, 2, . . . , 𝑛,
𝑦1󸀠󸀠 (𝑡) + 𝑡2 𝑦1 (𝑡) − 𝑦2󸀠󸀠 (𝑡)
where (2) is satisfied. The convergence was proved in the
approximation with Bezier curves when the degree of the 𝑡
approximate solution, 𝑁, tends to infinity (see [18]). + ∫ ((𝑡 − 𝑥) 𝑦1 (𝑥) + 𝑦2 (𝑥)) 𝑑𝑥 = 𝑔1 (𝑡) ,
0
Now, the residual function is defined over the interval (9)
[𝑡0 , 𝑡𝑓 ] as follows: 4𝑡3 𝑦1󸀠 (𝑡) + 6𝑡2 𝑦1 (𝑡) + 𝑦2󸀠󸀠󸀠 (𝑡)
𝑡 𝑛
𝑓
󵄩 󵄩2 𝑡
𝑅 (𝑡) = ∫ ∑ 󵄩󵄩󵄩𝑅𝑚 (𝑡)󵄩󵄩󵄩 𝑑𝑡, (6) + ∫ (𝑦1 (𝑥) + (𝑡 + 𝑥) 𝑦2 (𝑥)) 𝑑𝑥 = 𝑔2 (𝑡) ,
𝑡 0 𝑚=1 0
Table 1: Computed errors for Example 1. With 𝑁 = 5, the computed errors are shown in Table 2
which show the high accuracy of the proposed method.
𝑡 Absolute error for 𝑦1 (𝑡) Absolute error for 𝑦2 (𝑡)
0.0 0.000000 0.0000000000
0.2 1.4801 × 10−10 2.2475 × 10−11 4. Conclusions
0.4 0.162735585 × 10−5 3.12780502 × 10−7 In this paper, Bernstein’s approximation is used to approx-
0.6 0.251133963 × 10−5 0.1536077787 × 10−5 imate the solution of linear Volterra IDEs. In this method,
0.8 1.8337 × 10−10 0.8864238659 × 10−5 we approximate our unknown function with Bernstein’s
1.0 4.5905 × 10−10 7.897 × 10−12 approximation. The present results show that Bernstein’s
approximation method for solving linear Volterra IDEs is
Table 2: Computed errors for Example 2. very effective and simple, and the answers are trusty, and
their accuracy is high, and we can execute this method in a
t Absolute error for 𝑦1 (𝑡) Absolute error for 𝑦2 (𝑡) computer simply. The numerical examples support this claim.
0.0 0.000000 0.0000000000
0.2 3.840 × 10−11 1.5360 × 10−11
Acknowledgment
0.4 0.5791064832 × 10−3 0.156041748480 × 10−3
0.6 0.17373195072 × 10−2 0.156041748480 × 10−3 The authors are very grateful to the referees for their valuable
0.8 0.69492781056 × 10−2 1.5360 × 10−11 suggestions and comments that improved the paper.
1.0 0.000 0.000
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http://dx.doi.org/10.1155/2013/670847
Research Article
Analytical and Multishaped Solitary Wave Solutions for
Extended Reduced Ostrovsky Equation
Ben-gong Zhang
School of Mathematical and Computer Science, Wuhan Textile University, Wuhan 430200, China
Correspondence should be addressed to Ben-gong Zhang; benyan1219@126.com
Received 5 June 2013; Accepted 25 July 2013
Copyright © 2013 Ben-gong Zhang. This is an open access article distributed under the Creative Commons Attribution License,
We present the analytical and multishaped solitary wave solutions for extended reduced Ostrovsky equation (EX-ROE). The exact
solitary (traveling) wave solutions are expressed by three types of functions which are hyperbolic function solution, trigonometric
function solution, and rational solution. These results generalized the previous results. Multishape solitary wave solutions such as
loop-shaped, cusp-shaped, and hump-shaped can be obtained as well when the special values of the parameters are taken. The
(𝐺󸀠 /𝐺)-expansion method presents a wide applicability for handling nonlinear partial differential equations.
1. Introduction Just as it mentioned in [5] and the reference therein, when 𝛿 =

−1, (4) is referred to the Ostrovsky-Hunter equation (OHE).
The well-known Ostrovsky equation [1] When 𝛿 = 1, (4) is referred to the Vakhnenko equation
(𝑢𝑡 + 𝑐0 𝑢𝑥 + 𝛼𝑢𝑢𝑥 + 𝛽𝑢𝑥𝑥𝑥 )𝑥 = 𝛾𝑢, (1) (VE), which is in order to model the propagation of waves
in a relaxing medium [6, 7]. Parkes [3] pointed out that (4) is
where 𝑐0 is the velocity of dispersiveness linear waves, 𝛼 is the invariant under the transformation
nonlinear coefficient, and 𝛽 and 𝛾 are dispersion coefficients,
is a model for weakly nonlinear surface and internal waves in
a rotating ocean. 𝑢 󳨀→ −𝑢, 𝑡 󳨀→ −𝑡, 𝛿 󳨀→ −𝛿, (5)
For long waves, for which high-frequency dispersion is
negligible, 𝛽 = 0, and (1) becomes the so-called reduced
Ostrovsky equation (ROE) [2] so that the solutions of the OHE and VE are related in a simple
(𝑢𝑡 + 𝑐0 𝑢𝑥 + 𝛼𝑢𝑢𝑥 )𝑥 = 𝛾𝑢. (2) way.
The purpose of this paper is to study the extended reduced
Parkes [3] has studied (2) and found its periodic and Ostrovsky equation (EX-ROE):
solitary traveling wave solutions.
In fact, by applying the following transformation [4]:
𝑢 𝑡 (𝑥 + 𝑐0 𝑡) 𝜕 1
𝑢 󳨀→ , 𝑡 󳨀→ , 𝑥 󳨀→ (D2 𝑢 + 𝑝𝑢2 + 𝛽𝑢) + 𝑞D𝑢 = 0, (6)
(3) 𝜕𝑥 2
𝛼 √󵄨󵄨󵄨󵄨𝛾󵄨󵄨󵄨󵄨 √󵄨󵄨󵄨󵄨𝛾󵄨󵄨󵄨󵄨
to (2), we obtain the ROE in the neat form where D is defined previous, 𝑝, 𝑞, and 𝛽 are arbitrary nonzero
𝜕 constants. It is originally derived by Morrison and Parkes
D𝑢 + 𝛿𝑢 = 0,
𝜕𝑥 [8] which dubbed it as modified generalized Vakhnenko
(4) equation (mGVE) when 𝑝 = 2𝑞. They found that not only
𝜕 𝜕 𝛾
where D := +𝑢 , 𝛿 = 󵄨󵄨 󵄨󵄨 = ±1. does it have loop soliton solutions, hump-like and cusp-like
𝜕𝑡 𝜕𝑥 󵄨󵄨𝛾󵄨󵄨 soliton solutions, but it also has 𝑁-soliton solutions.
In order to investigate mGVE’s 𝑁-soliton solution, Morri- 2. Description of the Modified

son and Parkes [8] considered a Hirota-Satsuma-type shallow (𝐺󸀠 /𝐺)-Expansion Method
water wave equation [9] of the form
∞
The (𝐺󸀠 /𝐺)-expansion method is first proposed by Wang
𝑈𝑋𝑋𝑇 + 𝑝𝑈𝑈𝑇 − 𝑞𝑈𝑋 ∫ 𝑈𝑇 (𝑋󸀠 , 𝑇) d𝑋󸀠 + 𝛽𝑈𝑇 + 𝑞𝑈𝑋 = 0, et al. [28]. The useful (𝐺󸀠 /𝐺)-expansion method is then
𝑋 widely used by many authors [29–32]. Then it is modified in
(7) [33–35]. The main steps are as follows.
Suppose that a nonlinear equation is given by
where 𝑝 ≠0, 𝑞 ≠0, and 𝛽 is arbitrary constant. By using the
transformation 𝑃1 (𝑢, 𝑢𝑡 , 𝑢𝑥 , 𝑢𝑡𝑡 , 𝑢𝑥𝑡 , 𝑢𝑥𝑥 , . . .) = 0, (10)
𝑋
𝑥=𝑇+∫ 𝑈𝑇 (𝑋󸀠 , 𝑇) d𝑋󸀠 + 𝑥0 , where 𝑢 = 𝑢(𝑥, 𝑡) is an unknown function and 𝑃 is a poly-
−∞ (8) nomial in 𝑢 = 𝑢(𝑥, 𝑡) and its partial derivatives, in which the
𝑡 = 𝑋, 𝑢 (𝑥, 𝑡) = 𝑈 (𝑋, 𝑇) , highest-order derivatives and nonlinear terms are involved.
In the following we give the main steps of the (𝐺󸀠 /𝐺)-
where 𝑥0 is a constant, (7) yields (6). So (6) and (7) are equiv- expansion method.
alent to each other under the transformation (8). Specifically,
in (7), when 𝑝 = 2𝑞 and 𝛽 = −1, it was discussed by Ablowitz Step 1. The traveling wave variable 𝑢(𝑥, 𝑡) = 𝑢(𝜉), 𝜉 = 𝑥 − 𝑐𝑡,
et al. [10] and was shown to be integrable by inverse scatting where 𝑐 is a constant, permits us to reduce (10) to an ODE for
method. When 𝑝 = 𝑞 and 𝛽 = −1, it was discussed by 𝑢 = 𝑢(𝜉) in the form
Hirota and Satsuma [11] and was shown to be integrable using
Hirota’s bilinear technique. In [12], the authors referred to (6) 𝑃2 (𝑢, −𝑐𝑢󸀠 , 𝑢󸀠 , 𝑐2 𝑢󸀠󸀠 , −𝑐𝑢󸀠󸀠 , 𝑢󸀠󸀠 , . . .) = 0. (11)
with 𝑝 = 𝑞 = 1 and 𝛽 an arbitrary nonzero constant as the
generalized Vakhnenko equation (GVE). In fact, when 𝑝 = 𝑞 Step 2. Suppose that the solution of (10) can be expressed by
and 𝛽 = 0, (6) can be written as a polynomial in (𝐺󸀠 /𝐺) as follows:
−𝑚 −(𝑚−1)
𝜕𝑢 𝜕 𝐺󸀠 𝐺󸀠
( + D) ( D𝑢 + 𝑝𝑢) = 0. (9) 𝑢 (𝜉) = 𝛼−𝑚 ( ) + 𝛼−(𝑚−1) ( ) + ⋅⋅⋅
𝜕𝑥 𝜕𝑥 𝐺 𝐺
(12)
Clearly, solutions of the ROE are also solutions of (9) with 𝑚−1 𝑚
𝐺󸀠 𝐺󸀠
𝑝 = ±1. So for arbitrary 𝑝, 𝑞, and 𝛽, if we obtain the solutions + 𝛼𝑚−1 ( ) + 𝛼𝑚 ( ) ,
of EX-ROE, then we can also obtain the solutions of VE, GVE, 𝐺 𝐺
mGVE, ROE, and OHE by taking the special values of 𝑝, 𝑞,
and 𝛽. where 𝐺 = 𝐺(𝜉) satisfies the second-order linear ordinary
The EX-ROE has been studied by several researchers. For differential equation (LODE) in the form
example, Liu et al. [13] used Jacobi elliptic function method
𝐺󸀠󸀠 + 𝜆𝐺󸀠 + 𝜇𝐺 = 0, (13)
to obtain exact double periodic wave solutions and solitary
wave solutions. Parkes [4] constructed periodic and solitary where 𝛼−𝑚 , . . . , 𝛼𝑚 , 𝜆, and 𝜇 are constants to be determined
wave solutions of EX-ROE and gave the categorization of later. The unwritten part in (12) is also a polynomial in (𝐺󸀠 /𝐺),
the solutions. Xie and Cai [14] used the bifurcation method but the degree of which is generally equal to or less than 𝑚−1.
of dynamic systems and simulation method of differential The positive integer 𝑚 can be determined by considering the
equations to get exact compacton and generalized kink wave homogeneous balance between the highest-order derivatives
solutions of EX-ROE. Stepanyants [15] applied the qualitative and nonlinear terms appearing in (11).
theory of differential equations to give a full classification of
its solutions. Step 3. Substituting (12) into (11) and using (13), collecting
Recently, there are many methods being proposed to all terms with the same order of (𝐺󸀠 /𝐺) together, and then
study the traveling wave solutions of nonlinear partial differ- equating each coefficient of the resulting polynomial to zero
ential equations which are derived from physics, for example, yields a set of algebraic equations for 𝛼𝑚 , 𝛼𝑚−1 , . . . , 𝛼−𝑚 , 𝑐, 𝜆,
[16–27]. As well as these methods, there are still many other and 𝜇.
methods; we cannot list all of them. Here we will use modified
(𝐺󸀠 /𝐺)-expansion method to investigate EX-ROE. As a result, Step 4. Since the general solutions of (13) have been well
three types of traveling wave solutions are were obtained. known for us, then substituting 𝛼𝑚 , 𝛼𝑚−1 , . . . , 𝛼−𝑚 and 𝑐 and
When the special values of the parameters are taken, they are the general solutions of (13) into (12) we have more traveling
reduced to some previous results which obtained by an other wave solutions of the nonlinear differential equation (10).
method. The main idea of (𝐺󸀠 /𝐺)-expansion method is to use
The rest of the paper is organized as follows. In Section 2, an integrable ODE to expand a solution to a nonlinear
we present a methodology of the modified (𝐺󸀠 /𝐺)-expansion partial differential equation (PDE) as a polynomial or rational
method. In Section 3, we apply the method to the extended function of the solution of the ODE. However, such an idea
reduced Ostrovsky equation. In Section 4, some conclusions was also presented in [36–38]. The method used in this paper
are given. can be also thought of as the application of transformed
rational function method used in [37] in some sense. Maybe By using (18) and (19), it is derived that
the similar results can be obtained by using these very closely
related methods. We plan to further study the EX-ROE in
near future by using the methods proposed in [36–38]. We −2 −1
𝐺󸀠 𝐺󸀠
hope we can find much more interesting properties and new 𝑊𝜉 = 𝜇𝛼−1 ( ) + 𝜆𝛼−1 ( ) + 𝛼−1
phenomenon of this equation. 𝐺 𝐺
(20)
2
𝐺󸀠 𝐺󸀠
− 𝛼1 𝜇 − 𝜆𝛼1 ( ) − 𝛼1 ( ) ,
3. Exact Traveling Wave Solutions of 𝐺 𝐺
the Extended Reduced Ostrovsky Equation −4 −3
𝐺󸀠 𝐺󸀠
󸀠
In this section, we will use the (𝐺 /𝐺)-expansion method 𝑊𝜉2 = 𝜇2 𝛼−1
2
( ) 2
+ 2𝜆𝛼−1 𝜇( )
𝐺 𝐺
to the extended reduced Ostrovsky equation to get exact
−2
traveling wave solutions. 2 𝐺󸀠
First, in order to get traveling wave solutions, we need + (2𝛼−1 𝜇 + 𝜆2 𝛼−1
2
− 2𝜇𝛼−1 𝛼− 1) ( )
𝐺
some transformation. Recall that in Section 1 we have stated
that EX-ROE is equivalent to a Hirota-Satsuma-type shallow −1
2 𝐺󸀠 2
water wave equation (7) under the transformation of (8). So + (2𝜆𝛼−1 − 4𝜇𝜆𝛼−1 𝛼1 ) ( ) + 𝛼−1 − 4𝜇𝛼−1 𝛼1
here we introduce a new variable 𝑊 defined by 𝐺
𝐺󸀠
− 2𝜆2 𝛼−1 𝛼1 + 𝛼12 𝜇2 + 2𝜆𝛼12 𝜇 ( )
𝑈 = 𝑊𝑋 . (14) 𝐺
2 3 4
𝐺󸀠 𝐺󸀠 𝐺󸀠
Substituting (14) into (7) yields + (2𝛼12 𝜇 + 𝜆2 𝛼12 ) ( ) + 2𝜆𝛼12 ( ) + 𝛼12 ( ) ,
𝐺 𝐺 𝐺
(21)
𝑊𝑋𝑋𝑋𝑇 + 𝑝𝑊𝑋 𝑊𝑋𝑇 + 𝑞𝑊𝑋𝑋 𝑊𝑇 + 𝛽𝑊𝑋𝑇 + 𝑞𝑊𝑋𝑋 = 0. −4 −3
𝐺󸀠 𝐺󸀠
(15) 𝑊3𝜉 = 6𝛼−1 𝜇3 ( ) + 12𝛼−1 𝜆𝜇2 ( )
𝐺 𝐺
−2
Now giving the traveling wave transformation 𝑊(𝑋, 𝑇) = 𝐺󸀠
𝑊(𝜉), 𝜉 = 𝑋 − 𝑐𝑇, where 𝑐 is wave speed. Substituting them − (8𝜇2 𝛼−1 + 7𝛼−1 𝜆2 𝜇) ( )
𝐺
into (15) and integrating once, we have
−1
𝐺󸀠
− (8𝛼−1 𝜆𝜇 + 𝛼−1 𝜆3 ) ( )
1 𝐺
𝑐1 + 𝑐𝑊3𝜉 + 𝑐 (𝑝 + 𝑞) 𝑊𝜉2 + (𝑐𝛽 − 𝑞) 𝑊𝜉 = 0, (16)
2
− (2𝛼−1 𝜇 + 𝜆2 𝛼−1 ) − (2𝛼1 𝜇2 + 𝜆2 𝛼1 𝜇)
where 𝑐1 is integral constant that is to be determined later. 2
𝐺󸀠 𝐺󸀠
Considering the homogeneous balance between 𝑊3𝜉 and − (8𝛼1 𝜆𝜇 + 𝛼1 𝜆3 ) ( ) − (8𝜇𝛼1 + 7𝛼1 𝜆2 ) ( )
2 𝐺 𝐺
𝑊𝜉 , we have
3 4
𝐺󸀠 𝐺󸀠
− 12𝛼1 𝜆( ) − 6𝛼1 ( ) .
𝑚 + 3 = 2𝑚 + 2 󳨐⇒ 𝑚 = 1. (17) 𝐺 𝐺
(22)
We suppose that
By substituting (20)–(22) into (16) and collecting all terms
−1 with the same power of (𝐺󸀠 /𝐺) together, the left-hand sides of
𝐺󸀠 𝐺󸀠 (16) are converted into the polynomials in (𝐺󸀠 /𝐺). Equating
𝑊 (𝜉) = 𝛼−1 ( ) + 𝛼0 + 𝛼1 ( ), (18)
𝐺 𝐺 the coefficients of the polynomials to zero yields a set of
simultaneous algebraic equations for 𝛼−1 , 𝛼0 , 𝛼1 , 𝜆, 𝑐, 𝑐1 , and
𝜇 as follows (denote 𝐴 for (𝐺󸀠 /𝐺)):
where the 𝐺 = 𝐺(𝜉) satisfies the second-order LODE,
𝐺󸀠󸀠 + 𝜆𝐺󸀠 + 𝜇𝐺 = 0, (19) 𝑐 (𝑝 + 𝑞) 𝜇2 𝛼−1

2
𝐴−4 : 6𝑐𝛼−1 𝜇3 + = 0,
2
and 𝛼−1 , 𝛼0 , 𝛼1 , 𝜆, and 𝜇 are constants to be determined later. 𝐴−3 : 12𝑐𝜆𝜇2 𝛼−1 + 𝑐 (𝑝 + 𝑞) 𝜆𝜇𝛼−1
2
= 0,
𝐴−2 : (𝑞 − 𝑐𝛽) 𝜇𝛼−1 or

−1
𝑐 (𝑝 + 𝑞) (𝜆2 𝛼−1
2
+ 2𝜇𝜆2−1 − 2𝜇2 𝛼1 𝛼−1 ) 12𝜇 𝐺󸀠
− 𝑊 (𝑋, 𝑇) = 𝑊 (𝜉) = − ( ) + 𝛼0 , (27)
2 (𝑝 + 𝑞) 𝐺
− 𝑐 (7𝜆2 𝜇𝛼−1 + 8𝜇2 𝛼−1 ) = 0, where 𝐺 satisfies (19), 𝜉 = 𝑋 − 𝑞𝑇/(𝛽 + 𝜆2 − 4𝜇), and 𝛼0 is an
arbitrary constant.
𝐴−1 : (𝑐𝛽 − 𝑞) 𝜆𝛼−1 𝜆 + 𝑐 (𝑝 + 𝑞) (𝜆𝛼−1
2
− 2𝜆𝜇𝛼𝛼−1 ) Since the general solutions 𝐺 = 𝐺(𝜉) (hence 𝐺󸀠 =
d𝐺/d𝜉) of ODE (19) have been known for us, substituting
+ 𝑐 (𝜆3 𝛼−1 + 8𝜆𝜇𝛼−1 ) = 0, the solutions of (19) into (24) and (25), we have the general
traveling wave solutions of (15) as follows.
𝐴0 : 𝑐1 + 𝑐 (𝜆2 𝛼−1 + 2𝜇𝛼−1 )
2 2 2
Case 1. When 𝜆2 − 4𝜇 > 0, then we have the following exact
𝑐 (2𝛼1 𝜇 + 𝜆 𝛼1 𝜇) + 𝑐 (𝑝 + 𝑞) 𝛼−1 traveling wave solution of (15):
−
2
𝑊1 (𝑋, 𝑇)
𝑐𝛼2 𝜇2 (𝑝 + 𝑞)
− 2𝑐𝜇𝛼𝛼−1 (𝑝 + 𝑞) + 1 − (𝑐𝛽 − 𝑞) 𝜇𝛼1
2 6√𝜆2 − 4𝜇
= 𝑊1 (𝜉) =
− 𝑐𝜆2 𝛼1 𝛼−1 (𝑝 + 𝑞) + 𝛼−1 (𝑐𝛽 − 𝑞) = 0, 𝑝+𝑞
𝐴1 : (𝑞 − 𝑐𝛽) 𝛼1 𝜆 + 𝑐 (𝑝 + 𝑞) (𝜆𝜇𝛼12 − 2𝜆𝛼1 𝛼−1 ) 1

× ((𝐴 1 cosh ( √𝜆2 − 4𝜇𝜉))
2
− 𝑐 (𝜆3 𝛼1 + 8𝜆𝜇𝛼1 ) = 0, 1
+𝐴 2 sinh ( √𝜆2 − 4𝜇𝜉)) (28)
2
𝐴2 : (𝑞 − 𝑐𝛽) 𝛼1 + 𝑐 (𝑝 + 𝑞)
1
× (𝐴 1 sinh ( √𝜆2 − 4𝜇𝜉)
× (𝜆2 𝛼12 + 2𝑎𝛼12 𝜇 − 2𝛼1 𝛼−1 − 𝑐 (7𝜆2 𝛼1 + 8𝜇𝛼1 )) 2
−1
= 0, 1
+𝐴 2 cosh ( √𝜆2 − 4𝜇𝜉)) )
2
𝐴3 : − 12𝑐𝜆𝛼1 + 𝑐 (𝑝 + 𝑞) 𝜆𝛼12 = 0,
6𝜆
−
𝑐 (𝑝 + 𝑞) 𝛼12 (𝑝 + 𝑞) + 𝛼0
𝐴4 : − 6𝑐𝛼1 + = 0.
2 or
(23)
𝑊2 (𝑋, 𝑇)
Solving the algebraic equations above yields
= 𝑊2 (𝜉)
12
𝛼1 = ,
(𝑝 + 𝑞)
= −24𝜇 × ( (𝑝 + 𝑞)
𝑞 (24)
𝑐= ,
(𝛽 + 𝜆2 − 4𝜇)
× (√𝜆2 − 4𝜇
𝑐1 = 0, 𝛼−1 = 0,
1
or × ((𝐴 1 cosh ( √𝜆2 − 4𝜇𝜉)
2
12𝜇
𝛼−1 = − , 1
(𝑝 + 𝑞) +𝐴 2 sinh ( √𝜆2 − 4𝜇𝜉))
2
𝑞 (25)
𝑐= , 1
(𝛽 + 𝜆2 − 4𝜇) × (𝐴 1 sinh ( √𝜆2 − 4𝜇𝜉)
2
𝑐1 = 0, 𝛼1 = 0. −1
1
+𝐴 2 cosh ( √𝜆2 − 4𝜇𝜉)) )
Substituting system (24) and (25) into (18), we have the 2
formula of the solutions of (15) as follows: −1
12 𝐺 󸀠 −𝜆)) + 𝛼0 ,
𝑊 (𝑋, 𝑇) = 𝑊 (𝜉) = ( ) + 𝛼0 , (26)
(𝑝 + 𝑞) 𝐺 (29)
where 𝜉 = 𝑋 − 𝑞𝑇/(𝛽 + 𝜆2 − 4𝜇) and 𝛼0 , 𝐴 1 , 𝐴 2 are arbitrary or

constants.
24𝜇 (𝐴 1 + 𝐴 2 𝜉)
Case 2. When 𝜆2 − 4𝜇 < 0, then we have the following exact 𝑊6 (𝑋, 𝑇) = 𝑊6 (𝜉) = − + 𝛼0 ,
(𝑝 + 𝑞) [2𝐴 2 − 𝜆 (𝐴 1 + 𝐴 2 𝜉)]
traveling wave solution of (15):
(33)
𝑊3 (𝑋, 𝑇)
where 𝜉 = 𝑋 − 𝑞𝑇/(𝛽 + 𝜆2 − 4𝜇) and 𝛼0 , 𝐴 1 , 𝐴 2 are arbitrary
constants.
6√4𝜇 − 𝜆2 Now we will show how to get exact traveling wave
= 𝑊3 (𝜉) =
𝑝+𝑞 solutions of (6). From (8) and (14), the solution of EX-ROE
(6) is given in parametric form, with 𝑇 as the parameter, by
1 1
× ((−𝐴 1 sin ( √4𝜇 − 𝜆2 𝜉) + 𝐴 2 cos ( √4𝜇 − 𝜆2 𝜉))
2 2 𝑢 (𝑥, 𝑡) = 𝑈 (𝑡, 𝑇) , 𝑥 = 𝜃 (𝑡, 𝑇) , (34)
1
× (𝐴 1 cos ( √4𝜇 − 𝜆2 𝜉) where
2
−1
1 𝜃 (𝑋, 𝑇) = 𝑇 + 𝑊 (𝑋, 𝑇) + 𝑥0 .
+𝐴 2 sin ( √4𝜇 − 𝜆2 𝜉)) ) (35)
2
6𝜆 So by using (8), (14), (34), (35), (28), and (29), we obtain
− a parameterized hyperbolic-function-type traveling wave
(𝑝 + 𝑞) + 𝛼0
solution of (6) as follows:
(30)
or 𝑢1 (𝑥, 𝑡)
= 3 (𝐴22 − 𝐴21 ) (𝜆2 − 4𝜇)

𝑊4 (𝑋, 𝑇)
= 𝑊4 (𝜉) × ( (𝑝 + 𝑞)
= −24𝜇 (36)
1
× (𝐴 1 sinh ( √𝜆2 − 4𝜇𝜉)
2
× ((𝑝 + 𝑞) (√𝜆2 − 4𝜇
2 −1
1
1 +𝐴 2 cosh ( √𝜆2 − 4𝜇𝜉)) ) ,
× ((−𝐴 1 sin ( √4𝜇 − 𝜆2 𝜉) 2
2
𝑥 = 𝑇 + 𝑊1 (𝑡, 𝑇) + 𝑥0 ,
1
+𝐴 2 cos ( √4𝜇 − 𝜆2 𝜉))
2 or
1
× (𝐴 1 cos ( √4𝜇 − 𝜆2 𝜉) 𝑢2 (𝑥, 𝑡)
2
1 −1 = −12𝜇 (𝐴22 − 𝐴21 ) (𝜆2 − 4𝜇)
+𝐴 2 sin ( √4𝜇 − 𝜆2 𝜉)) )
2
−1 × ( (𝑝 + 𝑞)
− 𝜆)) + 𝛼0 ,
1
(31) × [(𝐴 1 √𝜆2 − 4𝜇 − 𝐴 2 𝜆) cosh ( √𝜆2 − 4𝜇𝜉)
2
where 𝜉 = 𝑋 − 𝑞𝑇/(𝛽 + 𝜆2 − 4𝜇) and 𝛼0 , 𝐴 1 , 𝐴 2 are arbitrary + (𝐴 2 √𝜆2 − 4𝜇 − 𝐴 1 𝜆)

constants.
2 −1
1
Case 3. When 𝜆2 − 4𝜇 = 0, then we have the following exact × sinh ( √𝜆2 − 4𝜇𝜉)] ) ,
rational solution of (15): 2
𝑥 = 𝑇 + 𝑊2 (𝑡, 𝑇) + 𝑥0 ,
12 𝐴2 6𝜆 (37)
𝑊5 (𝑋, 𝑇) = 𝑊5 (𝜉) = ( )− + 𝛼0
𝑝 + 𝑞 𝐴 1 + 𝐴 2𝜉 (𝑝 + 𝑞)
(32) where 𝜉 = 𝑡 − 𝑞𝑇/(𝛽 + 𝜆2 − 4𝜇) and 𝑥0 is an arbitrary constant.
By using (8), (14), (34), (35), (30), and (31), we obtain If we take 𝐴 1 = 0, 𝜇 = 0, 𝐴 2 ≠0 and 𝜆 > 0, then (36)
a parameterized trigonometric-function-type traveling wave yields the following solitary wave solution of (6):
solution of (6) as follows:
3𝜆2 1 𝑞
𝑢 (𝑥, 𝑡) = sech2 [ 𝜆 (𝑡 − 𝑇)] ,
𝑢3 (𝑥, 𝑡) 𝑝+𝑞 2 𝛽 + 𝜆2
(42)
6𝜆 1 𝑞
= −3 (𝐴22 + 𝐴21 ) (𝜆2 − 4𝜇) 𝑥=𝑇+ tanh [ 𝜆 (𝑡 − 𝑇)] + 𝑥0 .
(𝑝 + 𝑞) 2 𝛽 + 𝜆2
× ((𝑝 + 𝑞) Now we will give some discussion of the solitary wave
solution (42). Let 𝜆 = 2𝑘, 𝑥0 = 0; the solution (42) is reduced
1 (38) to the solution of (3.26) in [13] after correcting some minor
× (𝐴 1 cos ( √𝜆2 − 4𝜇𝜉) errors [4]. Now from (35), we introduce a new variable:
2
−1 6𝜆 1
1 2 𝜒 = 𝑥 − V𝑡 = −V (𝑋 − 𝑐𝑇) + tanh [ 𝜆 (𝑋 − 𝑐𝑇)] + 𝑥0 ,
+𝐴 2 sin ( √𝜆2 − 4𝜇𝜉)) ) , 𝑝+𝑞 2
2 (43)
𝑥 = 𝑇 + 𝑊3 (𝑡, 𝑇) + 𝑥0 , where V = 1/𝑐 = (𝜆2 + 𝛽)/𝑞. In [8], the authors considered
EX-ROE with 𝑝 = 2𝑞, 𝛽 ≠0 as mGVE and obtained 1-soliton
or
solution. In fact, if we take 𝑝 = 2𝑞, 𝜆 = 2𝑘, the solitary wave
solution (42) with (43) is reduced to the soliton solution (4.4)
𝑢4 (𝑥, 𝑡)
and (4.5) in [8]. From the above we can see that the solitary
= −12𝜇 (𝐴22 − 𝐴21 ) (𝜆2 − 4𝜇) wave solution (3.26) in [4] and the 1-soliton solution of mGVE
are just a special case of the solution (42) in this paper.
× ( (𝑝 + 𝑞)
4. Multishaped Solitary Wave Solutions
1 In [8, 13], the authors showed that the solutions of (4.4) and
× [ (−𝐴 2 √𝜆2 − 4𝜇 + 𝐴 1 𝜆) cos ( √𝜆2 − 4𝜇𝜉)
2 (4.5), (3.26) and (3.28) may be of different types, namely,
loops, cusps, or humps for different values of parameters 𝛽,
+ (𝐴 1 √𝜆2 − 4𝜇 + 𝐴 2 𝜆) 𝑘, 𝑝. Here we also show that by choosing different values of
the parameters 𝛽, 𝜆, 𝑝, 𝑞, different shape wave solutions can
2 −1 be obtained. As it is stated in Section 1, (9) reduces to VE
1
× sin ( √𝜆2 − 4𝜇𝜉)] ) , when 𝑝 = 𝑞 = 1, 𝛽 = 0. Taking solution (42) with (43), for
2
example, let 𝑝 = 𝑞 = 1, 𝛽 = 0, 𝜆 = 2𝑘; then it is reduced to
𝑥 = 𝑇 + 𝑊4 (𝑡, 𝑇) + 𝑥0 , one-loop soliton solution (3.4) and (3.5) in [39]. On the other
(39) hand, because the solutions of OHE and VE are connected in
a particularly simple way, if we take 𝑝 = 𝑞 = −1, 𝛽 = 0, 𝜆 = 2𝑘
in (42), we can obtain one-loop soliton solution of OHE.
where 𝜉 = 𝑡 − 𝑞𝑇/(𝛽 + 𝜆2 − 4𝜇) and 𝑥0 is an arbitrary constant.
From above analysis, one can clearly see that the solutions
By using (8), (14), (34), (35), (32), and (33), we obtain a
obtained in this paper are generalized for the previous results
parameterized rational-type traveling wave solution of (6) as
because here we only take the special case 𝐴 1 = 0, 𝜇 = 0,
follows:
𝐴 2 ≠0, 𝜆 > 0 and give special discussion of solution (42).
We conclude that if we take different values of the parameters
12𝐴22
𝑢5 (𝑥, 𝑡) = − , 𝐴 1 , 𝐴 2 , 𝜆, 𝜇, 𝑝, 𝑞, abundancy of types of exact solutions can
2
(𝑝 + 𝑞) (𝐴 1 + 𝐴 2 𝜉) (40) be obtained from solutions (36), (38), and (40). Here we omit
the detailed discussion.
𝑥 = 𝑇 + 𝑊5 (𝑡, 𝑇) + 𝑥0 , Instead, we give some discussion about solution (37). Sci-
ence from this solution, multishaped solitary wave solutions
or can be obtained. Suppose 𝐴 1 ≠0, 𝜇 < 0, 𝐴 2 = 0, 𝜆 = 0; we
reduce solution (37) to
48𝜇𝐴22
𝑢6 (𝑥, 𝑡) = − 2
, 12𝜇 𝑞
(𝑝 + 𝑞) (𝜆𝐴 1 + (−2 + 𝜆𝜉) 𝐴 2 ) (41) 𝑢 (𝑥, 𝑡) = sech2 [√−𝜇 (𝑡 − 𝑇)] ,
𝑝+𝑞 𝛽 − 4𝜇
𝑥 = 𝑇 + 𝑊6 (𝑡, 𝑇) + 𝑥0 , (44)
24√−𝜇 𝑞
𝑥=𝑇+ tanh [√−𝜇 (𝑡 − 𝑇)] + 𝑥0 .
(𝑝 + 𝑞) 𝛽 − 4𝜇
where 𝜉 = 𝑡 − 𝑞𝑇/(𝛽 + 𝜆2 − 4𝜇) and 𝑥0 is an arbitrary constant.
To our knowledge, these solutions are presented for the first We show that for different values of 𝛽, 𝜇, and 𝑝, the solution
time; they are new exact solutions of EX-ROE. (44) may be of different types. It also owns the property of
−1.0 −0.5−0.2 0.5 1.0 −3 −2 −1 −0.5 1 2 3 −3 −2 −1 −0.5 1 2 3

−0.4
−0.6 −1.0 −1.0
−0.8 −1.5 −1.5
−1.0 −2.0 −2.0
−1.2 −2.5 −2.5
(a) (b) (c)
Figure 1: The profile of solution (44) with 𝑝 = 3, 𝑞 = 1.5, 𝑡 = 0, and 𝑥0 = 0. For (a) loop-shaped 𝛽 = 0.005, 𝜇 = −0.5, (b) cusp-shaped
𝛽 = 0.05, 𝜇 = −1, and (c) hump-shaped 𝛽 = 0.5, 𝜇 = −1.
being loop-shaped, cusp-shaped and hump-shaped, as shown [6] V. A. Vakhnenko, “Solitons in a nonlinear model medium,” Jour-
in Figure 1. nal of Physics A, vol. 25, no. 15, pp. 4181–4187, 1992.
[7] V. O. Vakhnenko, “High-frequency soliton-like waves in a relax-
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5. Conclusion 2011–2020, 1999.
In this paper, we use (𝐺󸀠 /𝐺)-expansion method to study [8] A. J. Morrison and E. J. Parkes, “The 𝑁-soliton solution of the
extended reduced Ostrovsky equation. Several pairs of gen- modified generalised Vakhnenko equation (a new nonlinear
evolution equation),” Chaos, Solitons & Fractals, vol. 16, no. 1,
eralized traveling wave solutions are given directly. These
pp. 13–26, 2003.
solutions extend the previous results to more general cases. At
[9] A. Espinosa and J. Fujioka, “Hydrodynamic foundation and
the same time, multishaped wave solutions can be obtained if
Painlevé analysis of Hirota-Satsuma-type equations,” Journal of
the different parameters values are chosen. These explicit soli- the Physical Society of Japan, vol. 63, no. 4, pp. 1289–1294, 1994.
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[10] M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The
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[12] A. J. Morrison and E. J. Parkes, “The 𝑁-soliton solution of a
computer program such as MATHEMATICA and MAPLE.
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Many well-known nonlinear wave equations can be handled nal, vol. 43A, pp. 65–90, 2001, Integrable systems: linear and
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Acknowledgment solutions for a nonlinear evolution equation,” Chaos, Solitons &
Fractals, vol. 31, no. 5, pp. 1173–1180, 2007.
The authors thank anonymous referees for valuable sugges- [14] S. Xie and J. Cai, “Exact compacton and generalized kink wave
tions and comments which improve this paper readability solutions of the extended reduced Ostrovsky equation,” Com-
and convincibility. This paper is supported by the Starting munications in Nonlinear Science and Numerical Simulation, vol.
Research Founding of Wuhan Textile University. 14, no. 9-10, pp. 3561–3573, 2009.
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http://dx.doi.org/10.1155/2013/268902
Research Article
New Exact Solutions for a Generalized Double
Sinh-Gordon Equation
Gabriel Magalakwe and Chaudry Masood Khalique

International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences,
North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa
Correspondence should be addressed to Chaudry Masood Khalique; masood.khalique@nwu.ac.za
Received 26 June 2013; Revised 15 July 2013; Accepted 15 July 2013
Copyright © 2013 G. Magalakwe and C. M. Khalique. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We study a generalized double sinh-Gordon equation, which has applications in various fields, such as fluid dynamics, integrable
quantum field theory, and kink dynamics. We employ the Exp-function method to obtain new exact solutions for this generalized
double sinh-Gordon equation. This method is important as it gives us new solutions of the generalized double sinh-Gordon
equation.
1. Introduction (1) becomes the generalized sinh-Gordon equation [14, 15].

Furthermore, if 𝑛 = 𝑎 = 1 and 𝑏 = 2, (1) reduces to the sinh-
It is well known that finding exact travelling wave solutions of Gordon equation [16].
nonlinear partial differential equations (NLPDEs) is useful in
many scientific applications such as fluid mechanics, plasma Many authors have studied the generalized double sinh-
physics, and quantum field theory. Due to these applications Gordon equation (1). Travelling waves solutions of (1) were
many researchers are investigating exact solutions of NLPDEs obtained in [11] by using the tanh function method and the
since they play a vital role in the study of nonlinear phys- variable separable method. In [12] the method of bifurcation
ical phenomena. Finding exact solutions of such NLPDEs theory of dynamical system was used to prove the existence
provides us with a better understanding of the physical of periodic wave, solitary wave, kink and antikink wave, and
phenomena that these NLPDEs describe. Several techniques unbounded wave solutions of (1). It should be noted that
have been presented in the literature to find exact solutions solutions obtained in [12] were different the ones obtained
of the NLPDEs. These include the homogeneous balance in [11]. Recently, solitary and periodic waves solutions of
method, the Weierstrass elliptic function expansion method, (1) were found in [13] by employing (𝐺󸀠 /𝐺)-expansion
the 𝐹-expansion method, the (𝐺󸀠 /𝐺)-expansion method, method. It is further shown in [13] that solutions obtained by
the Exp-function method, the tanh function method, the using the (𝐺󸀠 /𝐺)-expansion method are more general than
extended tanh function method, and the Lie group method those given in [11], which were obtained by tanh function
[1–10]. method.
In this work, we study one such NLPDE, namely, the In this paper, we employ an entirely different method,
generalized double sinh-Gordon equation: known as the Exp-function method, to obtain new exact
solutions of the generalized sinh-Gordon equation (1). The
𝑢𝑡𝑡 − 𝑘𝑢𝑥𝑥 + 2𝛼 sinh (𝑛𝑢) + 𝛽 sinh (2𝑛𝑢) = 0, 𝑛 ≥ 1, paper is structured as follows. In Section 2, we obtain exact
(1) solutions of the generalized double sinh-Gordon equation (1)
which appears in many scientific applications [11–13]. It with the help of the Exp-function method. In Section 3 we
should be noted that when 𝑘 = 𝑎, 𝛼 = (1/2)𝑏, and 𝛽 = 0, present concluding remarks.
2. Exact Solutions of (1) Using where 𝑠𝑖 are coefficients only for simplicity. Balancing the
Exp-Function Method lowest order of Exp-function in (7), we have 2𝑑 + 3𝑞 = 4𝑑 + 𝑞,
which yields 𝑑 = 𝑞. For simplicity, we first set 𝑐 = 𝑝 = 1 and
In this section we employ the Exp-function method to 𝑑 = 𝑞 = 1. then (5) reduces to
solve the generalized double sinh-Gordon equation (1). This
method was introduced by He and Wu [17]. The Exp-function 𝑎1 exp (𝑧) + 𝑎0 + 𝑎−1 exp (−𝑧)
method results in the travelling wave solution based on 𝐻 (𝑧) = . (8)
𝑏1 exp (𝑧) + 𝑏0 + 𝑏−1 exp (−𝑧)
the assumption that the solution can be expressed in the
following form:
Inserting (8) into (4) and using Maple, we obtain
∑𝑑𝑛=−𝑐 𝑎𝑛 exp (𝑛𝑧) 1
𝐻 (𝑧) = , (2) [𝐶 exp (4𝑧) + 𝐶3 exp (3𝑧) + 𝐶2 exp (2𝑧)
∑𝑞𝑚=−𝑝 𝑏𝑚 exp (𝑚𝑧) 𝐵 4
+ 𝐶1 exp (𝑧) + 𝐶0 + 𝐶−1 exp (−𝑧)
where 𝑐, 𝑑, 𝑝, and 𝑞 are positive integers that can be
determined and 𝑎𝑛 and 𝑏𝑚 are unknown constants. According +𝐶−2 exp (−2𝑧) + 𝐶−3 exp (−3𝑧) + 𝐶−4 exp (−4𝑧)] = 0,
to Exp-function method, we introduce the travelling wave
(9)
substitution 𝑢(𝑥, 𝑡) = 𝑊(𝑧), where 𝑧 = 𝑥 − 𝑐𝑡. Then (1)
transforms to the nonlinear ordinary differential equation:
where
(𝑐2 − 𝑘) 𝑊󸀠󸀠 (𝑧) + 2𝛼 sinh (𝑛𝑊 (𝑧)) + 𝛽 sinh (2𝑛𝑊 (𝑧)) = 0. 4
𝐵 = (𝑏1 exp (𝑧) + 𝑏0 + 𝑏−1 exp (−𝑧)) ,
(3)
𝐶4 = 2𝛼𝑎13 𝑏1 𝑛 − 𝛽𝑏14 𝑛 + 𝛽𝑎14 𝑛 − 2𝛼𝑎1 𝑏13 𝑛,
Further, using the transformation 𝑊(𝑧) = (1/𝑛) ln(𝐻(𝑧)) on
(3), we obtain 𝐶3 = −2𝑎12 𝑏0 𝑏1 𝑐2 + 2𝑎1 𝑎0 𝑏12 𝑐2 + 6𝛼𝑎0 𝑎12 𝑏1 𝑛
2 (𝑐2 − 𝑘) 𝐻 (𝑧) 𝐻󸀠󸀠 (𝑧) − 2 (𝑐2 − 𝑘) 𝐻󸀠 (𝑧)2 + 2𝛼𝑛𝐻(𝑧)3 − 6𝛼𝑎1 𝑏0 𝑏12 𝑛 + 2𝑎12 𝑏0 𝑏1 𝑘 − 2𝑎0 𝑎1 𝑏12 𝑘
− 2𝛼𝑛𝐻 (𝑧) + 𝛽𝑛𝐻(𝑧)4 − 𝛽𝑛 = 0. + 2𝛼𝑎13 𝑏0 𝑛 − 2𝑎0 𝑎1 𝑏12 𝑘 + 2𝛼𝑎13 𝑏0 𝑛

(4)
+ 4𝛽𝑎0 𝑎13 𝑛 − 2𝛼𝑎0 𝑏13 𝑛 − 4𝛽𝑏0 𝑏13 𝑛,
We assume that the solution of (4) can be expressed as
𝐶2 = 4𝛽𝑎−1 𝑎13 𝑛 − 8𝑎12 𝑏−1 𝑏1 𝑐2 + 8𝑎−1 𝑎1 𝑏12 𝑐2
𝑎𝑐 exp (𝑐𝑧) + ⋅ ⋅ ⋅ + 𝑎−𝑑 exp (−𝑑𝑧)
𝐻 (𝑧) = . (5) + 8𝑎12 𝑏−1 𝑏1 𝑘 − 8𝑎−1 𝑎1 𝑏12 𝑘 + 2𝛼𝑎13 𝑏−1 𝑛
𝑏𝑝 exp (𝑝𝑧) + ⋅ ⋅ ⋅ + 𝑏−𝑞 exp (−𝑞𝑧)
− 2𝛼𝑎−1 𝑏13 𝑛 − 4𝛽𝑏−1 𝑏13 𝑛 + 6𝛼𝑎0 𝑎12 𝑏0 𝑛
The values of 𝑐 and 𝑑, 𝑝 and 𝑞 can be determined by balancing
the linear term of the highest order with the highest order + 6𝛼𝑎02 𝑎1 𝑏1 𝑛 − 6𝛼𝑎1 𝑏02 𝑏1 𝑛 − 6𝛽𝑏02 𝑏12 𝑛
of nonlinear term in (4), that is, 𝐻𝐻󸀠󸀠 and 𝐻4 . By straight
forward calculation, we have + 6𝛼𝑎−1 𝑎12 𝑏1 𝑛 − 6𝛼𝑎1 𝑏12 𝑏−1 𝑛 + 6𝛽𝑎02 𝑎12 𝑛
𝑐1 exp [(2𝑐 + 3𝑝) 𝑧] + ⋅ ⋅ ⋅ − 6𝛼𝑎0 𝑏0 𝑏12 𝑛,

𝐻𝐻󸀠󸀠 = ,
𝑐2 exp [5𝑝𝑧] + ⋅ ⋅ ⋅ 𝐶1 = −2𝑎02 𝑏0 𝑏1 𝑐2 + 2𝑎0 𝑎1 𝑏02 𝑐2 + 2𝑎02 𝑏0 𝑏1 𝑘
(6)
𝑐3 exp [4𝑐𝑧] + ⋅ ⋅ ⋅ 𝑐 exp [(4𝑐 + 𝑝) 𝑧] + ⋅ ⋅ ⋅
𝐻4 = = 3 , − 2𝑎0 𝑎1 𝑏02 𝑘 − 2𝑎12 𝑏0 𝑏−1 𝑐2 + 2𝑎−1 𝑎0 𝑏12 𝑐2
𝑐4 exp [4𝑝𝑧] + ⋅ ⋅ ⋅ 𝑐4 exp [5𝑝𝑧] + ⋅ ⋅ ⋅
− 2𝑎0 𝑎−1 𝑏12 𝑘 + 2𝛼𝑎03 𝑏1 𝑛 + 4𝛽𝑎03 𝑎1 𝑛 − 2𝛼𝑎1 𝑏03 𝑛
where 𝑐𝑖 are coefficients only for simplicity. Balancing the
highest order of Exp-function in (6), we have 2𝑐+3𝑝 = 4𝑐+𝑝, − 4𝛽𝑏03 𝑏1 𝑛 + 12𝑎−1 𝑎1 𝑏0 𝑏1 𝑐2 − 12𝑎−1 𝑎1 𝑏0 𝑏1 𝑘
which yields 𝑐 = 𝑝. Similarly, we balance the lowest order in
(4) to determine values of 𝑑 and 𝑞. We have + 12𝑎0 𝑎1 𝑏−1 𝑏1 𝑘 + 6𝛼𝑎02 𝑎1 𝑏0 𝑛 − 6𝛼𝑎0 𝑏02 𝑏1 𝑛
⋅ ⋅ ⋅ + 𝑠1 exp [− (2𝑑 + 3𝑞) 𝑧] + 12𝛼𝑎−1 𝑎0 𝑎1 𝑏1 𝑛 − 12𝛼𝑎1 𝑏−1 𝑏0 𝑏1 𝑛

𝐻𝐻󸀠󸀠 = ,
⋅ ⋅ ⋅ + 𝑠2 exp [−5𝑞𝑧] + 6𝛼𝑎−1 𝑎12 𝑏0 𝑛 − 6𝛼𝑎0 𝑏−1 𝑏12 𝑛 − 6𝛼𝑎−1 𝑏0 𝑏12 𝑛
⋅ ⋅ ⋅ + 𝑠3 exp [4𝑑𝑧] ⋅ ⋅ ⋅ + 𝑠3 exp [− (4𝑑 + 𝑞) 𝑧] + 6𝛼𝑎0 𝑎12 𝑏−1 𝑛 + 12𝛽𝑎−1 𝑎0 𝑎12 𝑛 − 12𝛽𝑏−1 𝑏0 𝑏12 𝑛
𝐻4 = = ,
⋅ ⋅ ⋅ + 𝑠4 exp [−4𝑞𝑧] ⋅ ⋅ ⋅ + 𝑠4 exp [−5𝑞𝑧]
(7) + 2𝑎12 𝑏0 𝑏−1 𝑘 − 12𝑎0 𝑎1 𝑏−1 𝑏1 𝑐2 ,
𝐶0 = 2𝛼𝑎03 𝑏0 𝑛 − 2𝛼𝑎0 𝑏03 𝑛 + 𝛽𝑎04 𝑛 Equating the coefficients of exp(𝑧) in (9) to zero, we obtain a
set of algebraic equations:
+ 6𝛼𝑎−1 𝑎12 𝑏−1 𝑛 + 6𝛼𝑎02 𝑎1 𝑏−1 𝑛 + 6𝛼𝑎−1
2
𝑎1 𝑏1 𝑛
𝐶4 = 0, 𝐶3 = 0, 𝐶2 = 0, 𝐶1 = 0, 𝐶0 = 0,
+ 6𝛼𝑎−1 𝑎02 𝑏1 𝑛 + 12𝛽𝑎−1 𝑎02 𝑎1 𝑛 − 6𝛼𝑎1 𝑏−1
2
𝑏1 𝑛
𝐶−1 = 0, 𝐶−2 = 0, 𝐶−3 = 0, 𝐶−4 = 0.
− 6𝛼𝑎1 𝑏−1 𝑏02 𝑛 − 6𝛼𝑎−1 𝑏−1 𝑏12 𝑛 − 𝛽𝑏04 𝑛 (11)
Solving the system (11) with the help of Maple, we obtain the
− 6𝛼𝑎−1 𝑏02 𝑏1 𝑛 − 12𝛽𝑏−1 𝑏02 𝑏1 𝑛 + 8𝑎−1 𝑎1 𝑏02 𝑐2 following three cases.
− 8𝑎02 𝑏−1 𝑏1 𝑐2 − 8𝑎−1 𝑎1 𝑏02 𝑘 + 8𝑎02 𝑏−1 𝑏1 𝑘 Case 1. We have the following:
2 2 2 2
+ 6𝛽𝑎−1 𝑎1 𝑛 − 6𝛽𝑏−1 𝑏1 𝑛 + 12𝛼𝑎−1 𝑎0 𝑎1 𝑏0 𝑛 𝛼𝑏02 − 4𝛼𝑏1 𝑏−1
𝑎−1 = 𝑏−1 , 𝑎0 = −𝑏0 , 𝑎1 = 𝑏1 , 𝛽= ,
− 12𝛼𝑎0 𝑏−1 𝑏0 𝑏1 𝑛, 4𝑏1 𝑏−1
𝛼𝑏02 𝑛 + 2𝑏−1 𝑏1 𝑐2
𝐶−1 = 12𝛼𝑎−1 𝑎0 𝑎1 𝑏−1 𝑛 − 12𝛼𝑎−1 𝑏−1 𝑏0 𝑏1 𝑛 𝑘= .
2𝑏−1 𝑏1
+ 2𝑎−1 𝑎0 𝑏02 𝑐2 − 2𝑎02 𝑏−1 𝑏0 𝑐2 + 2𝑎02 𝑏−1 𝑏0 𝑘 (12)
Case 2. We have the following:
− 2𝑎−1 𝑎0 𝑏02 𝑘 + 2𝑎0 𝑎1 𝑏−1
2 2 2
𝑐 − 2𝑎−1 𝑏0 𝑏1 𝑐2
2 2 𝑏−1 𝑏1 −𝛽 (𝑎12 + 𝑏12 )
− 2𝑎0 𝑎1 𝑏−1 𝑘 + 2𝑎−1 𝑏0 𝑏1 𝑘 + 2𝛼𝑎03 𝑏−1 𝑛 𝑎−1 = , 𝑎0 = 0, 𝑏0 = 0, 𝛼= ,
𝑎1 2𝑎1 𝑏1
+ 4𝛽𝑎−1 𝑎03 𝑛 − 2𝛼𝑎−1 𝑏03 𝑛 − 4𝛽𝑏−1 𝑏03 𝑛 −2𝛽𝑎12 𝑏12 𝑛 + 𝛽𝑎14 𝑛 + 𝛽𝑏14 𝑛 + 8𝑎12 𝑏12 𝑐2
𝑘= .
+ 12𝑎−1 𝑎1 𝑏−1 𝑏0 𝑐2 − 12𝑎−1 𝑎0 𝑏−1 𝑏1 𝑐2 8𝑎12 𝑏12
(13)
− 12𝑎−1 𝑎1 𝑏−1 𝑏0 𝑘 + 12𝑎−1 𝑎0 𝑏−1 𝑏1 𝑘
Case 3. We have the following:
+ 6𝛼𝑎−1 𝑎02 𝑏0 𝑛 − 6𝛼𝑎0 𝑏−1 𝑏02 𝑛 + 2
6𝛼𝑎−1 𝑎1 𝑏0 𝑛
−𝛽 (𝑎12 + 𝑏12 )
𝑎−1 = −𝜙𝑏1 , 𝑏−1 = −𝜙𝑎1 , 𝛼= ,
2
+ 6𝛼𝑎−1 2
𝑎0 𝑏1 𝑛 + 12𝛽𝑎−1 2
𝑎0 𝑎1 𝑛 − 6𝛼𝑎1 𝑏−1 𝑏0 𝑛 2𝑎1 𝑏1
2 2 −2𝛽𝑎12 𝑏12 𝑛 + 𝛽𝑎14 𝑛 + 𝛽𝑏14 𝑛 + 2𝑎12 𝑏12 𝑐2

− 6𝛼𝑎0 𝑏−1 𝑏1 𝑛 − 12𝛽𝑏−1 𝑏0 𝑏1 𝑛, 𝑘= ,
2𝑎12 𝑏12
3 2 2 2 (14)
𝐶−2 = 2𝛼𝑎−1 𝑏1 𝑛 + 8𝑎−1 𝑎1 𝑏−1 𝑐 + 8𝑎−1 𝑏−1 𝑏1 𝑘
2
− 8𝑎−1 𝑎1 𝑏−1 3
𝑘 + 4𝛽𝑎−1 3
𝑎1 𝑛 − 4𝛽𝑏−1 𝑏1 𝑛 where 𝜙 = (−𝑎0 𝑎12 𝑏0 + 𝑎02 𝑎1 𝑏1 + 𝑎1 𝑏02 𝑏1 − 𝑎0 𝑏0 𝑏12 )/(𝑎1 − 𝑏1 )2 (𝑎1 +
𝑏1 )2 .
3 2
− 2𝛼𝑎1 𝑏−1 𝑛 − 8𝑎−1 𝑏−1 𝑏1 𝑐2 + 6𝛼𝑎−1 𝑎02 𝑏−1 𝑛
Substituting values from (12) into (8), we obtain
2 2 2
+ 6𝛼𝑎−1 𝑎0 𝑏0 𝑛 − 6𝛼𝑎0 𝑏−1 𝑏0 𝑛 + 6𝛼𝑎−1 𝑎1 𝑏−1 𝑛
𝑏1 exp (𝑧) − 𝑏0 + 𝑏−1 exp (−𝑧)
2 2 2 2 2 𝐻 (𝑧) = . (15)
− 6𝛼𝑎−1 𝑏−1 𝑏1 𝑛 + 6𝛽𝑎−1 𝑎0 𝑛 − 6𝛽𝑏−1 𝑏0 𝑛 𝑏1 exp (𝑧) + 𝑏0 + 𝑏−1 exp (−𝑧)
− 6𝛼𝑎−1 𝑏−1 𝑏02 𝑛, As a result one of the solutions of (1) is given by
2 2 1 𝑏 exp (𝑧) − 𝑏0 + 𝑏−1 exp (−𝑧)

𝐶−3 = 6𝛼𝑎0 𝑎−1 𝑏−1 𝑛 − 6𝛼𝑎−1 𝑏−1 𝑏0 𝑛 𝑢1 (𝑥, 𝑡) = ln ( 1 ), (16)
𝑛 𝑏1 exp (𝑧) + 𝑏0 + 𝑏−1 exp (−𝑧)
2
− 2𝑎−1 𝑏−1 𝑏0 𝑐2 + 2𝑎−1 𝑎0 𝑏−1
2 2
𝑐
where 𝑧 = 𝑥−𝑐𝑡, 𝛽 = (𝛼𝑏02 −4𝛼𝑏1 𝑏−1 )/4𝑏1 𝑏−1 , and 𝑘 = (𝛼𝑏02 𝑛+
+ 2
2𝑎−1 𝑏0 𝑏−1 𝑘 − 2
2𝑎−1 𝑎0 𝑏−1 𝑘 2𝑏−1 𝑏1 𝑐2 )/2𝑏−1 𝑏1 .
As a special case, if we choose 𝑏0 = 2 and 𝑏−1 = 𝑏1 = 1 in
3 3
+ 2𝛼𝑎−1 𝑏0 𝑛 + 4𝛽𝑎0 𝑎−1 𝑛 (16), then we get 𝛽 = 0, 𝑘 = 2𝛼𝑛 + 𝑐2 and obtain the solution
of the generalized sinh-Gordon equation as
3 3
− 2𝛼𝑎0 𝑏−1 𝑛 − 4𝛽𝑏0 𝑏−1 𝑛,
1 1
4 4 3 3 𝑢1 (𝑥, 𝑡) = ln (tanh2 [( ) (𝑥 − 𝑐𝑡)]) , (17)
𝐶−4 = 𝛽𝑎−1 𝑛 − 𝛽𝑏−1 𝑛 + 2𝛼𝑎−1 𝑏−1 𝑛 − 2𝛼𝑎−1 𝑏−1 𝑛. 𝑛 2
(10) which is the solution obtained in [14, 15].
Now substituting the values from (13) (Case 2) into (8)

results in the second solution of (1) as
1 𝑎 exp (𝑧) + (𝑏−1 𝑏1 /𝑎1 ) exp (−𝑧)

𝑢2 (𝑥, 𝑡) = ln ( 1 ) , (18)
𝑛 𝑏1 exp (𝑧) + 𝑏−1 exp (−𝑧)
0.8
with 𝑧 = 𝑥 − 𝑐𝑡, 𝛼 = −𝛽(𝑎12 + 𝑏12 )/2𝑎1 𝑏1 , and 𝑘 = (−2𝛽𝑎12 𝑏12 𝑛 + 0.6
u 0.4
𝛽𝑎14 𝑛 + 𝛽𝑏14 𝑛 + 8𝑎12 𝑏12 𝑐2 )/8𝑎12 𝑏12 . 4
The third solution of (1) is obtained by using the values 0.2
0
from (14) (Case 3) and substituting them into (8). Conse- −10
quently, it is given by 2 t
−5
0
1 𝑎 exp (𝑧) + 𝑎0 − 𝑏1 𝜙 exp (−𝑧) x
5
𝑢3 (𝑥, 𝑡) = ln ( 1 ), (19)
𝑛 𝑏1 exp (𝑧) + 𝑏0 − 𝑎−1 𝜙 exp (−𝑧) 10
0
where 𝑧 = 𝑥−𝑐𝑡,𝜙 = (−𝑎0 𝑎12 𝑏0 +𝑎02 𝑎1 𝑏1 +𝑎1 𝑏02 𝑏1 −𝑎0 𝑏0 𝑏12 )/(𝑎1 − Figure 1: Profile of solution (16).
𝑏1 )2 (𝑎1 + 𝑏1 )2 , 𝛼 = −𝛽(𝑎12 + 𝑏12 )/2𝑎1 𝑏1 , and 𝑘 = (−2𝛽𝑎12 𝑏12 𝑛 +
𝛽𝑎14 𝑛 + 𝛽𝑏14 𝑛 + 2𝑎12 𝑏12 𝑐2 )/2𝑎12 𝑏12 .
To construct more solutions of (1), we now set 𝑐 = 𝑝 = 2
and 𝑑 = 𝑞 = 2. Then (5) reduces to
𝐻 (𝑧) = (𝑎2 exp (2𝑧) + 𝑎1 exp (𝑧) + 𝑎0 + 𝑎−1 exp (−𝑧)
+ 𝑎−2 exp (−2𝑧)) 0

(20) −0.02 2
× (𝑏2 exp (𝑧) + 𝑏1 exp (𝑧) + 𝑏0 u −0.04
1.5
−1 −0.06
+ 𝑏−1 exp (−𝑧) + 𝑏−2 exp (−2𝑧)) .
0 1 t
Proceeding as above, we obtain the following three solutions
0.5
of (1): 5
x
0
1 𝑎 𝑏 10
𝑢4 (𝑥, 𝑡) = ln (𝑎2 exp (2𝑧) + ( −1 1 ) exp (𝑧)
𝑛 𝑏−1
Figure 2: Profile of solution (23).
𝑎 𝑏
+ ( −1 0 ) + 𝑎−1 exp (−𝑧))
𝑏−1
(21)
𝑎𝑏 By taking 𝑛 = 2, 𝑏−1 = −1, 𝑏0 = 2, 𝑐 = 1, and 𝑏1 = −1 in
× ( 2 −1 exp (𝑧) + 𝑏1 exp (𝑧) the solution (16), we have its profile given in Figure 1.
𝑎−1
By taking 𝑛 = 3, 𝑏−2 = 1, 𝑏0 = 2, 𝑐 = 1, and 𝑎1 = 1 in the
−1
solution (23), we have its profile given in Figure 2.
+ 𝑏0 + 𝑏−1 exp (−𝑧)) ,
2 2
3. Concluding Remarks
where 𝑧 = 𝑥 − 𝑐𝑡, 𝛼 = −𝛽(𝑎−1 + 𝑏−1 )/2𝑎−1 𝑏−1 ,
In this paper we obtained new exact solutions of the gen-
1 𝑎 exp (2𝑧) + 𝑎1 exp (𝑧) + 𝑏0 eralized double sinh-Gordon equation (1) using the Exp-
𝑢5 (𝑥, 𝑡) = ln ( 2 ), (22) function method. We presented six different solutions of (1).
𝑛 −𝑎2 exp (𝑧) + 𝑏1 exp (𝑧) + 𝑏0
Earlier, the tanh function, the bifurcation, and the (𝐺󸀠 /𝐺)-
with 𝑧 = 𝑥 − 𝑐𝑡, 𝛽 = 𝛼(𝑏12 + 4𝑎2 𝑏0 )/4𝑎2 𝑏0 , and 𝑘 = (𝛼𝑛𝑏12 + expansion methods [11–13] were employed to obtain exact
solutions of (1). The solutions obtained in this paper were
2𝑎2 𝑏0 𝑐2 )/2𝑎2 𝑏0 , and
new and were different from the ones obtained in [11–13]. By
taking special values of the constants, we also retrieved the
1 𝑎 exp (2𝑧) − 𝑏0 + 𝑏−2 exp (−2𝑧)
𝑢6 (𝑥, 𝑡) = ln ( 2 ), (23) solution of the generalized sinh-Gordon equation, which was
𝑛 𝑎2 exp (2𝑧) + 𝑏0 + 𝑏−2 exp (−2𝑧) obtained in [14, 15]. The Exp-function method is very simple
and straightforward method for solving nonlinear partial
where 𝑧 = 𝑥 − 𝑐𝑡, 𝛼 = −(8𝑎2 𝑏−2 (𝑐2 − 𝑘)/𝑏02 𝑛), and 𝛽 = differential equations. Indeed this has some pronounced
2(4𝑎2 𝑏−2 𝑐2 − 4𝑎2 𝑏−2 𝑘 − 𝑏02 𝑐2 + 𝑏02 𝑘)/𝑏02 𝑛. merit as compared to the other methods. The correctness of
the solutions obtained here has been verified by substituting [15] K. Parand, J. A. Rad, and A. Rezaei, “Application of Exp-
them back into (1). function method for class of nonlinear PDE’s in mathematical
physics,” Journal of Applied Mathematics & Informatics, vol. 29,
pp. 763–779, 2011.
Acknowledgments [16] A.-M. Wazwaz, “The tanh method: exact solutions of the sine-
Gordon and the sinh-Gordon equations,” Applied Mathematics
Gabriel Magalakwe would like to thank SANHARP, NRF, and
and Computation, vol. 167, no. 2, pp. 1196–1210, 2005.
North-West University, Mafikeng Campus, South Africa, for
[17] J.-H. He and X.-H. Wu, “Exp-function method for nonlinear
their financial support.
wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp.
700–708, 2006.
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http://dx.doi.org/10.1155/2013/278097
Research Article
Optimal Homotopy Asymptotic Method for Solving
the Linear Fredholm Integral Equations of the First Kind
Mohammad Almousa and Ahmad Ismail

School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Gelugor, Penang, Malaysia
Correspondence should be addressed to Mohammad Almousa; mohammadalmousa12@yahoo.com
Received 20 April 2013; Accepted 16 June 2013
Copyright © 2013 M. Almousa and A. Ismail. This is an open access article distributed under the Creative Commons Attribution
cited.
The aim of this study is to present the use of a semi analytical method called the optimal homotopy asymptotic method (OHAM)
for solving the linear Fredholm integral equations of the first kind. Three examples are discussed to show the ability of the method
to solve the linear Fredholm integral equations of the first kind. The results indicated that the method is very effective and simple.
1. Introduction 2. Application of OHAM to

the Linear Fredholm Integral Equations of
Integral equations of the first kind arise in several applica-
tions. These include applications in biology, chemistry, phys- the First Kind
ics, and engineering. In recent years, much work has been In this section, we formulate the optimal homotopy asymp-
carried out by researchers in mathematics and engineering in totic method (OHAM) for solving the linear Fredholm inte-
applying and analyzing novel numerical and semi analytical gral equations of the first kind following the procedure as
methods for obtaining solutions of integral equations of the outlined in [6, 7] and other papers. Let us consider a form
first kind. Among these are the homotopy analysis method of the linear Fredholm integral equation of the first kind:
[1], operational Tau method [2], homotopy perturbation
method [3], Adomian decomposition [3], quadrature rule [4], 𝑏
and automatic augmented Galerkin algorithms [5]. 𝑓 (𝑠) − ∫ 𝐾 (𝑥, 𝑡) 𝑔 (𝑡) 𝑑𝑡 = 0. (2)
In this study, we develop the optimal homotopy asymp- 𝑎
totic method (OHAM), which was proposed by Marinca et al.
[6, 7], for solving the linear Fredholm integral equations of Using OHAM, we can obtain a family of equations as fol-
the first kind. This method is characterized by it is conver- lows:
gence criteria which are more flexible than other methods.
(1 − 𝑝) [𝐿 (𝑔 (𝑠, 𝑝)) + 𝑓 (𝑠)]
The general form of the linear Fredholm integral equa-
(3)
tions of the first kind is = 𝐻 (𝑝) [𝐿 (𝑔 (𝑠, 𝑝)) + 𝑓 (𝑠) + 𝑁 (𝑔 (𝑠, 𝑝))] ,
𝑏
𝑓 (𝑠) = ∫ 𝐾 (𝑠, 𝑡) 𝑔 (𝑡) 𝑑𝑡, (1) where 𝑝 ∈ [0, 1] is an embedding parameter, 𝑔(𝑠, 𝑝) is un-
𝑎 known function, and 𝐻(𝑝) is an (nonzero) auxiliary function
for 𝑝 ≠0 and 𝐻(0) = 0 and given as 𝐻(𝑝) = ∑𝑚 𝑗
𝑗=1 𝑐𝑗 𝑝 where
where 𝑎 and 𝑏 are constant and the functions 𝑘(𝑠, 𝑡) and 𝑓(𝑠)
are known. 𝑐𝑗 , 𝑗 = 1, 2, . . ., are auxiliary constants, and when 𝑝 = 0 and
It should be noted that OHAM has been applied to the 𝑝 = 1 it holds that
nonlinear Fredholm integral equations of the second kind by
[8]. 𝑔 (𝑠, 0) = 𝑔0 (𝑠) , 𝑔 (𝑠, 1) = 𝑔 (𝑠) , (4)
respectively. For obtaining the approximate solution, we use 3. Numerical Examples and Discussion
Taylor’s series expansion about 𝑝 as follows:
∞
In this section, three examples of the linear Fredholm integral
𝑔 (𝑠, 𝑝, 𝑐𝑗 ) = 𝑔0 (𝑠) + ∑ 𝑔𝑚 (𝑠, 𝑐𝑗 ) 𝑝 , 𝑚
𝑗 = 1, 2, . . . . equations of the first kind were solved to show the efficiency
𝑚=1
of the present method. Maple software with long format and
(5) double accuracy was used to carry out the computations.
If the series (5) convergence occurs when 𝑝 = 1, one has Example 1. We consider the following equation [9]:
∞ 𝜋/2
𝑔 (𝑠, 1, 𝑐𝑗 ) = 𝑔0 (𝑠) + ∑ 𝑔𝑚 (𝑠, 𝑐𝑗 ) , 𝑗 = 1, 2, . . . . 1 2
(6) sin (𝑠) = ∫ sin (𝑠) sin (𝑡) 𝑔 (𝑡) 𝑑𝑡, (13)
𝑚=1 2 0 𝜋
Substituting (5) in (3) and equating the coefficients of like for which the exact solution is 𝑔(𝑠) = sin(𝑠). Applying OHAM
powers of 𝑝, we get as follows: to the linear Fredholm integral equation of first kind yields
𝑂 (𝑝0 ) : 𝑔0 (𝑠) = −𝑓 (𝑠) , 𝐿 (𝑔 (𝑠, 𝑝)) = 𝑔 (𝑠) ,
𝜋/2
𝑏 2
1 𝑁 (𝑔 (𝑠, 𝑝)) = − ∫ sin (𝑠) sin (𝑡) 𝑔 (𝑡) 𝑑𝑡, (14)
𝑂 (𝑝 ) : 𝑔1 (𝑠) = −𝑐1 ∫ 𝐾 (𝑠, 𝑡) 𝑔0 (𝑡) 𝑑𝑡, 0 𝜋
𝑎
1
𝑏 𝑓 (𝑠) = sin (𝑠)
2
𝑂 (𝑝2 ) : 𝑔2 (𝑠) = (1 + 𝑐1 ) 𝑔1 (𝑠) − 𝑐1 ∫ 𝐾 (𝑠, 𝑡) 𝑔1 (𝑡) 𝑑𝑡
𝑎 which satisfies
𝑏 1
− 𝑐2 ∫ 𝐾 (𝑠, 𝑡) 𝑔0 (𝑡) 𝑑𝑡, (1 − 𝑝) [(𝑔0 (𝑠) + 𝑝𝑔1 (𝑠) + 𝑝2 𝑔2 (𝑠) + ⋅ ⋅ ⋅ ) + sin (𝑠)]
𝑎
2
𝑖−1
= (𝑝𝑐1 + 𝑝2 𝑐2 + 𝑝3 𝑐3 + ⋅ ⋅ ⋅ )
𝑂 (𝑝𝑖 ) : 𝑔𝑖 (𝑠) = (1 + 𝑐1 ) 𝑔𝑖−1 (𝑠) + ∑𝑐𝑗 𝑔𝑖−𝑗 (𝑠) 1
𝑗=2 × [ (𝑔0 (𝑠) + 𝑝𝑔1 (𝑠) + 𝑝2 𝑔2 (𝑠) + ⋅ ⋅ ⋅ ) + sin (𝑠)
2
𝜋/2
𝑖 𝑏 2
− ∑ 𝑐𝑘 ∫ 𝐾 (𝑠, 𝑡) 𝑔𝑖−𝑘 (𝑡) 𝑑𝑡. −∫ sin (𝑠) sin (𝑡) (𝑔0 (𝑡) + 𝑝𝑔1 (𝑡)
0 𝜋
𝑘=1 𝑎
(7) +𝑝2 𝑔2 (𝑡) + ⋅ ⋅ ⋅ ) 𝑑𝑡] .
For finding the constants 𝑐1 , 𝑐2 , 𝑐3 , . . ., we can get the result of (15)
the 𝑚th-order approximations as follows:
Now we use (7) to obtain a series of problems:
𝑚
𝑚
𝑔 (𝑠, 𝑐𝑗 ) = 𝑔0 (𝑠) + ∑ 𝑔𝑘 (𝑠, 𝑐𝑗 ) , 𝑗 = 1, 2, . . . , 𝑚. 1
(8) 𝑂 (𝑝0 ) : 𝑔0 (𝑠) = − sin (𝑠) ,
𝑘=1 2
𝜋/2
If we substitute (8) into (1) we obtain the residual equation 2
𝑂 (𝑝1 ) : 𝑔1 (𝑠) = −𝑐1 ∫ sin (𝑠) sin (𝑡) 𝑔0 (𝑡) 𝑑𝑡,
𝑏 0 𝜋
𝑅 (𝑠, 𝑐𝑗 ) = 𝐿 (𝑔𝑚 (𝑠, 𝑐𝑗 )) + 𝑓 (𝑠) − ∫ 𝐾 (𝑠, 𝑡) 𝑔𝑚 (𝑡, 𝑐𝑗 ) 𝑑𝑡. 𝑂 (𝑝2 ) : 𝑔2 (𝑠) = (1 + 𝑐1 ) 𝑔1 (𝑠)
𝑎
(9) 𝜋/2
2
𝑚
If 𝑅(𝑠, 𝑐𝑗 ) = 0, then 𝑔 (𝑠, 𝑐𝑗 ) will be the exact solution. The − 𝑐1 ∫ sin (𝑠) sin (𝑡) 𝑔1 (𝑡) 𝑑𝑡
0 𝜋
least squares method can be used to determine 𝑐1 , 𝑐2 , 𝑐3 , . . . .
𝜋/2
At first we consider the functional 2
− 𝑐2 ∫ sin (𝑠) sin (𝑡) 𝑔0 (𝑡) 𝑑𝑡.
𝑏
0 𝜋
𝐽 (𝑐𝑗 ) = ∫ 𝑅2 (𝑠, 𝑐𝑗 ) 𝑑𝑠. (10)
𝑎 (16)
By using Galerkin’s method we get the following system: Hence the solutions are
𝜕𝐽 𝑏
𝜕𝑅 1
= 2 ∫ 𝑅 (𝑠, 𝑐𝑗 ) 𝑑𝑠, (11) 𝑂 (𝑝0 ) : 𝑔0 (𝑠) = − sin (𝑠) ,
𝜕𝑐𝑗 𝑎 𝜕𝑐𝑗 2
1
and then minimizing it to obtain the values of 𝑐1 , 𝑐2 , . . . , 𝑚, we 𝑂 (𝑝1 ) : 𝑔1 (𝑠) = 𝑐1 sin (𝑠) ,
have 4
(17)
𝜕𝐽 𝜕𝐽 𝜕𝐽 2 1
= = ⋅⋅⋅ = = 0. (12) 𝑂 (𝑝 ) : 𝑔2 (𝑠) = (1 + 𝑐1 ) 𝑐1 sin (𝑠)
𝜕𝑐1 𝜕𝑐2 𝜕𝑐𝑚 4
With these constants, the approximate solution is deter- 1 1
− 𝑐12 sin (𝑠) + 𝑐2 sin (𝑠) .
mined. 8 4
By substituting 𝑔0 (𝑠), 𝑔1 (𝑠), and 𝑔3 (𝑠) solutions in (6), we Table 1: Numerical results of Example 1.
obtain
𝑠 𝑔exact 𝑔OHAM |𝑔exact − 𝑔OHAM |
1 1 0 0 0 0
𝑔 (𝑠) = − sin (𝑠) + 𝑐1 sin (𝑠) 0.1 0.09983341665 0.09983341665 0
2 4
0.2 0.1986693308 0.1986693308 0
1
+ (1 + 𝑐1 ) 𝑐1 sin (𝑠) (18) 0.3 0.2955202067 0.2955202067 0
4 0.4 0.3894183423 0.3894183423 0
1 1
− 𝑐12 sin (𝑠) + 𝑐2 sin (𝑠) . 0.5 0.4794255386 0.4794255386 0
8 4 0.6 0.5646424734 0.5646424734 0
0.7 0.6442176872 0.6442176872 0
For the calculations of the constants 𝑐1 and 𝑐2 , the use of the 0.8 0.7173560909 0.7173560909 0
technique mentioned in (8)–(12) yields 0.9 0.7833269096 0.7833269096 0
1.0 0.8414709848 0.8414709848 0
𝑐1 = 6.000000004, 𝑐2 = −24.00000002. (19)
Substituting values in (18), the final solution becomes Now we use (7) to obtain a series of problems:
1
𝑂 (𝑝0 ) : 𝑔0 (𝑠) = − 𝑠2 ,
𝑔 (𝑠) = sin (𝑠) . (20) 4
1
5 22
This is the exact solution. 𝑂 (𝑝1 ) : 𝑔1 (𝑠) = −𝑐1 ∫ 𝑠 𝑡 𝑔0 (𝑡) 𝑑𝑡,
0 2
Table 1 shows some numerical results of these solutions 𝑂 (𝑝2 ) : 𝑔2 (𝑠) = (1 + 𝑐1 ) 𝑔1 (𝑠)
calculated according to the present method.
The exact solution, OHAM solution and absolute error of 1 1
5 22 5
this example are shown in Figure 1. −𝑐1 ∫ 𝑠 𝑡 𝑔1 (𝑡) 𝑑𝑡−𝑐2 ∫ 𝑠2 𝑡2 𝑔0 (𝑡) 𝑑𝑡.
0 2 0 2
(24)
Example 2. We consider the following equation [10]:
Hence the solutions are
1
1 2 5 1
𝑠 = ∫ 𝑠2 𝑡2 𝑔 (𝑡) 𝑑𝑡, (21) 𝑂 (𝑝0 ) : 𝑔0 (𝑠) = − 𝑠2 ,
4 0 2 4
1
for which the exact solution is 𝑔(𝑠) = (1/2)𝑠2 . Applying 𝑂 (𝑝1 ) : 𝑔1 (𝑠) = 𝑐1 𝑠2 , (25)
8
OHAM to the linear Fredholm integral equation of first kind
1 1 1
yields 𝑂 (𝑝2 ) : 𝑔2 (𝑠) = (1 + 𝑐1 ) 𝑐1 𝑠2 − 𝑐12 𝑠2 + 𝑐2 𝑠2 .
8 16 8
𝐿 (𝑔 (𝑠, 𝑝)) = 𝑔 (𝑠) , By substituting 𝑔0 (𝑠), 𝑔1 (𝑠), and 𝑔3 (𝑠) solutions in (6), we
1
obtain
5 22
𝑁 (𝑔 (𝑠, 𝑝)) = − ∫ 𝑠 𝑡 𝑔 (𝑡) 𝑑𝑡, 1 1 1 1 1
0 2 (22) 𝑔 (𝑠) = − 𝑠2 + 𝑐1 𝑠2 + (1 + 𝑐1 ) 𝑐1 𝑠2 − 𝑐12 𝑠2 + 𝑐2 𝑠2 .
4 8 8 16 8
1 (26)
𝑓 (𝑠) = 𝑠2
4
For the calculations of the constants 𝑐1 and 𝑐2 , the use of the
technique mentioned in (8)–(12) yields
which satisfies
𝑐1 = 6, 𝑐2 = −24. (27)
1
(1 − 𝑝) [(𝑔0 (𝑠) + 𝑝𝑔1 (𝑠) + 𝑝 𝑔2 (𝑠) + ⋅ ⋅ ⋅ ) + 𝑠2 ]
2
Substituting values in (26), the final solution becomes
4
= (𝑝𝑐1 + 𝑝2 𝑐2 + 𝑝3 𝑐3 + ⋅ ⋅ ⋅ ) 1
𝑔 (𝑠) = 𝑠2 . (28)
2
1
× [ (𝑔0 (𝑠) + 𝑝𝑔1 (𝑠) + 𝑝2 𝑔2 (𝑠) + ⋅ ⋅ ⋅ ) + 𝑠2 This is the exact solution.
4
1 Table 2 shows some numerical results of these solutions
5 22
−∫ 𝑠 𝑡 (𝑔0 (𝑡) + 𝑝𝑔1 (𝑡) + 𝑝2 𝑔2 (𝑡) + ⋅ ⋅ ⋅ ) 𝑑𝑡] . calculated according to the present method.
0 2 The exact solution, OHAM solution and absolute error of
(23) this example are shown in Figure 2.
0.8 1
0.7
0.6 0.5
0.5
0.4 0
−1 −0.5 0 0.5 1
0.3
0.2 −0.5
0.1
0 −1
0 0.2 0.4 0.6 0.8 1
Absolute error for Example 1
Exact solution
OHAM solution
(a) Results for Example 1 (b) Absolute error for Example 1
Figure 1
0.5 1
0.4
0.5
0.3
0
0.2 −1 −0.5 0 0.5 1
0.1 −0.5
0
0 0.2 0.4 0.6 0.8 1 −1
Exact solution Absolute error for Example 2
OHAM solution
Figure 2
Table 2: Numerical results of Example 2. for which the exact solution is 𝑔(𝑠) = (1/2)𝑠2 . Applying
OHAM to the linear Fredholm integral equation of first kind
𝑠 𝑔exact 𝑔OHAM |𝑔exact − 𝑔OHAM |
yields
0 0 0 0
0.1 0.005 0.005 0 𝐿 (𝑔 (𝑠, 𝑝)) = 𝑔 (𝑠) ,
0.2 0.02 0.02 0 1
0.3 0.045 0.045 0 𝑁 (𝑔 (𝑠, 𝑝)) = − ∫ 2𝑠2 𝑡 𝑔 (𝑡) 𝑑𝑡, (30)
0
0.4 0.08 0.08 0
0.5 0.125 0.125 0 1
𝑓 (𝑠) = 𝑠2
0.6 0.18 0.18 0 2
0.7 0.245 0.245 0 which satisfies
0.8 0.32 0.32 0 1
0.9 0.405 0.405 0 (1 − 𝑝) [(𝑔0 (𝑠) + 𝑝𝑔1 (𝑠) + 𝑝2 𝑔2 (𝑠) + ⋅ ⋅ ⋅ ) + 𝑠2 ]
2
1.0 0.5 0.5 0
= (𝑝𝑐1 + 𝑝2 𝑐2 + 𝑝3 𝑐3 + ⋅ ⋅ ⋅ )
1
× [ (𝑔0 (𝑠) + 𝑝𝑔1 (𝑠) + 𝑝2 𝑔2 (𝑠) + ⋅ ⋅ ⋅ ) + 𝑠2
Example 3. We consider the following equation [9]: 2
1
1 − ∫ 2𝑠2 𝑡 (𝑔0 (𝑡) + 𝑝𝑔1 (𝑡) + 𝑝2 𝑔2 (𝑡) + ⋅ ⋅ ⋅ ) 𝑑𝑡] .
1 2 0
𝑠 = ∫ 2𝑠2 𝑡 𝑔 (𝑡) 𝑑𝑡, (29)
(31)
2 0
1
1
0.8
0.5
0.6
0.4 0
−1 −0.5 0 0.5 1
0.2
−0.5
0
0 0.2 0.4 0.6 0.8 1
−1
Exact solution
OHAM solution Absolute error for Example 3
Figure 3
Now we use (7) to obtain a series of problems: Table 3: Numerical results of Example 3.
1 𝑠 𝑔exact 𝑔OHAM |𝑔exact − 𝑔OHAM |

𝑂 (𝑝0 ) : 𝑔0 (𝑠) = − 𝑠2 ,
2 0 0 0 0
1 0.1 0.01 0.01 0
𝑂 (𝑝1 ) : 𝑔1 (𝑠) = −𝑐1 ∫ 2𝑠2 𝑡𝑔0 (𝑡) 𝑑𝑡, 0.2 0.04 0.04 0
0
0.3 0.09 0.09 0
2
𝑂 (𝑝 ) : 𝑔2 (𝑠) (32) 0.4 0.16 0.16 0
0.5 0.25 0.25 0
1
0.6 0.36 0.36 0
= (1 + 𝑐1 ) 𝑔1 (𝑠) − 𝑐1 ∫ 2𝑠2 𝑡𝑔1 (𝑡) 𝑑𝑡
0 0.7 0.49 0.49 0
1 0.8 0.64 0.64 0
− 𝑐2 ∫ 2𝑠2 𝑡𝑔0 (𝑡) 𝑑𝑡. 0.9 0.81 0.81 0
0
1.0 1.0 1.0 0
Hence the solutions are
1
𝑂 (𝑝0 ) : 𝑔0 (𝑠) = − 𝑠2 , Table 3 shows some numerical results of these solutions
2
calculated according to the present method.
1
𝑂 (𝑝1 ) : 𝑔1 (𝑠) = 𝑐1 𝑠2 , (33) The exact solution, OHAM solution and absolute error of
4 this example are shown in Figure 3.
1 1 1
𝑂 (𝑝2 ) : 𝑔2 (𝑠) = (1 + 𝑐1 ) 𝑐1 𝑠2 − 𝑐12 𝑠2 + 𝑐2 𝑠2 . 4. Conclusions
4 8 4
By substituting 𝑔0 (𝑠), 𝑔1 (𝑠), and 𝑔3 (𝑠) solutions in (6), we In this paper, we presented the application of the OHAM
obtain in solving the linear Fredholm integral equations of the first
kind. This method was tested on three different examples.
1 1 1
𝑔 (𝑠) = − 𝑠2 + 𝑐1 𝑠2 + (1 + 𝑐1 ) 𝑐1 𝑠2 This method proved to be an accurate and efficient technique
2 4 4 for finding approximate solutions for the linear Fredholm
(34)
1 1 integral equations of the first kind.
− 𝑐12 𝑠2 + 𝑐2 𝑠2 .
8 4
For the calculations of the constants 𝑐1 and 𝑐2 , the use of the References
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http://dx.doi.org/10.1155/2013/929478
Research Article
Numerical Study of Two-Dimensional Volterra Integral
Equations by RDTM and Comparison with DTM
Reza Abazari1 and Adem KJlJçman2

1
Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University, Ardabil, Iran
2
Department of Mathematics and Institute of Mathematical Research, University Putra Malaysia (UPM), 43400 Serdang, Malaysia
Correspondence should be addressed to Reza Abazari; abazari-r@uma.ac.ir
Copyright © 2013 R. Abazari and A. Kılıçman. This is an open access article distributed under the Creative Commons Attribution
cited.
The two-dimensional Volterra integral equations are solved using more recent semianalytic method, the reduced differential
transform method (the so-called RDTM), and compared with the differential transform method (DTM). The concepts of DTM
and RDTM are briefly explained, and their application to the two-dimensional Volterra integral equations is studied. The results
obtained by DTM and RDTM together are compared with exact solution. As an important result, it is depicted that the RDTM results
are more accurate in comparison with those obtained by DTM applied to the same Volterra integral equations. The numerical results
reveal that the RDTM is very effective, convenient, and quite accurate compared to the other kind of nonlinear integral equations.
It is predicted that the RDTM can be found widely applicable in engineering sciences.
1. Introduction where 𝐾 and 𝑓 are continuous functions and 𝐾 has the fol-
lowing form:
Mathematical modeling of many problems in science, engi-
neering, physics, and other disciplines leads to linear and 𝑚
nonlinear integrodifferential equations (IDE). The great use 𝐾 (𝑥, 𝑡, 𝑦, 𝑧, 𝑢 (𝑦, 𝑧)) = ∑𝑝𝑖 (𝑥, 𝑡) 𝑞𝑖 (𝑦, 𝑧, 𝑢 (𝑦, 𝑧)) . (2)
of mathematical models including integrodifferential equa- 𝑖=0
tions is one of the main reasons obtaining the solutions of this
kind of problems (see, e.g., [1–3] and the references therein).
So, it is very important to get some information about the The one-dimensional Volterra type of integral equation has
analytical solutions of these problems because these solutions been solved by many numerical methods, such as collocation
give significant information about the character of the mod- methods [1], Taylor-series expansion methods [2], Gauss-
eled event. But, in some cases, it is more difficult to obtain type quadratures method [3], spectral methods [6], Cheby-
analytical solutions of these models. These are usually difficult shev polynomial method [7], Tau method [8], sine-cosine
to solve analytically, and in many cases the solution must wavelets method [9], Monte Carlo method [10], and Haar
be approximated. To approximate the solutions of these functions method [11].
models, in recent years several numerical approaches have But in two-dimensional cases, a small amount of work
been proposed. has been done (see, e.g., [12–14]). Very recently, Tari et al.
In this paper, we consider the following Volterra type of in [4] employed the classic differential transform method for
integral equation [4, 5]: solving two-dimensional Volterra type of integral equations
(1), and Jang in [5] improved the proofs of the presented theo-
𝑡 𝑥
rems by Tari et al. in [4]. They derived fundamental properties
𝑢 (𝑥, 𝑡) − ∫ ∫ 𝐾 (𝑥, 𝑡, 𝑦, 𝑧, 𝑢 (𝑦, 𝑧)) 𝑑𝑦 𝑑𝑧 = 𝑓 (𝑥, 𝑡) , (1) of the differential transforms of some kernel functions 𝐾 in
0 0 Volterra integral equations.
However, the classic differential transform method, intro- Definition 1. If 𝑤(𝑥, 𝑡) is analytic and continuously differen-
duced by Zhou [15], is based on the definition of the differ- tiable with respect to time 𝑡 in the domain of interest, then
ential transform, which is a Taylor series. Thus, it requires
a cumbersome calculation to obtain the basic properties of 1 𝜕𝑚+𝑛
𝑊 (𝑚, 𝑛) = [ 𝑚 𝑛 𝑤 (𝑥, 𝑡)] 𝑥=𝑥 , (4)
the differential transforms. Some of DTM applications are 𝑚!𝑛! 𝜕𝑥 𝜕𝑡 𝑡=𝑡
0
0
mentioned in [16–21].
Recently, Keskin and Oturanç introduced a reduced form where the spectrum function 𝑊(𝑚, 𝑛) is the transformed
of DTM as reduced DTM (RDTM) and applied it to approxi- function, which is also called 𝑇-function in brief.
mate some PDE [22] and factional PDEs [23]. More recently,
Abazari and Ganji [24] extended RDTM to study the par- The differential inverse transform of 𝑊(𝑘, ℎ) is defined as
tial differential equation with proportional delay in 𝑡 and
∞ ∞
shrinking in 𝑥 and showed that, as a special advantage of 𝑚 𝑛
𝑤 (𝑥, 𝑡) = ∑ ∑ 𝑊 (𝑚, 𝑛) (𝑥 − 𝑥0 ) (𝑡 − 𝑡0 ) . (5)
RDTM rather than DTM, the reduced differential transform
𝑚=0 𝑛=0
recursive equations produce exactly all the Poisson series
coefficients of solutions based on the initial condition as Combining (4) and (5), it can be obtained that
weighted function, whereas the differential transform recur-
sive equations produce exactly all the Taylor series coeffi- ∞ ∞
1 𝜕𝑚+𝑛
cients of solutions. 𝑤 (𝑥, 𝑡) = ∑ ∑ [ 𝑚 𝑛 𝑤 (𝑥, 𝑡)] 𝑥=𝑥
𝑚=0 𝑛=0 𝑚!𝑛! 𝜕𝑥 𝜕𝑡
0
Here, we suggest the RDTM, for the approximating of the 𝑡=𝑡 0 (6)
solutions of the two-dimensional Volterra integral equations 𝑚 𝑛
× (𝑥 − 𝑥0 ) (𝑡 − 𝑡0 ) .
(1) with the same kernel functions in [4, 5]. In order to
demonstrate the effectiveness of the RDTM, the illustrative
When (𝑥0 , 𝑡0 ) are taken as (0, 0), then (5) can be expressed as
examples for the same kernel function of references [5] are
presented. These examples show that the RDTM produces ∞ ∞
exactly all the Poisson series coefficients (see Remark 5) 𝑤 (𝑥, 𝑡) = ∑ ∑ 𝑊 (𝑚, 𝑛) 𝑥𝑚 𝑡𝑛 . (7)
of the exact solutions, whereas, the classic DTM produces 𝑚=0 𝑛=0
exactly all the Taylor series coefficients of the exact solu-
tions. As an important result, notwithstanding the simplicity In real applications, the function 𝑤(𝑥, 𝑡) is represented by a
and robustness of RDTM, it is depicted that the RDTM results finite series of (7) that can be written as
are more accurate in comparison with those obtained by
𝑀 𝑁
classic DTM.
𝑤𝑀,𝑁 (𝑥, 𝑡) = ∑ ∑ 𝑊 (𝑚, 𝑛) 𝑥𝑚 𝑡𝑛 + 𝑅𝑀,𝑁 (𝑥, 𝑡) , (8)
𝑚=0 𝑛=0
2. Basic Definitions
and (7) implies that 𝑅𝑀,𝑁(𝑥, 𝑡) = ∑∞ ∞
𝑚=𝑀+1 ∑𝑛=𝑁+1 𝑊(𝑚,
𝑚 𝑛
With reference to the articles [16–21], the basic definitions of 𝑛)𝑥 𝑡 is negligibly small. Usually, the values of 𝑀 and 𝑁 are
two-dimensional differential transform method (DTM) and decided by convergency of the series coefficients.
their reduced form (RDTM) are introduced in the following From the above definitions, it can be found that the con-
two subsections, respectively. cept of the two-dimensional differential transform is derived
from the two-dimensional Taylor series expansion. With
(4) and (5), the fundamental mathematical operations per-
2.1. Two-Dimensional DTM. Consider a function of two formed using the two-dimensional differential transform
variables 𝑤(𝑥, 𝑡), and suppose that it can be represented as may be readily obtained, and these are listed in Table 1. (See
a product of two single-variable functions, that is, 𝑤(𝑥, 𝑡) = [4, 5, 15, 16].)
𝑓(𝑥)𝑔(𝑡). On the basis of the properties of the one-dimen- Recently, Jang [5] extended the two-dimensional DTM on
sional differential transform, the function 𝑤(𝑥, 𝑡) can be (1) as follows.
represented as
Theorem 2. Assume that 𝑈(𝑚, 𝑛), 𝑉(𝑚, 𝑛), 𝐻(𝑚, 𝑛), and
𝐺(𝑚, 𝑛) are the differential transforms of the functions 𝑢(𝑥, 𝑡),
V(𝑥, 𝑡), ℎ(𝑥, 𝑡), and 𝑔(𝑥, 𝑡), respectively; then we have the
∞ ∞ ∞ ∞
following:
𝑤 (𝑥, 𝑡) = ∑𝐹 (𝑖) 𝑥𝑖 ∑𝐺 (𝑗) 𝑡𝑗 = ∑ ∑𝑊 (𝑖, 𝑗) 𝑥𝑖 𝑡𝑗 , (3)
𝑖=0 𝑗=0 𝑖=0 𝑗=0 𝑡 𝑥
(a) if 𝑔(𝑥, 𝑡) = ∫0 ∫0 𝑢(𝑦, 𝑧)𝑑𝑦 𝑑𝑧, then
𝐺 (𝑚, 0) = 𝐺 (0, 𝑛) = 0, 𝑚, 𝑛 = 0, 1, . . . ,
where 𝑊(𝑖, 𝑗) = 𝐹(𝑖)𝐺(𝑗) is called the spectrum of 𝑤(𝑥, 𝑡).
1 (9)
The basic definitions and operations for two-dimensional 𝐺 (𝑚, 𝑛) = 𝑈 (𝑚 − 1, 𝑛 − 1) , 𝑚, 𝑛 = 1, 2, . . . ,
differential transform are introduced as follows. 𝑚𝑛
Table 1: The fundamental operations of two-dimensional differential transform method.
Original function Transformed function

𝑤(𝑥, 𝑡) = 𝑢(𝑥, 𝑡) ± V(𝑥, 𝑡) 𝑊(𝑚, 𝑛) = 𝑈(𝑚, 𝑛) ± 𝑉(𝑚, 𝑛)
𝑤(𝑥, 𝑡) = 𝑐 𝑢(𝑥, 𝑡) 𝑊(𝑚, 𝑛) = 𝑐 𝑈(𝑚, 𝑛)
𝜕
𝑤(𝑥, 𝑡) = 𝑢(𝑥, 𝑡) 𝑊(𝑚, 𝑛) = (𝑚 + 1)𝑈(𝑚 + 1, 𝑛)
𝜕𝑥
𝜕
𝑤(𝑥, 𝑡) = 𝑢(𝑥, 𝑡) 𝑊(𝑚, 𝑛) = (𝑚 + 1)𝑈(𝑚, 𝑛 + 1)
𝜕𝑡
𝜕𝑟+𝑠 (𝑚 + 𝑟)!(𝑛 + 𝑠)!
𝑤(𝑥, 𝑡) = 𝑢(𝑥, 𝑡) 𝑊(𝑚, 𝑛) = 𝑈(𝑚 + 𝑟, 𝑛 + 𝑠)
𝜕𝑥𝑟 𝜕𝑡𝑠 𝑚!𝑛!
𝑚 𝑛
𝑤(𝑥, 𝑡) = 𝑢(𝑥, 𝑡)V(𝑥, 𝑡) 𝑊(𝑚, 𝑛) = ∑ ∑ 𝑈(𝑟, 𝑛 − 𝑠)𝑉(𝑚 − 𝑟, 𝑠)
𝑟=0 𝑠=0
𝛼 𝛽
{1 𝑚 = 𝛼, 𝑛 = 𝛽
𝑤(𝑥, 𝑡) = 𝑥 𝑡 𝑊(𝑚, 𝑛) = 𝛿(𝑚 − 𝛼, 𝑛 − 𝛽) = {
{0 otherwise
𝑡 𝑥 𝑡 𝑥
(b) if 𝑔(𝑥, 𝑡) = ∫0 ∫0 𝑢(𝑦, 𝑧)V(𝑦, 𝑧)𝑑𝑦 𝑑𝑧, then (b) if 𝑔(𝑥, 𝑡) = (1/V(𝑥, 𝑡)) ∫0 ∫0 𝑢(𝑦, 𝑧)𝑑𝑦 𝑑𝑧, then
𝐺 (𝑚, 0) = 𝐺 (0, 𝑛) = 0, 𝑚, 𝑛 = 0, 1, . . . , 𝐺 (𝑚, 0) = 𝐺 (0, 𝑛) = 0, 𝑚, 𝑛 = 0, 1, . . . ,
1 𝑛−1 𝑚−1 𝑛−1 𝑚−1

𝐺 (𝑚, 𝑛) = ∑ ∑ 𝑈 (𝑘, ℓ) 𝑉 (𝑚 − 𝑘 − 1, 𝑛 − ℓ − 1) , ∑ ∑ 𝑉 (𝑘, ℓ) 𝐺 (𝑚 − 𝑘 + 1, 𝑛 − ℓ + 1)
𝑚𝑛 ℓ=0 𝑘=0 ℓ=0 𝑘=0
(13)
𝑚, 𝑛 = 1, 2, . . . , 1
= 𝑈 (𝑚, 𝑛) .
(10) (𝑚 + 1) (𝑛 + 1)
𝑡 𝑥 Proof. See [5].

(c) if 𝑔(𝑥, 𝑡) = ℎ(𝑥, 𝑡) ∫0 ∫0 𝑢(𝑦, 𝑧)𝑑𝑦 𝑑𝑧, then
2.2. Two-Dimensional Reduced DTM (RDTM). Consider a

𝐺 (𝑚, 0) = 𝐺 (0, 𝑛) = 0, 𝑚, 𝑛 = 0, 1, . . . ,
function of two variables 𝑤(𝑥, 𝑡), and suppose that it can
be represented as a product of two single-variable functions,
1 𝑛−1 𝑚−1 𝑉 (𝑚 − 𝑘 − 1, 𝑛 − ℓ − 1)
𝐺 (𝑚, 𝑛) = ∑ ∑ 𝐻 (𝑘, ℓ) , that is, 𝑤(𝑥, 𝑡) = 𝑓(𝑥)𝑔(𝑡). Based on the properties of one-
𝑚𝑛 ℓ=0 𝑘=0 (𝑚 − 𝑘) (𝑛 − ℓ) dimensional differential transform, the function 𝑤(𝑥, 𝑡) can
be represented as
𝑚, 𝑛 = 1, 2, . . . .
(11)
∞ ∞ ∞ ∞
𝑤 (𝑥, 𝑡) = ∑𝐹 (𝑖) 𝑥𝑖 ∑𝐺 (𝑗) 𝑡𝑗 = ∑ ∑𝑊 (𝑖, 𝑗) 𝑥𝑖 𝑡𝑗 , (14)
𝑖=0 𝑗=0 𝑖=0 𝑗=0
Proof. See [5].
Theorem 3. Assume that 𝑈(𝑚, 𝑛), 𝑉(𝑚, 𝑛), and G(𝑚, 𝑛) are where 𝑊(𝑖, 𝑗) = 𝐹(𝑖)𝐺(𝑗) is called the spectrum of 𝑤(𝑥, 𝑡).
the differential transforms of the functions 𝑢(𝑥, 𝑡), V(𝑥, 𝑡), and
𝑔(𝑥, 𝑡), respectively; then we have the following: Remark 4. The poisson function series generates a multi-
variate Taylor series expansion of the input expression 𝑤,
𝑡 𝑥 with respect to the variables 𝑋, to order 𝑛, using the variable
(a) if 𝑔(𝑥, 𝑡) = ∫0 ∫0 (𝑢(𝑦, 𝑧)/V(𝑦, 𝑧))𝑑𝑦 𝑑𝑧, then weights 𝑊.
𝐺 (𝑚, 0) = 𝐺 (0, 𝑛) = 0, 𝑚, 𝑛 = 0, 1, . . . , Remark 5. The relationship introduced in (14) is the poisson

series form of the input expression 𝑤(𝑥, 𝑡), with respect to
𝑛−1 𝑚−1 the variables 𝑥 and 𝑡, to order 𝑁, using the variable weights
∑ ∑ (𝑚 − 𝑘 + 1) (𝑛 − ℓ + 1) 𝑉 (𝑘, ℓ) 𝐺 (𝑚−𝑘+1, 𝑛−ℓ+1) 𝑊𝑘 (𝑥).
ℓ=0 𝑘=0
Similarly on previous section, the basic definitions of two-
= 𝑈 (𝑚, 𝑛) , differential reduced differential transformation are intro-
(12) duced as follows.
𝑡 𝑥
Definition 6. If 𝑤(𝑥, 𝑡) is analytical function in the domain of (b) if 𝑤(𝑥, 𝑡) = ℎ(𝑥, 𝑡) ∫𝑡 ∫𝑥 𝑢(𝑦, 𝑧)𝑑𝑦 𝑑𝑧, then
0 0
interest, then the spectrum function
𝑘 𝑥
1
1 𝜕 𝑘 𝑊𝑘 (𝑥) = ∑ 𝐻𝑟 (𝑥) ∫ 𝑈𝑘−𝑟−1 (𝑦) 𝑑𝑦, 𝑘 = 1, 2, . . . .
𝑊𝑘 (𝑥) = [ 𝑤 (𝑥, 𝑡)] (15) 𝑟=0 𝑘 − 𝑟 𝑥0
𝑘! 𝜕𝑡𝑘 𝑡=𝑡 (20)
0
is the reduced transformed function of 𝑤(𝑥, 𝑡).

Proof. (a) According to the fundamental operations of two-
Similarly on previous sections, the lowercase 𝑤(𝑥, 𝑡) dimensional RDTM listed in Table 2 and from Leibnitz for-
respects the original function while the uppercase 𝑊𝑘 (𝑥) mula, we get
stands for the reduced transformed function. The differential
inverse transform of 𝑊𝑘 (𝑥) is defined as
𝜕𝑘 𝜕𝑘 𝑡 𝑥
𝑘
𝑤 (𝑥, 𝑡) = 𝑘 (∫ ∫ 𝑢 (𝑦, 𝑧) V (𝑦, 𝑧) 𝑑𝑦 𝑑𝑧)
𝜕𝑡 𝜕𝑡 𝑡0 𝑥0
∞
𝑘
𝑤 (𝑥, 𝑡) = ∑ 𝑊𝑘 (𝑥) (𝑡 − 𝑡0 ) . (16)
𝑘=0
𝑥
𝜕𝑘−1
=∫ {𝑢 (𝑦, 𝑡) V (𝑦, 𝑡)} 𝑑𝑦
𝑥0 𝜕𝑡𝑘−1
Combining (15) and (16), it can be obtained that (21)
𝑘−1
𝑥
𝑘 − 1 𝜕𝑟
= ∫ {∑ ( ) 𝑟 𝑢 (𝑦, 𝑡)
𝑥0 𝑟=0 𝑟 𝜕𝑡
∞
1 𝜕𝑘 𝑘
𝑤 (𝑥, 𝑡) = ∑ [ 𝑘 𝑤 (𝑥, 𝑡)] (𝑡 − 𝑡0 ) . (17)
𝑘! 𝜕𝑡
𝑘=0 𝑡=𝑡 0 𝜕𝑘−𝑟−1
× V (𝑦, 𝑡) } 𝑑𝑦,
𝜕𝑡𝑘−𝑟−1
In real applications, the function 𝑤(𝑥, 𝑡) is represented by a
finite series of (16), around 𝑡0 = 0, and can be written as
therefore
𝑛
𝑥 𝑘−1
𝑤𝑛 (𝑥, 𝑡) = ∑ 𝑊𝑘 (𝑥) 𝑡𝑘 + 𝑅𝑛 (𝑥, 𝑡) , 𝜕𝑘 𝑘−1
(18) [ 𝑤 (𝑥, 𝑡)] = ∫ {∑ ( ) 𝑟! (𝑘 − 𝑟 − 1)!
𝑘=0 𝜕𝑡𝑘 𝑥 𝑟=0
𝑟
𝑡=𝑡0 0
and (18) implies that 𝑅𝑛 (𝑥, 𝑡) = ∑∞ 𝑘

𝑘=𝑛+1 𝑊𝑘 (𝑥)𝑡 is negligibly ×𝑈𝑟 (𝑦) 𝑉𝑘−𝑟−1 (𝑦) } 𝑑𝑦
small. Usually, the values of 𝑛 and 𝑚 are decided by conver-
gency of the series coefficients. From the above proposition, it
can be found that the concept of the reduced two-dimension- 𝑥 𝑘−1
al differential transform is derived from the two-dimensional = (𝑘 − 1)! ∫ { ∑ 𝑈𝑟 (𝑦) 𝑉𝑘−𝑟−1 (𝑦)} 𝑑𝑦,
differential transform method. With (15) and (16), the funda- 𝑥0 𝑟=0
mental mathematical operations performed by reduced two- (22)
dimensional differential transform can readily be obtained
and listed in Table 2.
and then, from using (15), for 𝑘 = 1, 2, . . ., we get
Similarly on previous subsection, we can extend the
RDTM on Volterra integral equations (1) as follow.
1 𝑥 𝑘−1
Theorem 7. Assume that 𝑈𝑘 (𝑥), 𝑉𝑘 (𝑥), 𝐻𝑘 (𝑥), and 𝑊𝑘 (𝑥) 𝑊𝑘 (𝑥) = ∫ { ∑ 𝑈 (𝑦) 𝑉𝑘−𝑟−1 (𝑦)} 𝑑𝑦. (23)
𝑘 𝑥0 𝑟=0 𝑟
are the reduced differential transforms of the functions 𝑢(𝑥, 𝑡),
V(𝑥, 𝑡), ℎ(𝑥, 𝑡), and 𝑤(𝑥, 𝑡), respectively; then we have the
following: (b) Analogous to part (a), we get
𝑡 𝑥
(a) if 𝑤(x, 𝑡) = ∫𝑡 ∫𝑥 𝑢(𝑦, 𝑧)V(𝑦, 𝑧)𝑑𝑦 𝑑𝑧, then 𝜕𝑘 𝜕𝑘 𝑡 𝑥
𝑘
𝑤 (𝑥, 𝑡) = 𝑘 (ℎ (𝑥, 𝑡) ∫ ∫ 𝑢 (𝑦, 𝑧) 𝑑𝑦 𝑑𝑧)
0 0
𝜕𝑡 𝜕𝑡 𝑡0 𝑥0
1 𝑥 𝑘−1 𝑘
𝑘 𝜕𝑟 𝑥
𝜕𝑘−𝑟−1
𝑊𝑘 (𝑥) = ∫ ( ∑ 𝑈 (𝑦) 𝑉𝑘−𝑟−1 (𝑦)) 𝑑𝑦, 𝑘 = 1, 2, . . . , = ∑ ( ) 𝑟 ℎ (𝑥, 𝑡) ∫ 𝑢 (𝑦, 𝑡) 𝑑𝑦,
𝑘 𝑥0 𝑟=0 𝑟 𝑟=0
𝑟 𝜕𝑡 𝑥0 𝜕𝑡
𝑘−𝑟−1
(19) (24)
Table 2: The fundamental operations of two-dimensional RDTM.
Original function Reduced transformed function

𝑤(𝑥, 𝑡) = 𝑢(𝑥, 𝑡) ± V(𝑥, 𝑡) 𝑊𝑘 (𝑥) = 𝑈𝑘 (𝑥) ± 𝑉𝑘 (𝑥)
𝜕 𝜕
𝑤(𝑥, 𝑡) = 𝑢(𝑥, 𝑡) 𝑊𝑘 (𝑥) = 𝑈 (𝑥)
𝜕𝑥 𝜕𝑥 𝑘
𝜕
𝑤(𝑥, 𝑡) = 𝑢(𝑥, 𝑡) 𝑊𝑘 (𝑥) = (𝑘 + 1)𝑈𝑘+1 (𝑥)
𝜕𝑡
𝜕𝑟+𝑠 (𝑘 + 𝑠)! 𝜕𝑟
𝑤(𝑥, 𝑡) = 𝑢(𝑥, 𝑡) 𝑊𝑘 (𝑥) = 𝑈 (𝑥)
𝜕𝑥𝑟 𝜕𝑡𝑠 𝑘! 𝜕𝑥𝑟 𝑘+𝑠
𝑘
𝑤(𝑥, 𝑡) = 𝑢(𝑥, 𝑡)V(𝑥, 𝑡) 𝑊𝑘 (𝑥) = ∑ 𝑈𝑟 (𝑥)𝑉𝑘−𝑟 (𝑥)
𝑟=0
{ 𝑥𝑚 𝑘=𝑛
𝑤(𝑥, 𝑡) = 𝑥𝑚 𝑡𝑛 𝑊𝑘 (𝑥) = 𝑥𝑚 𝛿(𝑘 − 𝑛) = {
{0 otherwise
therefore Proof. (a) By following the same manner as in the Theorem 7,

we get
𝑘
𝜕𝑘 𝑘 𝜕2 𝑤 (𝑥, 𝑡)
[ 𝑤 (𝑥, 𝑡)] = ∑ ( ) 𝑟! (𝑘 − 𝑟 − 1)! 𝐻𝑟 (𝑥) 𝑢 (𝑥, 𝑡) = V (𝑥, 𝑡) , (29)
𝜕𝑡𝑘 𝑟
𝑡=𝑡0 𝑟=0 𝜕𝑥 𝜕𝑡
𝑥
then
× ∫ 𝑈𝑘−𝑟−1 (𝑦) 𝑑𝑦,
𝑥0
𝜕𝑘 𝜕𝑘 𝜕2 𝑤 (𝑥, 𝑡)
𝑢 (𝑥, 𝑡) = ( V (𝑥, 𝑡))
𝑘
𝑘! 𝑥 𝜕𝑡𝑘 𝜕𝑡𝑘 𝜕𝑥 𝜕𝑡
=∑ 𝐻𝑟 (𝑥) ∫ 𝑈𝑘−𝑟−1 (𝑦) 𝑑𝑦, (30)
𝑟=0 𝑘 − 𝑟 𝑥0 𝑘
𝑘 𝜕𝑟+2 𝜕𝑘−𝑟
(25) = ∑( ) 𝑤 (𝑥, 𝑡) V (𝑦, 𝑡) ,
𝑟=0
𝑟 𝜕𝑥 𝜕𝑡𝑟+1 𝜕𝑡𝑘−𝑟
and then from using (15), for 𝑘 = 1, 2, . . ., we get therefore

𝑘
𝑘 𝜕𝑊
𝑘
1 𝑥 𝑘!𝑈𝑘 (𝑥) = ∑ ( ) (𝑟 + 1)! (𝑘 − 𝑟)! 𝑟+1 (𝑥) 𝑉𝑘−𝑟 (𝑥) , (31)
𝑟 𝜕𝑥
𝑊𝑘 (𝑥) = ∑ 𝐻 (𝑥) ∫ 𝑈𝑘−𝑟−1 (𝑦) 𝑑𝑦. (26) 𝑟=0
𝑟=0 𝑘 − 𝑟 𝑟 𝑥0
and then from using (15), for 𝑘 = 0, 1, 2, . . ., we get
Theorem 8. Assume that 𝑈𝑘 (𝑥), 𝑉𝑘 (𝑥), and 𝑊𝑘 (𝑥) are the
𝑘
reduced differential transforms of the functions 𝑢(𝑥, 𝑡), V(𝑥, 𝑡), 𝜕𝑊𝑟+1
𝑈𝑘 (𝑥) = ∑ (𝑟 + 1) (𝑥) 𝑉𝑘−𝑟 (𝑥) . (32)
and 𝑤(𝑥, 𝑡), respectively; then we have the following: 𝑟=0 𝜕𝑥
𝑡 𝑥 (b) Analogous to part (a), we get

(a) if 𝑤(𝑥, 𝑡) = ∫𝑡 ∫𝑥 (𝑢(𝑦, 𝑧)/V(𝑦, 𝑧))𝑑𝑦 𝑑𝑧, then
0 0
𝑡 𝑥
𝑤 (𝑥, 𝑡) V (𝑥, 𝑡) = ∫ ∫ 𝑢 (𝑦, 𝑧) 𝑑𝑦 𝑑𝑧, (33)
𝑘 𝑡0 𝑥0
𝜕𝑊𝑟+1 (𝑥)
𝑈𝑘 (𝑥) = ∑ (𝑟 + 1) 𝑉𝑘−𝑟 (𝑦) , 𝑘 = 0, 1, 2, . . . ,
𝑟=0 𝜕𝑥 then
(27) 𝜕𝑘 𝑥
𝜕𝑘−1
𝑘
{𝑤 (𝑥, 𝑡) V (𝑥, 𝑡)} = ∫ 𝑘−1
𝑢 (𝑦, 𝑡) 𝑑𝑦, (34)
𝜕𝑡 𝑥0 𝜕𝑡
𝑡 𝑥 therefore
(b) if 𝑤(𝑥, 𝑡) = (1/V(𝑥, 𝑡)) ∫𝑡 ∫𝑥 𝑢(𝑦, 𝑧)𝑑𝑦 𝑑𝑧, then
0 0
𝑘
𝑘 𝜕𝑟 𝜕𝑘−𝑟 𝑥
𝜕𝑘−1
∑ ( ) 𝑟 𝑤 (𝑥, 𝑡) 𝑘−𝑟 V (𝑥, 𝑡) = ∫ 𝑢 (𝑦, 𝑡) 𝑑𝑦,
𝑘 𝑥
𝑟=0
𝑟 𝜕𝑡 𝜕𝑡 𝑥0 𝜕𝑡
𝑘−1
𝑘∑𝑊𝑟 (𝑥) 𝑉𝑘−𝑟 (𝑥) = ∫ 𝑈𝑘−1 (𝑦) 𝑑𝑦, 𝑘 = 1, 2, . . . . (28) (35)
𝑟=0 𝑥0
and then from using (15), for 𝑘 = 1, 2, . . ., we get By applying the RDTM properties listed in Theorem 8, on
Volterra integral equation (38), for 𝑘 = 0, 1, 2, . . ., we get
𝑘 𝑥 𝑈𝑘 (𝑥)
𝑘!∑𝑊𝑟 (𝑥) 𝑉𝑘−𝑟 (𝑥) = (𝑘 − 1)! ∫ 𝑈𝑘−1 (𝑦) 𝑑𝑦, (36)
𝑟=0 𝑥0 𝑘
𝑑
= ∑ (𝑟 + 1)
𝑟=0 𝑑𝑥
and therefore
× {𝑈𝑟+1 (𝑥)
𝑘 𝑥
𝑘∑𝑊𝑟 (𝑥) 𝑉𝑘−𝑟 (𝑥) = ∫ 𝑈𝑘−1 (𝑦) 𝑑𝑦. (37)
𝑟=0 𝑥0 𝑟+1
− ∑ (𝑥𝛿ℓ,0 − 𝛿ℓ,1 )
ℓ=0
3. Numerical Results of DTM and RDTM 2 sin (𝑥 + ((𝑟+1−ℓ) 𝜋/2))

×[4𝛿𝑟+1−ℓ,0 +
(𝑟+1−ℓ)!
In this section, the reduced differential transform technique
is described to solve a class of Volterra integral equations
(1) with kernel functions of (2). In order to demonstrate the −𝑥𝛿𝑟+1−ℓ,1 ] }
effectiveness of the RDTM, the illustrative examples for the
same kernel function of [5] are presented. In each example,
(𝑘 − 𝑟) 𝜋
the numerical results of DTM, RDTM, and their comparisons × {2𝛿𝑘−𝑟,0 + sin (𝑥 + )} ,
with exact solution are given in separate tables. The results 2
of the test examples show that the RDTM results are more (41)
powerful than DTM results.
where 𝑈𝑖 (𝑥) is the reduced differential transform of 𝑢(𝑥, 𝑡).
Example 9. In the first example, consider the following two- After expanding the RDTM recurrence equations (41), with
dimensional Volterra integral equation [5]: initial value of (40), for 𝑘 = 0, 1, 2, 3, 4, the first five terms
of 𝑈𝑘 (𝑥) are obtained as follows:
𝑡 𝑥 𝑢 (𝑦, 𝑧) 𝑈1 (𝑥) = 2𝑥 cos (𝑥) − 4 − 2 sin (𝑥) ,
𝑢 (𝑥, 𝑡) − ∫ ∫ 𝑑𝑦 𝑑𝑧
0 0 2 + sin (𝑦 + 𝑧) 𝑈2 (𝑥) = − 𝑥 sin (𝑥) − 2 cos (𝑥) ,
(38)
= (𝑥 − 𝑡) (4 + 2 sin (𝑥 + 𝑡) − 𝑥𝑡) . 1
𝑈3 (𝑥) = sin (𝑥) − 𝑥 cos (𝑥) ,
3 (42)
1 1
𝑈4 (𝑥) = cos (𝑥) + 𝑥 sin (𝑥) ,
3 12
(a) DTM: Jang [5] solved this equation by using DTM and 1 1
obtained the following five-term DTM solution: 𝑈5 (𝑥) = 𝑥 cos (𝑥) − sin (𝑥) .
60 12
In the same manner, the rest of the components can be
𝑥42 2𝑥3 𝑥5 obtained by using the recursive equations (41). Substituting
𝑢5,5 (𝑥, 𝑡) = (4𝑥 + 2𝑥 − ) + (− 4 − + )𝑡
3 3 15 the quantities (41) in (18), the approximation solution of
Volterra integral equation (38) in the Poisson series form is
𝑥4 2 2𝑥 𝑥5 𝑈5 (𝑥, 𝑡) = 2𝑥 (2 + sin (𝑥)) + (2𝑥 cos (𝑥) − 4 − 2 sin (𝑥)) 𝑡
+ (− 2 + )𝑡 + ( − ) 𝑡3 (39)
12 3 180
𝑥 cos (𝑥) 3
1 𝑥2 𝑥 𝑥3 +(−𝑥 sin (𝑥)−2 cos (𝑥)) 𝑡2+(sin (𝑥)− )𝑡
+ ( − ) 𝑡4 + (− + ) 𝑡5 . 3
3 12 15 180
cos (𝑥) 𝑥 sin (𝑥) 4
+( + )𝑡
3 12
(b) RDTM: from Volterra integral equation (38), it is easy 𝑥 cos (𝑥) sin (𝑥) 5
to see that the 𝑢(𝑥, 0) = 𝑥(4 + 2 sin(𝑥)), and therefore +( − )𝑡 ,
60 12
RDTM version is (43)
which is the same as the first five terms of the Poisson series
𝑈0 (𝑥) = 𝑥 (4 + 2 sin (𝑥)) . (40) of the exact solution 𝑢(𝑥, 𝑡) = 2(𝑥 − 𝑡)(2 + sin(𝑥 + 𝑡)).
Table 3: Comparisons of the exact solution 𝑢(𝑥, 𝑡) = 2(𝑥 − 𝑡)(2 + sin(𝑥 + 𝑡)), with 𝑈5,5 (𝑥, 𝑡) obtained by classic DTM [5] and 𝑈5 (𝑥, 𝑡) obtained
by RDTM at some test points (𝑥, 𝑡) in Example 9.
Classic DTM [5] Reduced DTM

𝑥 𝑡 𝑢(𝑥, 𝑡)
𝑈5,5 (𝑥, 𝑡) |𝑢(𝑥, 𝑡) − 𝑈5,5 (𝑥, 𝑡)| 𝑈5 (𝑥, 𝑡) |𝑢(𝑥, 𝑡) − 𝑈5 (𝑥, 𝑡)|
0.1 +0.4591040413 +0.4591115320 7.4906677321𝑒 − 06 +0.4591040577 1.6393507773𝑒 − 08
0.4 −1.0258569894 −1.0257575253 9.9464024681𝑒 − 05 −1.0257906614 6.6327927673𝑒 − 05
0.2
0.7 −2.7833269096 −2.7813944067 1.9325029608𝑒 − 03 −2.7814532650 1.8736446239𝑒 − 03
1 −4.6912625375 −4.6755840000 1.5678537548𝑒 − 02 −4.6756686352 1.5593902330𝑒 − 02
0.1 +2.0517139787 +2.0516661667 4.7812049361𝑒 − 05 +2.0517139939 1.5172903822𝑒 − 08
0.4 +0.5566653819 +0.5573213333 6.5595140784𝑒 − 04 +0.5567258924 6.0510490391𝑒 − 05
0.5
0.7 −1.1728156344 −1.1698785000 2.9371343869𝑒 − 03 −1.1711316158 1.6840186272𝑒 − 03
1 −2.9974949866 −2.9817708333 1.5724153271𝑒 − 02 −2.9836944470 1.3800539634𝑒 − 02
0.1 +3.8966576735 +3.8929643413 3.6933321451𝑒 − 03 +3.8966576865 1.3025739598𝑒 − 08
0.4 +2.3456312688 +2.3439851520 1.6461167738𝑒 − 03 +2.3456822669 5.0998086549𝑒 − 05
0.8
0.7 +0.5994989973 +0.6014245867 1.9255893459𝑒 − 03 +0.6008906123 1.3916149372𝑒 − 03
1 −1.1895390524 −1.1754026667 1.4136385685𝑒 − 02 −1.1783733000 1.1165752332𝑒 − 02
0.1 +5.2041732481 +5.1881855000 1.5987748111𝑒 − 02 +5.2041732592 1.1127633925𝑒 − 08
0.4 +3.5825396760 +3.5680853333 1.4454342653𝑒 − 02 +3.5825824990 4.2822997450𝑒 − 05
1
0.7 +1.7949988863 +1.7840151667 1.0983719605𝑒 − 02 +1.7961453619 1.1464756153𝑒 − 03
1 +0.0000000000 −0.0000000000 −0.0000000000 +0.0090050384 9.0050384311𝑒 − 03
The numerical results obtained with RDTM are presented in (b) RDTM: it is easy to see that the 𝑢(𝑥, 0) = sin(𝑥), and
Table 3, in comparison with the classic DTM solution of [5] therefore RDTM version is
and the exact solution 𝑢(𝑥, 𝑡) = 2(𝑥 − 𝑡)(2 + sin(𝑥 + 𝑡)), for
some points of the intervals 0 ≤ 𝑥 ≤ 1 and 0 ≤ 𝑡 ≤ 1.
𝑈0 (𝑥) = sin (𝑥) . (46)
Example 10. In the second example, consider the following
two-dimensional Volterra integral equation [5]:
By applying the RDTM on nonlinear Volterra integral equa-

𝑡 𝑥 tion (44), for 𝑘 = 1, 2, . . ., we get
𝑢 (𝑥, 𝑡) − 2 ∫ ∫ 𝑒𝑦−𝑧 𝑢 (𝑦, 𝑧) 𝑑𝑦 𝑑𝑧
0 0
𝑘
sin (𝑥 + 𝑟𝜋/2) (−1)𝑘−𝑟 𝑒𝑥
= sin (𝑥 + 𝑡) (𝑒𝑥−𝑡 + 1) − 𝑒−𝑡 sin (𝑡) − 𝑒𝑥 sin (𝑥) . 𝑈𝑘 (𝑥) − {∑ ( + 𝛿𝑘−𝑟,0 )
(44) 𝑟=0 𝑟! (𝑘 − 𝑟)!
𝑘
(−1)𝑟 sin ((𝑘 − 𝑟) 𝜋/2)
−∑ − 𝛿𝑘,0 𝑒𝑥 sin (𝑥) }
𝑟=0 𝑟! (𝑘 − 𝑟)!
(a) DTM: the approximation solution of this equation is
also obtained by DTM in [5] as follows:
2 𝑥 𝑘−1 𝑦 (−1)𝑟
= ∫ {∑𝑒 𝑈 (𝑦)} 𝑑𝑦,
𝑥3 𝑥5 𝑥 2 𝑥4 𝑘 0 𝑟=0 𝑟! 𝑘−𝑟−1
𝑢5,5 (𝑥, 𝑡) = (𝑥 − + ) + (1 − + )𝑡 (47)
6 120 2 24
𝑥 𝑥3 𝑥5 1 𝑥2 𝑥4
+ (− + − ) 𝑡2 + (− + − ) 𝑡3 where 𝑈𝑖 (𝑥) is the reduced differential transform of 𝑢(𝑥, 𝑡).
2 24 240 6 12 144
After expanding the RDTM recurrence equations (47), with
𝑥 𝑥3 𝑥5 initial value of (46), for 𝑘 = 1, 2, 3, 4, 5, the first five terms of
+( − + ) 𝑡4 𝑈𝑘 (𝑥) are obtained as follows:
24 144 2880
1 𝑥2 𝑥4 𝑈1 (𝑥) = cos (𝑥) ,

+( − + ) 𝑡5 .
120 240 2880
1
(45) 𝑈2 (𝑥) = − sin (𝑥) ,
2
Table 4: Comparisons of the exact solution 𝑢(𝑥, 𝑡) = sin(𝑥 + 𝑡), with 𝑈5,5 (𝑥, 𝑡) obtained by classic DTM [5] and 𝑈5 (𝑥, 𝑡) obtained by reduced
DTM at some test points (𝑥, 𝑡) in Example 10.

0.1 0.2955202067 0.2955202184 1.1688660428𝑒 − 08 0.2955202070 2.9532337686𝑒 − 10
0.4 0.5646424734 0.5646439552 1.4818049646𝑒 − 06 0.5646439183 1.4448767682𝑒 − 06
0.2
0.7 0.7833269096 0.7833750550 4.8145422517𝑒 − 05 0.7833749959 4.8086256376𝑒 − 05
1 0.9320390860 0.9325020000 4.6291403277𝑒 − 04 0.9325019239 4.6283789648𝑒 − 04
0.1 0.5646424734 0.5646461680 3.6945737146𝑒 − 06 0.5646424741 6.8315986201𝑒 − 10
0.4 0.7833269096 0.7833397500 1.2840372517𝑒 − 05 0.7833299139 3.0042708277𝑒 − 06
0.5
0.7 0.9320390860 0.9321460859 1.0699997027𝑒 − 04 0.9321309857 9.1899716402𝑒 − 05
1 0.9974949866 0.9983398437 8.4485714595𝑒 − 04 0.9983208230 8.2583639762𝑒 − 04
0.1 0.7833269096 0.7834038828 7.6973172517𝑒 − 05 0.7833269106 1.0099717729𝑒 − 09
0.4 0.9320390860 0.9322215424 1.8245643277𝑒 − 04 0.9320433813 4.2953023213𝑒 − 06
0.8
0.7 0.9974949866 0.9978859380 3.9095139595𝑒 − 04 0.9976224907 1.2750404846𝑒 − 04
1 0.9738476309 0.9752880000 1.4403691218𝑒 − 03 0.9749626963 1.1150653929𝑒 − 03
0.1 0.8912073601 0.8915382743 3.3091424412𝑒 − 04 0.8912073612 1.1792198329𝑒 − 09
0.4 0.9854497300 0.9861662222 7.1649223376𝑒 − 04 0.9854546786 4.9486333631𝑒 − 06
1
0.7 0.9916648105 0.9928385451 1.1737346864𝑒 − 03 0.9918098802 1.4506974814𝑒 − 04
1 0.9092974268 0.9118055556 2.5081287299𝑒 − 03 0.9105512242 1.2537973843𝑒 − 03
1 Example 11. In the third example, consider the following two-

𝑈3 (𝑥) = − cos (𝑥) ,
6 dimensional Volterra integral equation [5]:
1 𝑡 𝑥
𝑈4 (𝑥) = sin (𝑥) , 𝑢 (𝑥, 𝑡) − 𝑒𝑡−𝑥 ∫ ∫ 𝑢 (𝑦, 𝑧) 𝑑𝑦 𝑑𝑧
24 0 0
1
𝑈5 (𝑥) = − cos (𝑥) . = sinh (𝑥 + 𝑡) (𝑒𝑡−𝑥 + 1) − 𝑒𝑡−𝑥 (sinh (𝑥) − sinh (𝑡)) .
120
(48) (50)
(a) DTM: the approximation solution of this equation is

In the same manner, the rest of the components were obtained also obtained by DTM in [5] as follows
by using the recursive equations (47). Substituting the quan-
tities (48) in (18), the approximation solution of Volterra 𝑥3 𝑥5 𝑥2 𝑥4
𝑢5,5 (𝑥, 𝑡) = (−𝑥 − − ) + (1 + + )𝑡
integral equation (44) in the Poisson series form is 6 120 2 24
𝑥 𝑥3 𝑥5 1 𝑥2 𝑥4
+ (− − − ) 𝑡2 + ( + + ) 𝑡3
sin (𝑥) 2 2 12 240 6 12 144
𝑈5 (𝑥, 𝑡) = sin (𝑥) + cos (𝑥) 𝑡 − 𝑡
2
(49) 𝑥 𝑥3 𝑥5
cos (𝑥) 3 sin (𝑥) 4 cos (𝑥) 5 + (− − − ) 𝑡4
− 𝑡 + 𝑡 − 𝑡, 24 144 2880
6 24 120
1 𝑥2 𝑥4
+( + + ) 𝑡5 .
120 240 2880
which is the same as the first five terms of the Poisson series (51)
of the exact solution 𝑢(𝑥, 𝑡) = sin(𝑥 + 𝑡). The numerical
results obtained with reduced DTM are presented in Table 4, (b) RDTM: it is easy to see that the 𝑢(𝑥, 0) = − sinh(𝑥),
in comparison with the classic DTM solution of [5] and the and therefore RDTM version is
exact solution 𝑢(𝑥, 𝑡) = sin(𝑥 + 𝑡), for some points of the
intervals 0 ≤ 𝑥 ≤ 1 and 0 ≤ 𝑡 ≤ 1. 𝑈0 (𝑥) = − sinh (𝑥) . (52)
Table 5: Comparisons of the exact solution 𝑢(𝑥, 𝑡) = sinh(𝑡 − 𝑥), with 𝑈5,5 (𝑥, 𝑡) obtained by classic DTM [5] and 𝑈5 (𝑥, 𝑡) obtained by reduced
DTM at some test points (𝑥, 𝑡) in Example 11.

0.1 −0.1001667500 −0.1001667561 6.0968226301𝑒 − 09 −0.1001667498 2.5944103810𝑒 − 10
0.4 +0.2013360025 +0.2013367851 7.8252557270𝑒 − 07 +0.2013368189 8.1631586063𝑒 − 07
0.2
0.7 +0.5210953055 +0.5211116472 1.6341756253𝑒 − 05 +0.5211117115 1.6406043554𝑒 − 05
1 +0.8881059822 +0.8881853333 7.9351145710𝑒 − 05 +0.8881854339 7.9451747271𝑒 − 05
0.1 −0.4107523258 −0.4107529453 6.1950968455𝑒 − 07 −0.4107523251 7.0149602793𝑒 − 10
0.4 −0.1001667500 −0.1001714167 4.6666468226𝑒 − 06 −0.1001641445 2.6055545215𝑒 − 06
0.5
0.7 +0.2013360025 +0.2013887643 5.2761781823𝑒 − 05 +0.2014033478 1.7345289013𝑒 − 05
1 +0.5210953055 +0.5215820313 4.8672575625𝑒 − 04 +0.5216052465 5.0994098756𝑒 − 04
0.1 −0.7585837018 −0.7585783977 5.3041062001𝑒 − 06 −0.7585837006 1.2071608158𝑒 − 09
0.4 −0.4107523258 −0.4108535808 1.0125499718𝑒 − 04 −0.4107476947 4.6310571234𝑒 − 06
0.8
0.7 −0.1001667500 −0.1002690359 1.4228584682𝑒 − 04 −0.1000423588 1.2439120515𝑒 − 04
1 +0.2013360025 +0.2019546667 6.1866412557𝑒 − 04 +0.2023226727 9.8667016118𝑒 − 04
0.1 −1.0265167257 −1.0264561563 6.0569458175𝑒 − 05 −1.0265167241 1.6018943949𝑒 − 09
0.4 −0.6366535821 −0.6370106667 3.5708451843𝑒 − 04 −0.6366473802 6.2019846800𝑒 − 06
1
0.7 −0.3045202934 −0.3051720521 6.5175863619𝑒 − 04 −0.3043519610 1.6833243259𝑒 − 04
1 +0.0000000000 −0.0000000000 −0.0000000000 +0.0013512390 1.3512390404𝑒 − 03
By applying the RDTM on nonlinear Volterra integral equa- In the same manner, the rest of the components were obtained
tion (50), for 𝑘 = 1, 2, . . ., we get by using the recursive equations (47). Substituting the quan-
tities (48) in (18), the approximation solution of Volterra
integral equation (44) in the Poisson series form is
𝑘
𝛿𝑟,0 (1 + 𝑒−𝑥 )
∑ {𝑈𝑟 (𝑥) +
2 sinh (𝑥) 2
𝑟=0 𝑈5 (𝑥, 𝑡) = − sinh (𝑥) + cosh (𝑥) 𝑡 − 𝑡
2
(55)
1 (−1)𝑘−𝑟 𝑥 cosh (𝑥) 3 sinh (𝑥) 4 cosh (𝑥) 5
+ (1+(−1)𝑟 𝑒𝑥 − (1+2𝑟 ) (𝑒−𝑥 +𝑒−2𝑥 )) } { 𝑒 } + 𝑡 − 𝑡 + 𝑡,
2𝑟! (𝑘 − 𝑟)! 6 24 120
1 𝑥 which is same as the first five terms of the Poisson series
= ∫ 𝑈 (𝑦) 𝑑𝑦,
𝑘 0 𝑘−𝑟 of the exact solution 𝑢(𝑥, 𝑡) = sinh(𝑡 − 𝑥). The numerical
(53) results obtained with reduced DTM are presented in Table 5,
in comparison with the classic DTM solution of [5] and the
where 𝑈𝑖 (𝑥) is the reduced differential transform of 𝑢(𝑥, 𝑡). exact solution 𝑢(𝑥, 𝑡) = sinh(𝑡 − 𝑥), for some points of the
After expanding the RDTM recurrence equations (53), with intervals 0 ≤ 𝑥 ≤ 1 and 0 ≤ 𝑡 ≤ 1.
initial value of (52), for 𝑘 = 1, 2, 3, 4, 5, the first five terms of
𝑈𝑘 (𝑥) are obtain as follows: 4. Conclusions
In this study, we presented the definition and operation of
𝑈1 (𝑥) = cosh (𝑥) ,
both two-dimensional differential transformation method
1 (DTM) and their reduced form, the so-called reduced-DTM
𝑈2 (𝑥) = − sinh (𝑥) , (RDTM) for finding the solutions of a class of Volterra inte-
2
gral equations. For illustration purposes, we consider three
1 different examples. It is worth pointing out that both DTM
𝑈3 (𝑥) = cosh (𝑥) , (54)
6 and RDTM have convergence for the solutions; actually, the
1 accuracy of the series solution increases when the number of
𝑈4 (𝑥) = sinh (𝑥) , terms in the series solution is increased. From the computa-
24
tional process of DTM and RDTM, we find that the RDTM
1 is easier to apply. In other words, it is obvious that DTM has
𝑈5 (𝑥) = cosh (𝑥) .
120 very complicated computational process rather than RDTM.
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http://dx.doi.org/10.1155/2013/161873
Research Article
A Pressure-Stabilized Lagrange-Galerkin Method in a Parallel
Domain Decomposition System
Qinghe Yao and Qingyong Zhu

School of Engineering, Sun Yat-Sen University, 510275 Guangzhou, China
Correspondence should be addressed to Qingyong Zhu; mcszqy@mail.sysu.edu.cn
Copyright © 2013 Q. Yao and Q. Zhu. This is an open access article distributed under the Creative Commons Attribution License,
A pressure-stabilized Lagrange-Galerkin method is implemented in a parallel domain decomposition system in this work, and the
new stabilization strategy is proved to be effective for large Reynolds number and Rayleigh number simulations. The symmetry of
the stiffness matrix enables the interface problems of the linear system to be solved by the preconditioned conjugate method, and
an incomplete balanced domain preconditioner is applied to the flow-thermal coupled problems. The methodology shows good
parallel efficiency and high numerical scalability, and the new solver is validated by comparing with exact solutions and available
benchmark results. It occupies less memory than classical product-type solvers; furthermore, it is capable of solving problems of
over 30 million degrees of freedom within one day on a PC cluster of 80 cores.
1. Introduction The present study is concentrated on improving the

solvability of the Lagrange-Galerkin method on large scale
The Lagrange-Galerkin method raises wide concern about and complex problems by domain decompositions. Piecewise
the finite-element simulation of fluid dynamics. Based on the linear interpolations are thus employed for velocity, pressure,
approximation of the material derivative along the trajectory and temperature; therefore, the so-called inf-sup condition
of fluid particle, the method is natural in the simulation to [11] should be satisfied, which is the first difficulty to be
physical phenomena, and it is demonstrated to be uncondi- overcome in this work. Stabilization methods for incom-
tionally stable for a wide class of problems [1–5]. A number of pressible flow problems were reported by many researchers
researches about the Lagrange-Galerkin method were done in (cf. [12–15]). Park and Sung proposed a stabilization for
the case of single processor element (PE) (cf. [6–8]); the sym- Rayleigh-Bénard convection by using feedback control [16];
metry of the matrices and good stability of the scheme were
for consistently stabilized finite element methods, Barth et al.
reported; using a numerical integration based on a division of
classified the stabilization techniques and studied influence of
each element, Rui and Tabata [9] developed a second scheme
the stabilization parameter in convergence [17]; Bochev et al.
for convection-diffusion problem; Massarotti et al. [10] used
a second-order characteristic curve method, and a special stated the requirements on choice of stabilization parameter
iteration was used to keep the symmetry of the stiffness if time step and mesh are allowed to vary independently
matrix. The Lagrange-Galerkin method uses an implicit time [18]. As far as we know, it may not be enough to investigate
discretization, and therefore an element searching algorithm what stabilization techniques are efficient for nonsteady
is necessary to implement it. The element searching may and nonlinear flow problems approximated by Lagrange-
become very expensive when the geometry is complicated or Galerkin methods in a domain decomposition system, where
the mesh size is very small. Due to its doubtable efficiency the interface problem can be solved by preconditioned
and feasibility for complex simulations in the case of single conjugate gradient (PCG) method. In this paper, a pressure-
PE, rare research has been done to implement it in parallel, by stabilization method, which keeps the symmetry of the
which the enormous computation power enables us to solve linear system and is effective for high Reynolds number and
more challenging simulation problems. Rayleigh number simulations, is introduced to implement
the Lagrange-Galerkin method in a domain decomposition The fluid is assumed to be incompressible according to
system. Boussinesq approximation, and the density is assumed to be
The element searching algorithm in a domain decompo- constant except in the gravity force term where it depends on
sition system using unstructured grids is the second difficulty temperature according to the indicated linear law; see (2). The
to implement the Lagrange-Galerkin method in a domain energy equation is
decomposition system (cf. [5, 19]). Minev et al. reported an
optimized binary searching algorithm for single PE by storing 𝜕𝑇
+ 𝑢 ⋅ ∇𝑇 − 𝑎Δ𝑇 = 𝑆 in Ω × (0, 𝑡) ,
the necessary data structures in a way similar to the CSR com- 𝜕𝑡
pact storage format; however, the element information data ̂
is stored distributedly in the domain decomposition system 𝑇=𝑇 on Γ2 × (0, 𝑡) ,
by the skyline format, and a different way needs to be found (4)
𝜕𝑇 𝜕Ω
to overcome the extra difficulty caused by the parallel com- 𝑎 =0 on ,
𝜕𝑛 Γ2 × (0, 𝑡)
puting algorithm. This step is critical, in the sense that it can
be very computationally expensive and can thus make the 𝑇 = 𝑇0 in Ω, at 𝑡 = 0,
entire algorithm impractical.
The remainder of this paper is organized as fol- where Γ2 ⊂ 𝜕Ω, 𝑎 is the thermal diffusion coefficient [m2 /s],
lows: in Section 2, the formula of the governing equation and 𝑆 is the source term with the unit of [𝐾/s].
and the pressure-stabilization Lagrange-Galerkin method is
described; Section 3 focuses on the parallel implementation
of this scheme. Numerical results and comparisons with 2.2. The Lagrange-Galerkin Finite-Element Method. Some
classical asymmetric product type methods in [20] are shown preliminaries are arranged for the derivation of a finite
in Section 4. Conclusions are drawn in Section 5. element scheme of (1) and (4). Let the subscript ℎ denote
the representative length of the triangulation, and let Iℎ ≡
{𝐾} denote a triangulation of Ω consisting of tetrahedral
2. Formulation elements. Given that 𝑔 is a vector valued function on Γ1 , the
finite element spaces are as follows:
2.1. The Governing Equations. Let Ω be a three-dimensional
polyhedral domain with the boundary 𝜕Ω. The conservation 3 󵄨
equations of mass and momentum are governed by 𝑋ℎ ≡ {Vℎ ∈ 𝐶0 (Ω) ; Vℎ 󵄨󵄨󵄨𝐾 ∈ 𝑃1 (𝐾)3 , ∀𝐾 ∈ Iℎ } ,
𝜕𝑢 󵄨
+ (𝑢 ⋅ ∇) 𝑢 − 2]∇ ⋅ 𝐷 (𝑢) + ∇𝑝 = 𝑓buoyancy 𝑀ℎ ≡ {𝑞ℎ ∈ 𝐶0 (Ω) ; 𝑞ℎ 󵄨󵄨󵄨𝐾 ∈ 𝑃1 (𝐾) , ∀𝐾 ∈ Iℎ } ,
𝜕𝑡
𝑉ℎ (𝑔) ≡ {Vℎ ∈ 𝑋ℎ ; Vℎ (𝑃) = 𝑔 (𝑃) , ∀𝑃 ∈ Γ1 } , (5)
in Ω × (0, 𝑡) ,
Θℎ (𝑏) ≡ {𝜃ℎ ∈ 𝑀ℎ ; 𝜃ℎ (𝑃) = 𝑏 (𝑃) , ∀𝑃 ∈ Γ2 } ,
∇⋅𝑢=0 in Ω × (0, 𝑡) ,
(1) 𝑉ℎ ≡ 𝑉ℎ (0) , Θℎ ≡ Θℎ (0) , 𝑄ℎ = 𝑀ℎ .
𝑢 = 𝑢̂ on Γ1 × (0, 𝑡) ,
3
Let (⋅, ⋅) defines the 𝐿 2 inner product; the continuous
𝜕Ω bilinear forms 𝑎 and 𝑏 are introduced by
∑ 𝜎𝑖𝑗 𝑛𝑗 = 0 on ,
𝑗=0 Γ1 × (0, 𝑡)
𝑎 (𝑢, V) ≡ 2] (𝐷 (𝑢) , 𝐷 (V)) ,
𝑢 = 𝑢0 in Ω, at 𝑡 = 0, (6)
𝑏 (𝑢, V) ≡ − (∇ ⋅ 𝑢, 𝑞) ,
where Γ1 ⊂ 𝜕Ω and
respectively.
𝑓buoyancy = 𝛽 (𝑇𝑟 − 𝑇) 𝑔 (2) Let Δ𝑡 be the time increment, and let 𝑁𝑡 ≡ [𝑡/Δ𝑡] be the
is the gravity force per unit mass derived on the basis of total step number. Let the superscript 𝑛 denote the time step;
a finite element approximation of (1) is described as follows:
Boussinesq approximation. 𝑔 is the gravity [m/s2 ], 𝛽, 𝑇, 𝑁𝑡
and 𝑇𝑟 are the thermal expansion coefficient [1/𝐾], the find {(𝑢ℎ𝑛 , 𝑝ℎ𝑛 )}𝑛=1 ∈ 𝑉ℎ (𝑔)×𝑄ℎ , such that for (Vℎ , 𝑞ℎ ) ∈ 𝑉ℎ ×𝑄ℎ ,
temperature [𝐾], and the reference temperature [𝐾], and
𝑢, 𝑡, ], and 𝑝 are velocity vector [m/s], time [s], kinematic 𝑢ℎ𝑛 − 𝑢ℎ𝑛−1 ∘ 𝑋1 (𝑢ℎ𝑛−1 , Δ𝑡)
( , Vℎ ) + 𝑎 (𝑢ℎ𝑛 , Vℎ )
viscosity coefficient [m2 /s], and kinematic pressure [m2 /s2 ], Δ𝑡
respectively. 𝜎𝑖𝑗 is the stress tensor [N/m2 ] defined by (7)
+ 𝑏 (Vℎ , 𝑝ℎ𝑛 ) = (𝑓𝑛 , Vℎ ) ,
𝜎𝑖𝑗 (𝑢, 𝑝) ≡ −𝑝𝛿𝑖𝑗 + 2]𝐷𝑖𝑗 (𝑢) ,
𝑏 (𝑢ℎ𝑛 , 𝑞ℎ ) = 0,
1 𝜕𝑢𝑖 𝜕𝑢𝑗 (3)
𝐷𝑖𝑗 (𝑢) ≡ ( + ), 𝑖, 𝑗 = 1, 2, 3, where 𝑋1 (⋅, ⋅) denotes a first-order approximation of a parti-
2 𝜕𝑥𝑗 𝜕𝑥𝑖
cle’s position [5], and the notation ∘ denotes the composition
with the Kronecker delta 𝛿𝑖𝑗 . of functions.
For the purpose of large scale computation, a piecewise Step 4. Compute the relative error by a 𝐻1 × 𝐿2 × 𝐻1 norm
equal-order interpolation for velocity and pressure is used, defined by
as can be seen from (5). Pressure stabilization is thus
needed to keep the necessary link between 𝑉ℎ and 𝑄ℎ . A 󵄩󵄩 󵄩 1
󵄩󵄩(𝑢, 𝑝, 𝑇)󵄩󵄩󵄩𝐻1 ×𝐿2 ×𝐻1 ≡ ‖𝑢‖ 1 3
penalty Galerkin least-squares (GLS) stabilization method for √Re 𝐻 (Ω) (13)
pressure is proved in [12] to hold the same asymptotic error 󵄩 󵄩
+ 󵄩󵄩󵄩𝑝󵄩󵄩󵄩𝐿2 + ‖𝑇‖𝐻1 (Ω) ,
estimates as the method of Hughes et al. [21] and it is com-
putationally cheap. For P1/P1 elements, the stabilization is where Re denotes the Reynolds number, and set
reduced to
󵄩󵄩 𝑛 𝑛 𝑛 󵄩
2 󵄩󵄩(𝑢 , 𝑝 , 𝑇 ) − (𝑢𝑛−1 , 𝑝𝑛−1 , 𝑇𝑛−1 )󵄩󵄩󵄩 1 2 1
∑ 𝛿𝐾 ℎ𝐾 (∇𝑝ℎ𝑛 , −∇𝑞ℎ )𝐾 , 󵄩 󵄩𝐻 ×𝐿 ×𝐻
(8) diff = 󵄩󵄩 𝑛−1 𝑛−1 𝑛−1 󵄩󵄩
𝐾∈Iℎ
󵄩󵄩 (𝑢 , 𝑝 , 𝑇 )󵄩󵄩𝐻1 ×𝐿2 ×𝐻1 (14)
which does no modification to the momentum equation ≤ ErrNS
because of vanishing of the second-order derivate term. Here,
ℎ𝐾 denotes the maximum diameter of an element 𝐾. Unlike as the steady-state criterion; if (14) is satisfied or the number
[6, 12], where a constant 𝛿 (>0) is used as the stabilization of loops reaches the maximum, then stop the iteration;
parameter, an element-wise stabilization parameter otherwise, repeat Steps 1–3.
{ 𝛼, As can be seen from Steps 2 and 3, both (1) and (4)

{
{ 󵄩 󵄩 4
{for log10 [Max {󵄩󵄩󵄩󵄩∇𝑝ℎ𝑛−1 󵄩󵄩󵄩󵄩2 }𝑖=1 ] ≤ 1,
{ are approximated by the Lagrange-Galerkin method, and the
𝛿𝐾 = { 󵄩 󵄩 4 (9) searching algorithm only needs to be performed once in a
{
{
{ 𝛼 × log10 [Max {󵄩󵄩󵄩󵄩∇𝑝ℎ𝑛−1 󵄩󵄩󵄩󵄩2 } ] , nonsteady loop. It can also be seen that the solver is also
{ 𝑖=1
flexible, and it can solve pure Navier-Stokes problems by
{ otherwise
setting the body force in (2) to external force and omitting
is used in this work, where ∇𝑝ℎ𝑛−1 is gradient of the FEM Step 2.
approximated pressure at 𝑡𝑛−1 and 𝑖 is the number of the nodal
point in a tetrahedral element. Since 𝛼 is very important to 3. Implementation
balance the accuracy and convergence of the scheme, it is
discussed in Section 4.1. The localized stabilization parameter 3.1. A Parallel Domain Decomposition System. To begin with
is designed to be adaptive to the pressure gradient, and thus the parallel domain decomposition method, the domain
it has a better control on the pressure field. decomposition is introduced briefly as follows. The whole
By adding (8) to (7), a pressure-stabilized FEM scheme for domain is decomposed into a number of subdomains without
Navier-Stokes problems is achieved. The nonsteady iteration overlapping, and the solution of each subdomain is super-
loops for solving (1) and (4) and then reads the following. imposed on the equation of the inner boundary of the
subdomains. By static condensation, the linear system
Step 1. Compute the particle’s coordinates by
𝐾𝑢 = 𝑓 (15)
𝑋1 (𝑢ℎ𝑛−1 , Δ𝑡) ≡ 𝑥 − 𝑢ℎ𝑛−1 Δ𝑡, (10)
is written as
𝑛−1
and search the element holding the particle at 𝑡 . 𝐾𝐼𝐼(1)
0 ⋅⋅⋅ 0 (1) (1)
𝐾𝐼𝐵 𝑅𝐵 𝑢(1)
[ ][ 𝐼 ]
Step 2. Find 𝑇ℎ𝑛 by [ .. .. ] [ .. ]
[ 0 d . . ][ . ]
[ ][ ]
[ .. .. .. ] [ .. ]
𝑇ℎ𝑛 − 𝑇ℎ𝑛−1 ∘ 𝑋1 (𝑢ℎ𝑛−1 , Δ𝑡) [ . d . . ][ . ]
[ ][ ]
( , 𝜃ℎ ) + (𝑎∇𝑇ℎ𝑛 , ∇𝜃ℎ ) = (𝑆𝑛 , 𝜃ℎ ) . [ 0 ⋅⋅⋅ ⋅⋅⋅ (𝑁)
𝐾𝐼𝐼 (𝑁) (𝑁) ] [ (𝑁) ]
𝐾𝐼𝐵 𝑅𝐵 ] [𝑢𝐼 ]
Δ𝑡 [
(1)𝑇 (1)𝑇
(11) [𝑅𝐵 𝐾𝐼𝐵 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 𝑅𝐵(𝑁)𝑇 𝐾𝐼𝐵
(𝑁)𝑇
𝐾𝐵𝐵 ] [ 𝑢𝐵 ]
(16)
Step 3. Find (𝑢ℎ𝑛 , 𝑝ℎ𝑛 ) by (1)
𝑓𝐼
[ . ]
[ . ]
𝑢ℎ𝑛 − 𝑢ℎ𝑛−1 ∘ 𝑋1 (𝑢ℎ𝑛−1 , Δ𝑡) [ . ]
( , Vℎ ) + 𝑎0 (𝑢ℎ𝑛 , Vℎ ) [ . ]
Δ𝑡 =[ ]
[ .. ] ,
[ (𝑁) ]
[𝑓 ]
+ 𝑏 (Vℎ , 𝑝ℎ𝑛 ) + 𝑏 (𝑢ℎ𝑛 , 𝑞ℎ ) [ 𝐼 ]
(12)
2 [ 𝑓𝐵 ]
+ ∑ 𝛿𝐾 ℎ𝐾 (∇𝑝ℎ𝑛 , −∇𝑞ℎ )𝐾
𝐾∈Iℎ where 𝐾 is the stiffness matrix, 𝑢 denotes the unknowns (𝑢
and 𝑝), and 𝑓 is the force vector. 𝑅 is the restriction operator
= (𝑓𝑛 , Vℎ ) + (𝛽 (𝑇𝑟 − 𝑇ℎ𝑛 ) 𝑔, Vℎ ) . consists of 0-1 matrix. The superscripts (𝑁) means the
𝑁th subdomain, and subscript 𝐼 and 𝐵 relate to the element Part interface
R2
of the inner boundary, and interface boundary respectively.
From (16), it can be observed that the interface problems
ne (𝜆 2 )
𝑁 𝑇
(𝑖)𝑇 (𝑖)−1 (𝑖)
∑𝑅𝐵(𝑖) (𝑖)
(𝐾𝐵𝐵 − 𝐾𝐼𝐵 𝐾𝐼𝐼 𝐾𝐼𝐵 ) 𝑅𝐵(𝑖) 𝑢𝐵 ecurrent
ei
𝑖=1
(17)
𝑁 ne (𝜆 1 ) ne (𝜆 3 )
𝑇(𝑖) 𝑇 −1(𝑖)
= ∑𝑅𝐵(𝑖) (𝑓𝐵 − (𝑖)
𝐾𝐼𝐵 (𝑖)
𝐾𝐼𝐼 𝑓𝐼 )
𝑖=1
Figure 1: A searching algorithm.
and the inner problems
−1 (𝑖) 𝑇 (1) initialize: 𝑒0 = 𝑒current ;

𝑢(𝑖) (𝑖) (𝑖) (𝑖)
𝐼 = 𝐾𝐼𝐼 (𝑓𝐼 − 𝐾𝐼𝐵 𝑅𝐵 𝑢𝐵 ) , 𝑖 = 1, . . . , 𝑁 (18)
(2) iterate 𝑖 = 0, 1, . . . , Maxloops;
If 𝜆 1 , 𝜆 2 , 𝜆 3 > 0, return 𝑒𝑖 ;
can be solved separately [22]. In this work, the interface
problems are solved first iteratively, and the inner problems else if 𝑛𝑒(Min{𝜆 1 , 𝜆 2 , 𝜆 3 }) ≠
boundary
are then solved by substituting 𝑢𝐵 in to (18). 𝑒𝑖+1 = 𝑛𝑒(Min{𝜆 1 , 𝜆 2 , 𝜆 3 });
The Lagrange-Galerkin method keeps the symmetry of else break;
the stiffness matrix, and the GLS pressure-stabilization term
in (8) also produces a symmetric matrix; therefore, 𝐾 is sym- (3) return 𝑒𝑖 .
metric in (15), and a PCG method is employed to get the 𝑢𝐼
The request of the old solutions, which is the 𝑢ℎ𝑛−1 in (10),
in (18), and to avoid drawback of the classical domain decom-
is relatively trivial when using single PEs or simply solving the
position method, such as Neumann-Neumann and diagonal-
problem parallel using symmetric multiprocessing; however,
scaling, a balanced domain decomposition preconditioner is
in the domain decomposition system, the particle is not
used to prevent the growing of condition number with the
limited within one part; it may pass the interface of different
number of subdomains. An identity matrix is chosen as the
parts, as can be seen from Figure 1. Because one PE only has
coarse matrix, and the coarse problem is solved incompletely
the elements information that belongs to the current part,
by omitting the fill-ins in some sensitive places during
communications between PEs are necessary. However, the
the Cholesky factorization. By using this inexact balanced
number of total elements in one subdomain may be different,
domain decomposition preconditioning, the coarse matrix is
which means that some point to point communication
sparser and thus easier to be solved; therefore, the new solver
techniques, such as MPI Send/MPI Recv or MPI Sendrecv
is expected to have better solvability on large scale computa-
in MPICH, cannot be used in element wise computation. In
tion models.
the previous research [23], a global variable to store all the
old solutions is constructed. This method maintains a high
3.2. The Lagrange-Galerkin Method in Parallel. The element computation speed but costs a huge memory usage. To reduce
searching algorithm requires a global-wise element informa- the memory consumption, a request-response system is used
tion to determine the position of one particle in the previous in this work. In the computation, the searching algorithm is
time step. However, in the parallel domain decomposition performed first, and the element that contains the current
system, the whole domain is split into several parts one particle in the previous time step is thus known; therefore,
processor element (PE) works only on the current part, and the PE to get 𝑢ℎ𝑛−1 from is also known. However, as the
it does not contain any element information of other parts. sender does not know which PE requires message from itself,
Each part is further divided into many subdomains, and the the receiver has to send its request to the sender first; after
domain decomposition is performed by the PE in charge of the request is detected, the sender sends the message to the
the part. This parallelity causes a computational difficulty: for receiver. The procedure is as follows:
each time step, the particle is not limited within one part;
therefore, exchanging the data between different PEs is (1) by scanning all the particles in the current subdomain,
necessary, which demands the PEs to communicate in the an array including all the data that is needed by the
subdomain-wise computation. current PE is sent to all the other PEs.
In order to know the position of a particle at 𝑡𝑛−1 , a (2) All PEs check if there is any request to itself. If it exists,
neighbour elements list is created at the beginning of the PEs will prepare an array of the needed data and send
analysis. Based on the information of neighbour elements and it.
the coordinates calculated by (10), it is possible to find the (3) The current PE receives the data sent by other PEs.
element holding this particle at 𝑡𝑛−1 . A 2-dimensional search-
ing algorithm is present as follows (𝜆 𝑖 is the barycentric coor- Data transferred by MPI communication should be pack-
dinates, and 𝑛𝑒(𝜆 𝑖 ) is the neighbour element; see Figure 1): aged properly to avoid the overflow of MPI buffer in case
1E+04 1E+04
1E+03 1E+03
1E+02 1E+02
1E+01 1E+01
1E+00 1E+00
Residual
Residual
1E−01 1E−01
1E−02 1E−02
1E−03 1E−03
1E−04 1E−04
1E−05 1E−05
1E−06 1E−06
1E−07 1E−07
0 1000 2000 3000 4000 5000 0 2000 4000 6000 8000 10000
Iteration times Iteration times
𝛿= 0.001 𝛿=1 Re = 10E6 (𝛿 = 0.005) Re = 10E3 (𝛿 K)

𝛿= 0.005 𝛿 = 10 Re = 10E5 (𝛿 = 0.005) Re = 10E4 (𝛿 K)
𝛿= 0.01 𝛿 = 100 Re = 10E3 (𝛿 = 0.005) Re = 10E5 (𝛿 K)
𝛿= 0.1 Re = 10E4 (𝛿 = 0.005) Re = 10E6 (𝛿 K)
(a) (b)
Figure 2: Convergence of (a) different constant 𝛿 at Re = 103 ; (b) different Re for 𝛿 = 0.005 and localized 𝛿𝐾 .
800
700
Number of iterations
600
500
400
300
200
100
0
90 180 270 360 450 540 630 720 810 900 990 1080
Number of subdomains
None
Current
Figure 3: Numerical scalability.
of large-scale computation. Nonblocking communication is It was set as the default preconditioner for all the following
employed, and as the 3 steps are performed subsequently, computations of this research.
thus the computation time and communication time will be The penalty methods are not consistent since the sub-
overlapped. stitution of an exact solution into the discrete equations
(12) leaves a residual that is proportional to the penalty
parameter (cf. [17]); therefore, 𝛿𝐾 should be determined
4. Numerical Results and Discussion carefully. Numerical experiments of a lid-driven cavity flow
The parallel efficiency of new solver is firstly evaluated in were tested, and the mesh size was 62 × 62 × 62. The total
this section, and to validate the scheme, exact solutions and degrees of freedom (DOF) are 1,000,188, and the results
available benchmark results classical computational models are given by Figure 2. For the purpose of higher accuracy,
are compared. The CG convergence is judged by Euclidian 𝛿𝐾 is expected to be small; however, the convergence turns
worse when 𝛿𝐾 goes small, as can be seen from Figure 2(a).
norm with a tolerance of 10−6 , and for nonsteady iteration,
In Figure 2(b), a constant 𝛿 = 0.005 is used for different
ErrNS = 10−4 is set as the criterion, using the 𝐻1 × 𝐿2 × 𝐻1 Reynolds numbers, and no convergence is achieved within
norm defined in (13). For pure Navier-Stokes problems, a
10000 PCG loops for Re = 106 ; and the comparison shows
similar 𝐻1 × 𝐿2 norm, which is related to velocity and
that the 𝛿𝐾 performs much better than a constant 𝛿 = 0.005
pressure, is employed to judge the steady state.
when 𝛼 is set to 0.005.
The parallel efficiency is assessed firstly by freezing the
4.1. Efficiency Test. The BDD serious preconditioners were mesh size of test problem and refining the domain decom-
employed in this work; they are very efficient, and their position by decreasing the subdomain size and therefore
iteration numbers are about 1âĄĎ10 of the normal domain increasing the number of subdomains; the comparison of the
decomposition preconditioners (cf. [23]). The inexact pre- numerical scalability of the current scheme with and without
conditioner mentioned in Section 3 also shows good conver- the preconditioner is assessed by Figure 3. It can be seen that
gence and is more suitable for large scale computations [24]. with the preconditioning technique, the iterative procedure
2,800
500
2,400
Total memory usage (GB)

2,000 400
Elapsed time (s)
1,600
300
1,200
200
800
100
400
0 0
5,000 35,000 65,000 95,000 125,000 155,000 185,000 5,000 35,000 65,000 95,000 125,000 155,000 185,000
DOF DOF
BiCGSTAB (ADV sFlow 0.5) BiCGSTAB (ADV sFlow 0.5)

BiCGSTAB2 (ADV sFlow 0.5) BiCGSTAB2 (ADV sFlow 0.5)
GPBiCG (ADV sFlow 0.5) GPBiCG (ADV sFlow 0.5)
Current Current
(a) (b)
Figure 4: Time and memory usages.
8 The test problem was solved by the new solver and

7 the ADV sFlow 0.5 [25], which contains some nonsym-
6 metric product-type solvers like GPBiCG, BiCGSTAB, and
Speed-up
5 BiCGSTAB2 [20]. The ADV sFlow 0.5 employed a domain

4 decomposition system similar to the work; however, no
3 precondition technique is used because of the non-symmetry
2 of the stiffness matrix in (15). The comparisons of elapsed
1 time and memory occupation of the new solver and that of
8 16 24 32 40 48 56 64 product-type solvers in ADV sFlow 0.5 are show in Figures
Number of PEs 4(a) and 4(b). As can be seen, the current scheme reduces
Model1 (DOF: 148,955)
the demand of computation time and memory consumption
Model2 (DOF: 1,134,905) remarkably, and it is more suitable for large scale problems
Model3 (DOF: 12,857,805) than product-type solvers.
Ideal The parallel scalability of the searching algorithm is
also a concern for us, as it characterizes the ability of an
Figure 5: Numerical scalability. algorithm to deliver larger speed-up using a larger number
of PEs. To know this, the number of subdomains in one part
is fixed, and computations on the test problem of various
mesh sizes are performed by the new scheme. The speed-
up is shown in Figure 5. Three models were tested by the
searching algorithm. With an increase in the mesh size
of current scheme converges rapidly, and the convergence is of the computation model, the parallel scalability of the
independent of the number of subdomains. searching algorithm tends to be better. An explanation to this
Based on the paralyzed Lagrange-Galerkin method, the is that when the DOF increase, the number of elements in
new solver makes a symmetric stiffness matrix, therefore one subdomain is also increasing; therefore, the searching
only the lower/upper triangular matrix needs to be saved. algorithm is accelerated more efficiently. However, too many
Moreover, nonblocking MPI communication is used instead elements in one subdomain will occupy more memory, and a
of constructing global arrays to keep the old solutions, trade-off strategy is necessary for parameterization.
and the current solver is expected to reduce the memory
consumption without sacrificing the computation speed. The
usage of time and memory of solving the thermal driven 4.2. Validation Tests. In this section, a variety of test problems
cavity problem by different solvers is compared, and the have been presented in order to prove the capability of the
results are given by Figure 4. parallel Lagrange-Galerkin algorithm. Benchmarks test of
x3
x3
1 1
x2
b 1
0 4 x1
x2
Figure 6: A plane Couette flow model.
1
1
0.9
0.8 0 1 x1
0.7
0.6 Figure 8: A lid-driven cavity model.
x3 /b
0.5
0.4
0.3
0.2 lines are used to present the results, and they are named as
0.1
0
“Num Res 1(𝐵),” where 𝐵 stands for the Brinkman number.
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Crossed lines in Figure 7 present the computation results with
u3 /U no exact solutions setting on the left and right faces, and they
are named as “Num Res 0(𝐵).”
Exac Val(−3) Exac Val(−2)
It can be seen form Figure 7 that both of these two sets
Exac Val(−1) Exac Val(0)
Exac Val(1) Exac Val(2)
of computation results show good agreement with the exact
Exac Val(3) Num Res 0(−3) solution, and dotted lines are closer to the exact solution,
Num Res 0(−2) Num Res 0(−1) representing a better simulation to the ideal condition (cf.
Num Res 0(0) Num Res 0(1) [27, 28]).
Num Res 0(2) Num Res 0(3)
Num Res 1(−3) Num Res 1(−2)
Num Res 1(−1) Num Res 1(0) 4.2.2. A Lid-Driven Cavity. The Navier-Stokes problems
Num Res 1(1) Num Res 1(2) solver was then verified by a lid-driven cavity flow. The
Num Res 1(3) ideal gas flows over the upper face of the cube, and no-slip
conditions are applied to all other faces, as in Figure 8.
Figure 7: Numerical results versus exact solutions.
All the faces of the cube were set with Dirichlet boundary
conditions, and a zero reference pressure was at the centre of
the cube to keep the simulation stable. The pressure profiles of
Navier-Stokes problems are in Sections 4.2.1, 4.2.2, and 4.2.3, the scheme using localized stabilization parameter in (9) and
and flow-thermal coupled problems are in Sections 4.2.4 and the scheme using constant (𝛿 = 1) parameter are compared,
4.2.5. and the results are show in Figure 9.
Figure 9(a) shows the pressure counters of the scheme
4.2.1. A Plane Couette Flow. The solver for Navier-Stokes with the localized stabilization parameter in (9) and the
equations in (1) was first tested with a 3D plane Couette Figure 9(b), shows scheme with a constant parameter. The
flow. Under ideal conditions, the model is of infinite length; model was run at Re = 104 , and oscillations are viewed
therefore, 4 times of the height is used as the length of the in Figure 9(b); however, the isolines in Figure 9(a) is quite
model see Figure 6. A constant velocity (̂ 𝑢, 0, 0) is applied smooth, showing that the pressure-stabilization term has a
on the upper horizontal face, and no-slip conditions are set better control on the pressure field at high Reynolds number.
on the lower horizontal face. A pressure gradient is imposed The model was run at different Reynolds numbers with a
along 𝑥1 for all the faces as essential boundary conditions. 128×128×128 mesh to test the solvability of the new scheme.
An unstructured 3D mesh was generated by ADVEN- As shown in Figure 10, when the Reynolds number increases,
TURE TetMesh [25], and the local density around the plane the eddy at right bottom of plane 𝑥1 𝑥3 vanishes, while the
of 𝑥1 = 2, where the data was picked from, was enriched. eddy at the left bottom appears due to the increasing in the
The total DOF is around 1,024,000. The so-called Brinkman speed, and the flow goes more likely around the wall. The
number [26] is believed to be the dominating parameter of primary eddy goes lower and lower when Reynolds number
the flow, and a serious of numerical experiments is done at becomes higher, and the particle is no longer limited to a
various Brinkman number. To simulate the infinity length single side of the cavity; it can pass from one side to the other,
better, the exact solution is enforced on both the left face and back again violating the mirror symmetry, as is seen from
(𝑥1 = 0) and the right face (𝑥1 = 4) as Dirichlet bound- other planes of Figure 10. Similar 3D results for high Reynolds
ary conditions. The comparisons between the computation number were reported by [29], and the solvability of the new
results and exact solutions are given by Figure 7. Dotted solver for high Reynolds number was confirmed.
0.15 0.21
0.12 0.16
0.09 0.12
0.06 0.07
0.03 0.02
0 −0.03
(a) (b)
Figure 9: Pressure counters (Re = 104 ).
Figure 10: Velocity and pressure profiles for different Reynolds number: Re = 1,000 (top), Re = 3,200 (middle), and Re = 12,000 (bottom)
along different middle planes: plane 𝑥1 𝑥2 (left), plane 𝑥1 𝑥3 (middle), and plane 𝑥2 𝑥3 (right).
4.2.3. Backward Facing Step. The solver for Navier-Stokes nodal points and 2,417,575 tetrahedral elements, and the local
equations was then tested with backward facing step, the density of mesh was increased around the step.
fluid considered was air. The problem definition is shown in A laminar flow is considered to enter the domain at
Figure 11, and the height of the step ℎ is the characteristic inlet section, the inlet velocity profile is parabolic, and the
length. An unstructured 3D mesh was generated with 419,415 Reynolds number is based on the average velocity at the inlet.
x2
x3 L = 1.5
H = 0.101
0.9
l = 0.05
=
W
h = 0.052
S = 0.049 x1
Figure 11: Backward facing steps.
x3
(a) 1
r Tlow Thigh
4
(b)
Figure 12: Pressure counters (a) and velocity vectors (b) (Re = 200).
x2
16
x1
14
Reattachment length (r/S)
12 1
10 Figure 14: The model of infinite plates.
8
6
4
The comparison of primary reattachment length between
2 current results and other available benchmark results are
0 show in Figure 13. It is seen that the agreement is excellent
0 100 200 300 400 500 600 700 800 900
at different Rayleigh numbers (cf. [31, 32]).
Re
Exp. (Armaly et al.) Williams et al. 4.2.4. Natural Convection of Flat Plates. In order to test
Jiang et al. Current
Ku et al.
the coupled solver of Navier-Stokes equations and the
convection-diffusion equation, the third application model
Figure 13: Primary reattachment lengths. was the natural convection between two infinite flat plates.
The geometry is given in 3-dimensional by Figure 14. No-
slip boundary conditions applied on the left and right vertical
walls. The temperature on the left wall is assumed to be lower
The total length of the domain is 30 times the step height, so and set at 5[𝐾]; the right wall is set at 6[𝐾]. An unstructured
that the zero pressure is set at the outlet. A full 3D simulation 3D mesh about 1 million tetrahedral elements was generated,
of the step geometry for 100 ≤ Re ≤ 800 is present in this and the local grid density around the mid-plane was enriched.
paper, and the primary reattachment lengths are predicted. The model was run at the size of 20 × 20 × 80 to get
To determine the reattachment length, the position of the numerical solutions, and it was compared with the exact
the zero-mean-velocity line was measured. The points of solutions in Figure 15. To simulate the infinity length of the
detachment and reattachment were taken as the extrapolated plate better, the exact solution is enforced on both the upper
zero-velocity line down the wall. The pressure contour in face (𝑥3 = 4) and the lower face (𝑥3 = 0) as what is
Figure 12(a) confirms the success of the pressure-stabilization done in Section 4.2.1, and the results are present by a dotted
method; velocity vectors and the primary attachment are line (“Num Res 1”) in Figure 12. And the model without
demonstrated in Figure 12(b); similar results have been doc- exact solution set as boundary is named as “Num Res 0” in
umented by many, like in [10, 30]. Figure 15.
u3 (·, 0, 2) coupled problems described by (1) and (4). Similar three-

0.2 dimensional results can also be found in [33, 34]. The pressure
0.15 profile in Figure 17(d) is smooth and symmetric, implying
0.1 that the stabilization item in (8) works well.
0.05
In order to further validate the new solver, a comparison
of temperature and velocity profiles of the current solver and
0 x1
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 other benchmark results was made. The centreline velocity
−0.05 results 𝑤(⋅, 0.5, 0.5) and the temperature results 𝑇(⋅, 0.5, 0.5),
−0.1 which are believed to be very sensitive in this simulation,
−0.15 are present in diagrams (a) and (b) of Figure 18, respectively.
−0.2 The velocity results share close resemblance to that of the
ADV sFlow 0.5, and they both show the more end-wall effects
Exact value compared with the results of 2D case. The three temperature
Num Res 0
results show good agreement with each other, and the line
Num Res 1
representing the current results is the smoothest, as the
Figure 15: Numerical results versus exact solutions. mesh is the finest among the three. Similar results have been
reported by other researchers (cf. [10, 33, 34]).
Thermal convection problems are believed to be domi-
nated by two dimensionless numbers by many researchers,
x3 the Prandtl number and the Rayleigh number. To acquaint
ourselves with the solvability of the new solver and to
challenge applications of higher difficulty, a wide range of
Tlow Thigh
Rayleigh numbers from 103 to 107 is studied with Pr = 0.71,
1 and the results for the steady-state solution are presented in
x2
Figure 19. Dimensionless length is used and the variation of
Rayleigh number is determined by changing the characteris-
1 tic length of the model.
The local Nusselt number (Nu = 𝜕𝑇/𝜕𝑥1 ) is a concern
x1
1 of many researchers, as they are sensitive to the mesh size.
In Figure 19, the diagram (a) and the diagram (b) represent
Figure 16: A thermal-driven cavity.
the local Nusselt number at the hot wall and the cold
wall, respectively. Similar results can also be found in [10,
30, 35, 36]. The capability of the solver based on domain
With the parameter setting of ] = 0.5, 𝑇𝑟 = 5.5, 𝛽 = 1.0, decomposed Lagrange-Galerkin scheme for high Rayleigh
and 𝑎 = 1, the numerical experiment was performed. Results number is also confirmed by this figure.
of the profile on 𝑢3 (⋅, 0, 2), which are believed by many to be The new solver enables the simulation of large scale
very sensitive, are shown in Figure 15. Both “Num Res 0” and problems, thus models of Rayleigh number up to 107 can be
“Num Res 1” are in great agreement with the exact solutions, run on small PC cluster. In this simulation, an unstructured
and “Num Res 1” is closer to the exact solution, producing a mesh of 30,099,775 DOF is generated, the time step, is 0.01 s
better simulation to the ideal condition. Similar results have and it takes about 24 hours to finish, using the a small Linux
been documented in [33]. cluster of 64 PEs (64 cores@2.66 GHz).
4.2.5. Thermal-Driven Cavity. The new solver is also applied 5. Conclusions

to a 3-dimensional nonlinear thermal driven cavity flow
problem, which is cavity full of ideal gas; see Figure 16. A pressure-stabilized Lagrange-Galerkin method is imple-
No-slip boundary conditions are assumed to prevail on all mented in a domain decomposition system in this research.
the walls of the cavity. Both the horizontal walls are assumed By using localized stabilization parameter, the new scheme
to be thermally insulated, and the left and right sides are kept shows better control in the pressure field than constant sta-
at different temperatures. The cube is divided into 120 × 120 × bilization parameter; therefore it has good solvability at high
120 small cubes, and each small cube contains six tetrahedral Reynolds number and high Rayleigh number. The reliability
elements. The time step is set to 0.01 s, with Pr = 0.71 and and accuracy of the present numerical results are validated by
Ra = 104 ; the steady state is achieved after 0.39 s, as in comparing with the exact solutions and recognized numerical
Figure 17. results. Based on a domain decomposition method, the ele-
Figures 17(a) and 17(b) show the contour of vorticity and ment searching algorithm shows good numerical scalability
the velocity vectors at the steady stage, respectively, from the and parallel efficiency. The new solver reduces the memory
front view. The temperature contour is shown in Figure 17(c), consumption and is faster than classical product-type solvers.
and pressure profiles are show in Figure 17(d). The previous It is able to solve large scale problems of over 30 million
results convince us of the success in solving flow-thermal degrees of freedom within one day by a small PC cluster.
369 19
271 15
173 12
75 8
−24 4
−122 0
(a) (b)
1 709.4
0.8 448.9
0.6 188.4
0.4 −72.1
0.2 −332.6
0 −593.1
(c) (d)
Figure 17: Steady state of the thermal-driven cavity (Ra = 104 ).
25 1
20 0.9
15 0.8
10 0.7
5 0.6
Temperature
Velocity (w)
0 x1 0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−5 0.4
−10 0.3
−15 0.2
−20 0.1
−25 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Current D. C. Wan (2D) x1
ADV sFlow 0.5 (3D) N. Massarotti (2D)
Current
ADV sFlow 0.5 (3D)
K. L. Wong (3D)
(a) (b)
Figure 18: Centerline temperature velocity profiles of the symmetry plane (Ra = 104 ).
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5
x3
x3
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 10 20 30 40 0 10 20 30 40
Nu Nu
Nu0 10E3 Nu0 10E6 Nu1 10E3 Nu1 10E6

Nu0 10E4 Nu0 10E7 Nu1 10E4 Nu1 10E7
Nu0 10E5 Nu1 10E5
(a) (b)
Figure 19: Local Nusselt number along the hot wall (a) and the cold wall (b).
Acknowledgments for the Navier-Stokes equations,” Transactions of the Japan

Society for Industrial and Applied Mathematics, vol. 18, no. 3, pp.
This work was supported by the National Science Foun- 427–445, 2008.
dation of China (NSFC), Grants 11202248, 91230114, and [8] M. Tabata and S. Fujima, “Robustness of a characteristic finite
11072272; the China Postdoctoral Science Foundation, Grant element scheme of second order in time increment,” in Compu-
2012M521646, and the Guangdong National Science Founda- tational Fluid Dynamics 2004, pp. 177–182, 2006.
tion, Grant S2012040007687. [9] H. Rui and M. Tabata, “A second order characteristic finite ele-
ment scheme for convection-diffusion problems,” Numerische
Mathematik, vol. 92, no. 1, pp. 161–177, 2002.
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http://dx.doi.org/10.1155/2013/769102
Research Article
A Note on the Triple Laplace Transform and Its Applications to
Some Kind of Third-Order Differential Equation
Abdon Atangana
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State,
Bloemfontein 9300, South Africa
Correspondence should be addressed to Abdon Atangana; abdonatangana@yahoo.fr
Received 25 March 2013; Accepted 20 May 2013
Copyright © 2013 Abdon Atangana. This is an open access article distributed under the Creative Commons Attribution License,
We introduced a relatively new operator called the triple Laplace transform. We presented some properties and theorems about the
relatively new operator. We examine the triple Laplace transform of some function of three variables. We make use of the operator
to solve some kind of third-order differential equation called “Mboctara equations.”
1. Introduction the concept of double Laplace transform. This new operator

has been intensively used to solve some kind of differential
The topic of partial differential equations is one of the most equation [11] and fractional differential equations. The aim
important subjects in mathematics and other sciences. The of this work is to extend the Laplace transform to the triple
behaviour of the solution very much depends essentially Laplace transform. We will start with the definition of the
on the classification of PDEs therefore the problem of triple Laplace transform.
classification for partial differential equations is very natural
and well known since the classification governs the sufficient
number and the type of the conditions in order to determine 2. Definitions and Theorems
whether the problem is well posed and has a unique solution. Definition 1. Let 𝑓 be a continuous function of three vari-
The Laplace transform has been intensively used to solve ables; then, the triple Laplace transform of 𝑓(𝑥, 𝑦, 𝑡) is defined
nonlinear and linear equations [1–7]. The Laplace transform by
is used frequently in engineering and physics; the output of a
linear time invariant system can be calculated by convolving 𝐿 𝑥,𝑦,𝑡 [𝑓 (𝑥, 𝑦, 𝑡)]
its unit impulse response with the input signal. Performing
this calculation in Laplace space turns the convolution into ∞
a multiplication; the latter is easier to solve because of its = 𝐹 (𝑝, 𝑠, 𝑘) ∭ exp [−𝑝𝑥] exp [−𝑠𝑦] (1)
0
algebraic form. The Laplace transform can also be used
× exp [−𝑘𝑡] 𝑓 (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡,
to solve differential equations and is used extensively in
electrical engineering [1–7]. The Laplace transform reduces where, 𝑥, 𝑦, 𝑡 > 0 and 𝑝, 𝑠, 𝑘 are Laplace variables, and
a linear differential equation to an algebraic equation, which
can then be solved by the formal rules of algebra. The original 𝑓 (𝑥, 𝑦, 𝑡)
differential equation can then be solved by applying the
inverse Laplace transform. The English electrical engineer 1 𝛼+𝑖∞ 𝑝𝑥
Oliver Heaviside first proposed a similar scheme, although = ∫ 𝑒
2𝜋𝑖 𝛼−𝑖∞
without using the Laplace transform, and the resulting oper-
ational calculus is credited as the Heaviside calculus. Recently 1 𝛽+𝑖∞ 𝑠𝑦
×[ ∫ 𝑒
Kılıçman et al. [8–11] extended the Laplace transform to 2𝜋𝑖 𝛽−𝑖∞
1 𝜇+𝑖∞ 𝑘𝑡 Theorem 3. Let 𝑓(𝑥, 𝑦, 𝑡) and 𝑔(𝑥, 𝑦, 𝑡) be continuous func-

×[ ∫ 𝑒
2𝜋𝑖 𝜇−𝑖∞ tions defined for 𝑥, 𝑦, 𝑡 ≥ 0 and having Laplace transforms,
𝐹(𝑝, 𝑠, 𝑘) and 𝐺(𝑝, 𝑠, 𝑘), respectively. If 𝐹(𝑝, 𝑠, 𝑘) = 𝐺(𝑝, 𝑠, 𝑘),
× 𝐹(𝑝, 𝑠, 𝑘) 𝑑𝑘] 𝑑𝑠] 𝑑𝑝 then 𝑓(𝑥, 𝑦, 𝑡) = 𝑔(𝑥, 𝑦, 𝑡).
Proof. From the definition of the inverse Laplace transform, if

(2)
𝛼, 𝛽, and 𝜇 are sufficiently large, then the integral expression,
is the inverse triple Laplace transform. by
Property 2. Assuming that the continuous function 𝑓(𝑥, 𝑦, 𝑡) 𝑓 (𝑥, 𝑦, 𝑡)

is triple Laplace transformable, then,
1 𝛼+𝑖∞ 𝑝𝑥
𝜕3 𝑓 (𝑥, 𝑦, 𝑡) = ∫ 𝑒
𝐿 𝑡,𝑦,𝑥 [ ] 2𝜋𝑖 𝛼−𝑖∞
𝜕𝑥𝜕𝑦𝜕𝑡
1 𝛽+𝑖∞ 𝑠𝑦
×[ ∫ 𝑒
= 𝑝𝑠𝑘𝐹 (𝑝, 𝑠, 𝑘) − 𝑝𝑠𝐹 (𝑝, 𝑠, 0) − 𝑝𝑠𝐹 (𝑝, 0, 𝑘) 2𝜋𝑖 𝛽−𝑖∞
+ 𝑝𝐹 (𝑝, 0, 0) − 𝑠𝑘𝐹 (0, 𝑠, 𝑘) + 𝑠𝐹 (0, 𝑠, 0) 1 𝜇+𝑖∞ 𝑘𝑡
×[ ∫ 𝑒
2𝜋𝑖 𝜇−𝑖∞
+ 𝑘𝐹 (0, 0, 𝑘) − 𝐹 (0, 0, 0) ,
𝜕3 𝑓 (𝑥, 𝑦, 𝑡) × 𝐹(𝑝, 𝑠, 𝑘) 𝑑𝑘] 𝑑𝑠] 𝑑𝑝,

𝐿 𝑥,𝑥,𝑡 [ ]
𝜕𝑡𝜕𝑥2
(6)
𝜕𝐹 (0, 𝑦, 𝑘) (3)
= 𝑘𝑝2 𝐹 (𝑝, 𝑦, 𝑘) − 𝑝𝑘𝐹 (0, 𝑦, 𝑘) − for the triple inverse Laplace transform, can be used to obtain
𝜕𝑥
𝜕𝐹 (0, 𝑦, 0) 𝑓 (𝑥, 𝑦, 𝑡)
− 𝑝2 𝐹 (𝑝, 𝑦, 0) + 𝑝𝐹 (0, 𝑦, 0) + ,
𝜕𝑥
3
𝜕 𝑓 (𝑥, 𝑦, 𝑡) = ∫ 𝑒
𝐿 𝑥𝑥𝑥 [ ] 2𝜋𝑖 𝛼−𝑖∞
𝜕𝑥3
3 2 ×[ ∫ 𝑒
= 𝑝 𝐹 (𝑝, 𝑦, 𝑡) − 𝑝 𝐹 (0, 𝑦, 𝑡) 2𝜋𝑖 𝛽−𝑖∞
𝜕𝐹 (0, 𝑦, 𝑡) 𝜕2 𝐹 (0, 𝑦, 𝑡) 1 𝜇+𝑖∞ 𝑘𝑡

−𝑝 − . ×[ ∫ 𝑒
𝜕𝑥 𝜕𝑥2 2𝜋𝑖 𝜇−𝑖∞
3. Uniqueness and Existence of the Triple × 𝐹(𝑝, 𝑠, 𝑘) 𝑑𝑘] 𝑑𝑠] 𝑑𝑝.

Laplace Transform
(7)
In this section, we will study the uniqueness and existence
of triple Laplace transform. First of all, let 𝑓(𝑥, 𝑦, 𝑡) be By hypothesis, we have that 𝐹(𝑝, 𝑠, 𝑘) = 𝐺(𝑝, 𝑠, 𝑘). then
a continuous function on the interval [0,∞) which is of replacing this in the previous expression, we have the follow-
exponential order, that is, for some 𝑎, 𝑏, 𝑐 ∈ 𝑅. Consider ing:
󵄨󵄨 𝑓 (𝑥, 𝑦, 𝑡) 󵄨󵄨
󵄨 󵄨󵄨
sup 󵄨󵄨󵄨󵄨 󵄨󵄨 < 0.
󵄨 (4) 𝑓 (𝑥, 𝑦, 𝑡)
𝑥,𝑦,𝑡>0 󵄨󵄨 exp [𝑎𝑥 + 𝑏𝑦 + 𝑐𝑡] 󵄨󵄨
Under the previous condition, the triple Laplace transform, = ∫ 𝑒
2𝜋𝑖 𝛼−𝑖∞
∞
𝐹 (𝑝, 𝑠, 𝑘) = ∭ exp [−𝑝𝑥] exp [−𝑠𝑦] 1 𝛽+𝑖∞ 𝑠𝑦
0
×[ ∫ 𝑒
(5) 2𝜋𝑖 𝛽−𝑖∞
× exp [−𝑘𝑡] 𝑓 (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡,
1 𝜇+𝑖∞ 𝑘𝑡
×[ ∫ 𝑒
exists for all 𝑝 > 𝑎, 𝑠 > 𝑏, and 𝑘 > 𝑐 and is in actuality 2𝜋𝑖 𝜇−𝑖∞
infinitely differentiable with respect to 𝑝 > 𝑎, 𝑠 > 𝑏 and 𝑘 > 𝑐.
All functions in this study are assumed to be of exponential × 𝐺(𝑝, 𝑠, 𝑘) 𝑑𝑘] 𝑑𝑠] 𝑑𝑝,
order. The following theorem shows that 𝑓(𝑥, 𝑦, 𝑡) can be
uniquely obtained from 𝐹(𝑝, 𝑠, 𝑡). (8)
which boil down to Theorem 5. A function 𝑓(𝑥, 𝑦, 𝑡) which is continuous on

𝑓 (𝑥, 𝑦, 𝑡) [0, ∞) and satisfies the growth condition (4) can be recovered
from only 𝐹(𝑝, 𝑠, 𝑘) as
1 𝛼+𝑖∞ 𝑝𝑥 𝑛 +1
= ∫ 𝑒 (−1)𝑛1 +𝑛2 +𝑛3 𝑛1 𝑛1 +1 𝑛2 2
2𝜋𝑖 𝛼−𝑖∞ 𝑓 (𝑥, 𝑦, 𝑡) = lim ( ) ( )
𝑛1 → ∞ 𝑛 !𝑛 !𝑛 ! 𝑥 𝑦
𝑛 →∞ 1 2 3
2
𝑛3 → ∞
×[ ∫ 𝑒 (15)
2𝜋𝑖 𝛽−𝑖∞ 𝑛3 +1
𝑛3 𝑛1 𝑛2 𝑛3
×( ) Χ𝑛1 +𝑛2 +𝑛3 [ , , ].
1 𝜇+𝑖∞ 𝑡 𝑥 𝑦 𝑡
×[ ∫ 𝑒𝑘𝑡
2𝜋𝑖 𝜇−𝑖∞
Evidently, the main difficulty in using Theorem 5 for com-
puting the inverse Laplace transform is the repeated symbolic
× 𝐺(𝑝, 𝑠, 𝑘) 𝑑𝑘] 𝑑𝑠] 𝑑𝑝, differentiation of 𝐹(𝑝, 𝑠, 𝑘).
Let us see how Theorem 5 can be applicable. Let us
= 𝑔 (𝑥, 𝑦, 𝑡) , consider the following functions:
(9)
𝑓 (𝑥, 𝑦, 𝑡) = exp [−𝑎𝑥 − 𝑏𝑦 − 𝑐𝑡] . (16)
and this proves the uniqueness of the triple Laplace trans-
form. Naturally the triple Laplace transform of the previous func-
tion is given later as
Theorem 4. If, at the point (𝑝, 𝑠, 𝑘), the integrals
1
∞ 𝐹 (𝑝, 𝑠, 𝑘) = . (17)
𝐹1 (𝑝, 𝑠, 𝑘) = ∭ exp [−𝑝𝑥] exp [−𝑠𝑦] (𝑝 − 𝑎) (𝑠 − 𝑏) (𝑘 − 𝑐)
0
Now applying the high-order mixed derivative to the previous
× exp [−𝑘𝑡] 𝑓1 (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡 expression, we obtain the following:
(10)
∞
𝐹2 (𝑝, 𝑠, 𝑘) = ∭ exp [−𝑝𝑥] exp [−𝑠𝑦] 𝜕𝑛1 +𝑛2 +𝑛3 [𝐹 (𝑝, 𝑠, 𝑘)]
0 = 𝑛1 !𝑛2 !𝑛3 !(−1)𝑛1 +𝑛2 +𝑛3
𝜕𝑝𝑛1 𝜕𝑠𝑛2 𝜕𝑘𝑛3
× exp [−𝑘𝑡] 𝑓2 (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡
× (𝑎+𝑃)−1−𝑛1 (𝑠+𝑏)−1−𝑛2 (𝑐+𝑘)−1−𝑛3 .
are convergent and in addition if (18)
∞
𝐹3 (𝑝, 𝑠, 𝑘) = ∭ exp [−𝑝𝑥] exp [−𝑠𝑦] Applying Theorem 5 in the previous expression, we obtain the
0 (11) following result:
× exp [−𝑘𝑡] 𝑓3 (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡
𝑛1 1+𝑛1 𝑛2 1+𝑛2 𝑛3 1+𝑛3 𝑛1 −𝑛1 −1
𝑓 (𝑥, 𝑦, 𝑡) = lim (𝑎 + )
is absolutely convergent, then, the following expression: 𝑛1 → ∞ 𝑥𝑛1 +1 𝑦𝑛2 +1 𝑡𝑛3 +1 𝑥
𝑛 →∞
2
𝑛3 → ∞
𝐹 (𝑝, 𝑠, 𝑘) = 𝐹1 (𝑝, 𝑠, 𝑘) 𝐹2 (𝑝, 𝑠, 𝑘) 𝐹3 (𝑝, 𝑠, 𝑘) (12) (19)
−𝑛2 −1 −𝑛3 −1
𝑛2 𝑛3
is the Laplace transform of the function × (𝑏 + ) (𝑐 + ) .
𝑦 𝑡
𝑓 (𝑥, 𝑦, 𝑡)
Making a change of variable in the previous expression, we
𝑡 𝑦 𝑥
= ∫ ∫ ∫ 𝑓3 (𝑥 − (𝑥1 + 𝜌) , 𝑦 − (𝑦1 + 𝜎) , obtain the following simplified result:
0 0 0
𝑎𝑛1 −𝑛1 −1 𝑏𝑛 −𝑛2 −1
𝑡 − (𝑡1 + 𝜏)) 𝑓2 (𝑥1 − 𝜌, 𝑦1 − 𝜎, 𝑡1 − 𝜏) 𝑓 (𝑥, 𝑦, 𝑡) = lim (1 + ) (1 + 2 )
𝑛1 → ∞
𝑛2 → ∞
𝑥 𝑦
𝑛3 → ∞
× 𝑓1 (𝜌, 𝜎, 𝜏) 𝑑𝜌 𝑑𝜎 𝑑𝜏, (20)
(13) 𝑐𝑛3 −𝑛3 −1
× (1 + ) .
and the integral 𝑡
∞ Using together, the application of logarithm and the
𝐹 (𝑝, 𝑠, 𝑘) = ∭ exp [−𝑝𝑥] exp [−𝑠𝑦] L’Hôpital’s rule on the previous expression, we arrive at the
0 (14) following result:
× exp [−𝑘𝑡] 𝑓 (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡
ln (𝑓 (𝑥, 𝑦, 𝑡)) = −𝑎𝑥 − 𝑏𝑦 − 𝑐𝑡 󳨐⇒ 𝑓 (𝑥, 𝑦, 𝑡)
is convergent at the point (𝑝, 𝑠, 𝑘); for the readers who are (21)
interested, they can see the proof in [11, 12]. = exp [−𝑎𝑥 − 𝑏𝑦 − 𝑐𝑡] .
4. Some Properties of Triple We will present the proof

Laplace Transform
𝐿 𝑥,𝑦,𝑡 [𝑓 (𝛼𝑥, 𝛽𝑦, 𝛾𝑡)] (𝑝, 𝑠, 𝑘)
In this section, we present some properties of the triple ∞
Laplace transform. Note that these properties follow from = ∭ exp [−𝑝𝑥] exp [−𝑠𝑦] exp [−𝑘𝑡]
those of the double Laplace transform introduced by 0
Kılıçman and Eltayeb [8]. The properties of the triple Laplace × 𝑓 (𝛼𝑥, 𝛽𝑦, 𝛾𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡,
transform will enable us to find further transform pairs (27)
{𝑓(𝑥, 𝑦, 𝑡), 𝐹(𝑝, 𝑠, 𝑘)}: ∞ ∞
∫ exp [−𝑝𝑥] (∬ exp [−𝑠𝑦] exp [−𝑘𝑡]
0 0
(i) 𝐹 (𝑝 + 𝑎, 𝑠 + 𝑏, 𝑘 + 𝑑)
(22) × 𝑓 (𝛼𝑥, 𝛽𝑦, 𝛾𝑡) 𝑑𝑦 𝑑𝑡) 𝑑𝑥.
−𝑎𝑥−𝑦𝑏−𝑐𝑡
= 𝐿 𝑥,𝑦,𝑡 [𝑒 𝑓 (𝑥, 𝑦, 𝑡)] (𝑝, 𝑠, 𝑘) .
Note that the double integral inside the bracket satisfies the
property of the double Laplace transform as [11]
We will present the proof ∞
(∬ exp [−𝑠𝑦] exp [−𝑘𝑡] 𝑓 (𝛼𝑥, 𝛽𝑦, 𝛾𝑡) 𝑑𝑦 𝑑𝑡)
0
𝐿 𝑥,𝑦,𝑡 [𝑒−𝑎𝑥−𝑦𝑏−𝑐𝑡 𝑓 (𝑥, 𝑦, 𝑡)] (𝑝, 𝑠, 𝑘) (28)
1 𝑠 𝑘
∞
= 𝐹 (𝛼𝑥, , ) .
𝛽𝛾 𝛽 𝛾
= ∭ exp [−𝑝𝑥] exp [−𝑠𝑦] exp [−𝑘𝑡] exp [−𝑎𝑥]
0
Thus
× exp [−𝑏𝑦] exp [−𝑐𝑡] 𝑓 (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡, 𝐿 𝑥,𝑦,𝑡 [𝑓 (𝛼𝑥, 𝛽𝑦, 𝛾𝑡)] (𝑝, 𝑠, 𝑘)
∞
∞
∫ exp [−𝑝𝑥] exp [−𝑎𝑥] 1 𝑠 𝑘
0 = ∫ exp [−𝑝𝑥] 𝐹 (𝛼𝑥, , ) 𝑑𝑥
0 𝛽𝛾 𝛽 𝛾 (29)
∞
× (∬ exp [−𝑠𝑦] exp [−𝑘𝑡] exp [−𝑏𝑦] 1 𝑝 𝑠 𝑘
0 = 𝐹( , , ),
𝛼𝛽𝛾 𝛼 𝛽 𝛾
× exp [−𝑐𝑡] 𝑓 (𝑥, 𝑦, 𝑡) 𝑑𝑡 𝑑𝑦) 𝑑𝑡. and this completes the proof.
(23) (iii) The following property can also be observed:
𝜕𝑛+𝑚+V [𝐹 (𝑝, 𝑠, 𝑘)]

Note that the integral inside the bracket satisfies the proper- 𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V (30)
ties of the double Laplace transform and is given as [11] 𝑛+𝑚+V 𝑛 𝑚 V
= 𝐿 𝑥,𝑦,𝑡 [(−1) 𝑥 𝑦 𝑡 𝑓 (𝑥, 𝑦, 𝑡)] (𝑝, 𝑠, 𝑘) .
∞ We will present the proof
(∬ exp [−𝑠𝑦] exp [−𝑘𝑡] exp [−𝑏𝑦] exp [−𝑐𝑡]
0 ∞
(24) 𝐹 (𝑝, 𝑠, 𝑘) = ∭ exp [−𝑝𝑥] exp [−𝑠𝑦] exp [−𝑘𝑡]
0 (31)
× 𝑓 (𝑥, 𝑦, 𝑡) 𝑑𝑡 𝑑𝑦) = 𝐹 (𝑥, 𝑠 + 𝑏, 𝑘 + 𝑑) .
× 𝑓 (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡.
Thus Then,
𝜕𝑛+𝑚+V [𝐹 (𝑝, 𝑠, 𝑘)]
∞
𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V
∫ exp [−𝑝𝑥] exp [−𝑎𝑥] 𝐹 (𝑥, 𝑠 + 𝑏, 𝑘 + 𝑑) 𝑑𝑡
0 (25) 𝜕𝑛+𝑚+V ∞
= (∭ exp [−𝑝𝑥] exp [−𝑠𝑦]
= 𝐹 (𝑝 + 𝑎, 𝑠 + 𝑏, 𝑘 + 𝑑) , 𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V 0
× exp [−𝑘𝑡] 𝑓 (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡) .

and this completes the proof.
(ii) The following can also be observed: (32)
Now making use of the convergence properties of the

1 𝑝 𝑠 𝑘 improper integral involved, we can interchange the opera-
𝐹 ( , , ) = 𝐿 𝑥,𝑦,𝑡 [𝑓 (𝛼𝑥, 𝛽𝑦, 𝛾𝑡)] (𝑝, 𝑠, 𝑘) . (26)
𝛼𝛽𝛾 𝛼 𝛽 𝛾 tion of differentiation and integration and differentiate with
respect to 𝑝, 𝑠, and 𝑘 under the integral sign. Thus, we arrive where 𝜓(𝑥, 𝑦, 𝑡) is any continuous function. Let Ψ(𝑝, 𝑠, 𝑘)
at the following expression: denote the triple Laplace transform of the continuous
function 𝜓(𝑥, 𝑦, 𝑡). However, if one defines the function
𝜕𝑛+𝑚+V [𝐹 (𝑝, 𝑠, 𝑘)] 𝑀(𝑥, 𝑦, 𝑡) = 𝑓(𝑥𝛼, 𝑦𝛽, 𝑡𝛾), making use of the second pro-
𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V perty established in (29), we arrive at the following:
𝜕𝑛 ∞ 1 𝑝 𝑠 𝑘
= ∫ exp [−𝑝𝑥] 𝐹 ( , , ) = 𝐿 𝑥,𝑦,𝑡 [𝑓 (𝛼𝑥, 𝛽𝑦, 𝛾𝑡)] (𝑝, 𝑠, 𝑘) . (40)
𝜕𝑝𝑛 0 𝛼𝛽𝛾 𝛼 𝛽 𝛾
(33) Here if one applies the third property, in particular by rep-
𝜕𝑚+V ∞
× ( 𝑛 V ∬ exp [−𝑠𝑦] exp [−𝑘𝑡] lacing 𝑝 = 𝑚/𝑥, 𝑠 = 𝑛/𝑦, 𝑘 = V/𝑡 as follows:
𝜕𝑠 𝜕𝑘 0
1 𝑝 𝑠 𝑘
𝐿 𝑥𝑦𝑡 (𝑀 (𝑥, 𝑦, 𝑡)) = 𝐹( , , ), (41)
× 𝑓 (𝑥, 𝑦, 𝑡) 𝑑𝑦 𝑑𝑡) 𝑑𝑥. 𝛼𝛽𝛾 𝛼 𝛽 𝛾
𝜕𝑛+𝑚+V [𝐿 𝑥𝑦𝑡 (𝑀 (𝑥, 𝑦, 𝑡))]
Note that the expression in the bracket satisfies the property
of the double Laplace transform as [11]
𝜕𝑛+𝑚+V [(1/𝛼𝛽𝛾) 𝐹 (𝑝/𝛼, 𝑠/𝛽, 𝑘/𝛾)]
𝜕𝑚+V ∞ =
∬ exp [−𝑠𝑦] exp [−𝑘𝑡] 𝑓 (𝑥, 𝑦, 𝑡) 𝑑𝑦 𝑑𝑡 𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V
𝑛
𝜕𝑠 𝜕𝑘 V (42)
0 (34)
1
= 𝐿 𝑦,𝑡 [(−1)𝑚+V 𝑦𝑚 𝑡V 𝑓 (𝑥, 𝑦, 𝑡)] (𝑠, 𝑘) . = 𝑚+1 𝑛+1 V+1
𝛼 𝛽 𝛾
Thus 𝜕𝑛+𝑚+V [𝐹 (𝑝/𝛼, 𝑠/𝛽, 𝑘/𝛾)]
𝑛+𝑚+V × .
𝜕 [𝐹 (𝑝, 𝑠, 𝑘)] 𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V
Now let us put 𝜓(𝑥, 𝑦, 𝑡) = 𝑒−𝑝𝑥−𝑠𝑦−𝑘𝑡 𝑀(𝑥, 𝑦, 𝑡). Now if we
𝜕𝑛 ∞ make use of (38), we obtain the following
= ∫ exp [−𝑝𝑥]
𝜕𝑝𝑛 0 𝜓 (1, 1, 1) = 𝑒−𝑝−𝑠−𝑘 𝑀 (1, 1, 1) = 𝑒−𝑝−𝑠−𝑘 𝑓 (𝛼, 𝛽, 𝛾)
× (𝐿 𝑦,𝑡 [(−1)𝑚+V 𝑦𝑚 𝑡V 𝑓 (𝑥, 𝑦, 𝑡)] (𝑠, 𝑘)) 𝑑𝑥. 𝑚𝑚+1 𝑛𝑛+1 VV+1 ∞
(35) = lim ∭ 𝑥𝑚 𝑦𝑛 𝑡V 𝑒−𝑝𝑥−𝑠𝑦−𝑘𝑡

𝑚→∞
𝑛→∞ 𝑚!𝑛!V! 0
V→∞
And finally, we obtain
× 𝑒−𝑚𝑥−𝑛𝑦−V𝑡 Ψ (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡
𝑛+𝑚+V
𝜕 [𝐹 (𝑝, 𝑠, 𝑘)]
𝑚𝑚+1 𝑛𝑛+1 VV+1
𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V = lim 𝐿 𝑥𝑦𝑡 [𝑥𝑚 𝑦𝑛 𝑡V 𝑒−𝑚𝑥−𝑛𝑦−V𝑡 Ψ (𝑥, 𝑦, 𝑡)] .
(36) 𝑚→∞
𝑛→∞ 𝑚!𝑛!V!
𝑛+𝑚+V 𝑛 𝑚 V V→∞
= 𝐿 𝑥,𝑦,𝑡 [(−1) 𝑥 𝑦 𝑡 𝑓 (𝑥, 𝑦, 𝑡)] (𝑝, 𝑠, 𝑘) ,
(43)
and this completes the proof.
Now using the previous three properties, we will show the Now taking into account properties (i) and (ii), (42) together
proof of Theorem 5. with the function 𝑀(𝑥, 𝑦, 𝑡), we arrive at the following:
Proof of Theorem 5. Let us define the set of functions depend- 𝐿 𝑥𝑦𝑡 [𝑥𝑚 𝑦𝑛 𝑡V 𝑒−𝑚𝑥−𝑛𝑦−V𝑡 Ψ (𝑥, 𝑦, 𝑡)]
ing on parameters 𝑚, 𝑛, and V as 𝜕𝑛+𝑚+V [𝐿 𝑥𝑦𝑡 (𝑒−𝑚𝑥−𝑛𝑦−𝑘𝑡 Ψ (𝑥, 𝑦, 𝑡)) (𝑝, 𝑠, 𝑘)]
𝑚+1 𝑛+1 V+1 = (−1)𝑚+𝑛+V
𝑚 𝑛 V 𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V
ℎ𝑚,𝑛,V (𝑥, 𝑦, 𝑡) = 𝑥𝑚 𝑦𝑛 𝑡V 𝑒−𝑚𝑥−𝑛𝑦−V𝑡 . (37)
𝑚!𝑛!V! 1
= (−1)𝑚+𝑛+V
It worth noting that the previous function is a kind of three- 𝛼𝑚 𝛽𝑛 𝛾V
dimensional density of probability, and it therefore follows
𝜕𝑛+𝑚+V [𝐿 𝑥𝑦𝑡 (Ψ (𝑥, 𝑦, 𝑡)) (𝑝 + 𝑚, 𝑠 + 𝑛, 𝑘 + V)]
that ×
∞ 𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V
∭ ℎ𝑚,𝑛,V (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡 = 1. (38) 1
0 = (−1)𝑚+𝑛+V
𝛼𝑚 𝛽𝑛 𝛾V
In addition of this, we will have that
∞ ×((𝜕𝑛+𝑚+V [𝐿 𝑥𝑦𝑡 (𝑓 (𝛼𝑥, 𝛽𝑦, 𝛾𝑡))
lim ∭ ℎ𝑚,𝑛,V (𝑥, 𝑦, 𝑡) 𝜓 (𝑥, 𝑦, 𝑡) 𝑑𝑥 𝑑𝑦 𝑑𝑡 = 𝜓 (1, 1, 1) ,
𝑚→∞ 0
𝑛→∞ 𝑝+𝑚 𝑠+𝑛 𝑘+V −1
V→∞
×( , , )]) (𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V ) )
(39) 𝛼 𝛽 𝛾
= (−1)𝑚+𝑛+V Now applying the inverse triple Laplace transform on the

𝑛+𝑚+V
previous equation we obtain the following solution:
1 𝜕 [𝐹 ((𝑝+𝑚) /𝛼, (𝑠+𝑛) /𝛽, (𝑘+V) /𝛾)]
× .
𝛼𝑚 𝛽𝑛 𝛾V 𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V 𝐺 (𝑝, 𝑠, 𝑘)
𝑢 (𝑥, 𝑦, 𝑡) = 𝐿−1
𝑥𝑦𝑡 [ ] = 𝑒𝑥+𝑦−𝑡 . (51)
(44) 1 + 𝑝𝑠𝑘
Now observe that from (44) with the fact that 𝑓(𝛼, 𝛽, 𝑡) = This is the exact solution for Mboctara equation.
𝜓(1, 1, 1)𝑒𝑝+𝑠+𝑘 , we arrive at the following:
Example 2. Let us consider the following nonhomogeneous
𝑓 (𝛼, 𝛽, 𝑡) Mboctara equation
𝑚𝑚+1 𝑛𝑛+1 VV+1 𝜕𝑥𝑦𝑡 𝑢 (𝑥, 𝑦, 𝑡) + 𝑢 (𝑥, 𝑦, 𝑡) = −𝑒𝑥−2𝑦+𝑡
= 𝑒𝑝+𝑠+𝑘 lim (52)
𝑚→∞
𝑛→∞ 𝑚!𝑛!V!
V→∞ subjected to the following initial and boundaries conditions:
𝑚 𝑚+1 𝑛 𝑛+1 V V+1
×( ) ( ) ( ) 𝑢 (𝑥, 0, 0) = 𝑒𝑥 , 𝜕𝑡 𝑢 (𝑥, 0, 𝑡) = 𝑒𝑥+𝑡 , 𝜕𝑥 𝑢 (𝑥, 0, 𝑡) = 𝑒𝑥+𝑡 ,
𝛼 𝛽 𝛾
𝜕𝑛+𝑚+V [𝐹 ((𝑝 + 𝑚) /𝛼, (𝑠 + 𝑛) /𝛽, (𝑘 + V) /𝛾)] 𝑢 (0, 0, 0) = 1, 𝑢 (𝑥, 0.5, 𝑡) = 𝑒𝑥+𝑡−1 .
× .
𝜕𝑝𝑛 𝜕𝑠𝑛 𝜕𝑘V (53)
(45)
Now applying the triple Laplace transform on both sides
The previously mentioned is true for any 𝑝, 𝑠, 𝑘 in the of (52), we obtain the following:
complete space, in particular, for 𝑝 = 0, 𝑠 = 0, 𝑘 = 0, and
in this case Theorem 5 is covered. 𝑝𝑠𝑘𝑈 (𝑝, 𝑠, 𝑘) + 𝑈 (𝑝, 𝑠, 𝑘)
1 (54)
5. Application to Third-Order Partial = 𝐺 (𝑝, 𝑠, 𝑘) − .
(1 + 𝑝) (2 + 𝑠) (1 + 𝑘)
Differential Equation
Factorising the right side of (54), we obtain the following:
In this section, we present the application of this operator
for solving some kind of third-order partial differential 𝐺 (𝑝, 𝑠, 𝑘) − 1/ (1 + 𝑝) (2 + 𝑠) (1 + 𝑘)
equations. 𝑈 (𝑝, 𝑠, 𝑘) = .
1 + 𝑝𝑠𝑘
Example 1. consider the following third-order partial differ- (55)
ential equation:
Now applying the inverse triple Laplace transform on the
𝜕𝑥𝑦𝑡 𝑢 (𝑥, 𝑦, 𝑡) + 𝑢 (𝑥, 𝑦, 𝑡) = 0. (46) previous equation, we obtain the following solution
The previous equation is called the Mboctara equation and is 𝐺 (𝑝, 𝑠, 𝑘) − 1/ (1 + 𝑝) (2 + 𝑠) (1 + 𝑘)
subjected to the following boundaries and initial conditions: 𝑢 (𝑥, 𝑦, 𝑡) = 𝐿−1
𝑥𝑦𝑡 [ ]
1 + 𝑝𝑠𝑘
𝑢 (𝑥, 𝑦, 0) = 𝑒𝑥+𝑦 , 𝑢 (𝑥, 0, 𝑡) = 𝑒𝑥−𝑡 ,
(47) = 𝑒𝑥−2𝑦+𝑡 .
𝑦−𝑡 𝑥+𝑦−1
𝑢 (0, 𝑦, 𝑡) = 𝑒 , 𝑢 (𝑥, 𝑦, 1) = 𝑒 . (56)
Now applying the triple Laplace transform on both sides of This is the exact solution for nonhomogeneous Mboctara
(46), we obtain the following: equation.
𝑝𝑠𝑘𝑈 (𝑝, 𝑠, 𝑘) + 𝑈 (𝑝, 𝑠, 𝑘) = 𝐺 (𝑝, 𝑠, 𝑘) . (48) Example 3. Let us consider the following nonhomogeneous
Here Mboctara equation
𝐺 (𝑝, 𝑠, 𝑘) = 𝑝𝑠𝑈 (𝑝, 𝑠, 0) + 𝑝𝑠𝑈 (𝑝, 0, 𝑘) − 𝑝𝑈 (𝑝, 0, 0) 𝜕𝑥𝑦𝑡 𝑢 (𝑥, 𝑦, 𝑡) + 𝑢 (𝑥, 𝑦, 𝑡) = cos (𝑥) cos (𝑦) cos (−𝑡)
+ 𝑠𝑘𝑈 (0, 𝑠, 𝑘) − 𝑠𝑈 (0, 𝑠, 0) − 𝑘𝑈 (0, 0, 𝑘) − sin (𝑥) sin (𝑦) sin (−𝑡) ,
(57)
+ 𝑈 (0, 0, 0) .
(49) subjected to the following initial and boundaries conditions:
Factorising the right side of equation (49), we obtain the 𝑢 (𝑥, 𝑦, 0) = cos (𝑥) cos (𝑦) ,
following:
𝜕𝑡 𝑢 (𝑥, 𝑦, 0) = 𝜕𝑥 𝑢 (0, 𝑦, 𝑡) = 𝜕𝑦 𝑢 (𝑥, 0, 𝑡) = 0, (58)
𝐺 (𝑝, 𝑠, 𝑘) 𝜋 𝜋 𝜋
𝑈 (𝑝, 𝑠, 𝑘) = . (50) 𝑢 (𝑥, , 𝑡) = 𝑢 (𝑥, 𝑦, ) = 𝑢 ( , 𝑦, 𝑡) = 0.
1 + 𝑝𝑠𝑘 2 2 2
Table 1: Table of triple Laplace transform for some function of three variables.
Functions𝑓(𝑥, 𝑦, 𝑡) Triple laplace transform 𝐹(𝑝, 𝑠, 𝑘)

𝑎𝑏𝑐
𝑎𝑏𝑐
𝑝𝑠𝑘
1
𝑥𝑦𝑡 𝑝2 𝑠2 𝑘2
𝑥𝑛 𝑦𝑚 𝑡V , 𝑛, 𝑚, V are natural numbers 𝑘−1−V 𝑠−1−𝑚 𝑝−𝑛−1 Γ (1 + 𝑛) Γ (1 + 𝑚) Γ (1 + V)
𝑥𝑛 𝑦𝑚 𝑡V 𝑒−𝑎𝑥−𝑏𝑦−𝑐𝑡 (𝑘 + 𝑐)−1−V (𝑠 + 𝑏)−1−𝑚 (𝑝 + 𝑎)
−𝑛−1
Γ (1 + 𝑛) Γ (1 + 𝑚) Γ (1 + V)
1
𝑒−𝑎𝑥−𝑏𝑦−𝑐𝑡 (𝑎 + 𝑝) (𝑏 + 𝑠) (𝑐 + 𝑘)
𝑘𝑠𝑝
cos(𝑥) cos(𝑦) cos(𝑡)
(1 + 𝑝2 ) (1 + 𝑠2 ) (1 + 𝑘2 )
1
sin(𝑥) sin(𝑦) sin(𝑡) (1 + 𝑝2 ) (1 + 𝑠2 ) (1 + 𝑘2 )
−1 + 𝑝𝑠 + 𝑘 (𝑝 + 𝑠)
sin (𝑥 + 𝑦 + 𝑡)
(1 + 𝑝2 ) (1 + 𝑠2 ) (1 + 𝑘2 )
𝑘 + 𝑝 + 𝑠 − 𝑘𝑝𝑠
cos(𝑥 + 𝑦 + 𝑡) −
(1 + 𝑝2 ) (1 + 𝑠2 ) (1 + 𝑘2 )
𝜋√𝜋
√𝑥𝑦𝑡
83 √𝑘𝑠𝑝
(𝑏) (𝑐) (𝑎)
𝑒𝑎𝑥+𝑦𝑏+𝑐𝑡 sinh(𝑎𝑥)sinh(𝑏𝑦) sinh(𝑐𝑡)
(−2𝑎𝑝 + 𝑝2 ) (−2𝑏𝑠 + 𝑠2 ) (−2𝑐𝑘 + 𝑘2 )
(𝑏 − 𝑠) (𝑐 − 𝑘) (𝑎 − 𝑝)
𝑒𝑎𝑥+𝑦𝑏+𝑐𝑡 cosh(𝑎𝑥) cosh(𝑏𝑦) cosh(𝑐𝑡)
(−2𝑎𝑝 + 𝑝2 ) (−2𝑏𝑠 + 𝑠2 ) (−2𝑐𝑘 + 𝑘2 )
√𝑐2 𝑘−√𝑏2 𝑠
𝑎 𝑏 𝑐 𝑒− √ 𝑐2 𝑘 √𝑎2 𝑝 √𝑏2 𝑠
Erf [ ] Erf [ ] Erf [ ] (−1 + 𝑒− ) (1 − 𝑒− ) (−1 + 𝑒− )
2√𝑥 2√𝑦 √
2 𝑡 𝑘𝑝𝑠
sin(𝑎𝑥) sin(𝑏𝑦) sin(𝑐𝑡) √𝑎2 √𝑏2 √𝑐2
arctan ( ) arctan ( ) arctan ( )
𝑥 𝑦 𝑡 𝑝 𝑠 𝑘
1/2(−1+𝑚)
𝑏2 √𝑏2
𝑘−1+V (1 + ) 𝑠−1+𝑚 cos [𝑐𝑡] cos ((−1 + 𝑚) arctan [ ]) Γ (1 − 𝑚) Γ (1 − V)
cos(𝑎𝑥) cos(𝑏𝑦) cos(𝑐𝑡) 𝑠2 𝑠
1/2(−1+𝑛)
𝑥𝑛 𝑦𝑚 𝑡V 𝑎2 |𝑎|
× (1 + ) 𝑝−1+𝑛 cos ((𝑛 − 1) arctan ( )) Γ (1 − 𝑛)
𝑝2 𝑝
1/2(−1+𝑚) 1/2(−1+𝑛)
𝑏2 𝑎2
𝑘−1+V (1 + ) 𝑠 −1+𝑚
Γ (1 − 𝑚) Γ (1 − V) (1 + )
sin(𝑎𝑥) sin(𝑏𝑦) sin(𝑐𝑡) 𝑠2 𝑝2
𝑥𝑛 𝑦𝑚 𝑡V |𝑎|
× 𝑝−1+𝑛 Γ (1 − 𝑛) sign (𝑎) sin ((𝑛 − 1) arctan ( 𝑝 )) Γ (1 − 𝑛)
−1−𝑛 1+𝑛 2+𝑛 1

8−𝑛 (𝑘𝑠𝑝) Hypergeometric2𝐹1 (
, ,− 2)
2 2 𝑘
1+𝑛 2+𝑛 1
𝐽𝑛 (𝑥) 𝐽𝑛 (𝑦) 𝐽𝑛 (𝑡) Hypergeometric2𝐹1 ( , ,− 2)
2 2 𝑠
1+𝑛 2+𝑛 1
[Hypergeometric2𝐹1 ( , ,− 2)
2 2 𝑝
−1−𝑛 1+𝑛 2+𝑛 1
8−𝑛 (𝑘𝑠𝑝) Hypergeometric2𝐹1 (
, , 1 + 𝑛, 2 )
2 2 𝑘
1+𝑛 2+𝑛 1
𝐼𝑛 (𝑥) 𝐼𝑛 (𝑦) 𝐼𝑛 (𝑡) Hypergeometric2𝐹1 ( , , 1 + 𝑛, 2 )
2 2 𝑠
1+𝑛 2+𝑛 1
Hypergeometric2𝐹1 ( , , 1 + 𝑛, 2 )
2 2 𝑝
Exact solution of the nonhomogeneous Mboctara equation (4.1) Exact solution of the nonhomogeneous Mboctara equation (4.6)
500 500
400 400
300 300
200 200
100 100
0 0
6 6
6 6
4 4
4 4
2 2
2 x 2
Time y x
0 0 0 0
(a) (b)
Exact solution of the nonhomogeneous Mboctara equation (4.6) Exact solution of the nonhomogeneous Mboctara equation (4.11)
60 1
0.5
40
20
−0.5
0 −1
6 6
4 6 6
4
4 4
2 2
Time 2 x y 2 x
0 0 0 0
(c) (d)
Figure 1: Numerical simulation of the exact solutions of the Homogeneous and non-homogeneous Mboctara equations.
Now applying the triple Laplace transform on both sides of Factorising the right side of (59), we obtain the following:
(57), we obtain the following:
𝑝𝑠𝑘𝑈 (𝑝, 𝑠, 𝑘) + 𝑈 (𝑝, 𝑠, 𝑘) 2 2 2

𝐺 (𝑝, 𝑠, 𝑘) 𝑘𝑠𝑝/ (1 + 𝑝 ) (1 + 𝑠 ) (−1 + 𝑘 )
𝑈 (𝑝, 𝑠, 𝑘) = +
𝑘𝑠𝑝 1 + 𝑝𝑠𝑘 1 + 𝑝𝑠𝑘
= 𝐺 (𝑝, 𝑠, 𝑘) +
(1 + 𝑝2 ) (1 + 𝑠2 ) (−1 + 𝑘2 ) (59)
1/ (1 + 𝑝2 ) (1 + 𝑠2 ) (−1 + 𝑘2 )
− .
1 1 + 𝑝𝑠𝑘
− .
(1 + 𝑝 ) (1 + 𝑠2 ) (−1 + 𝑘2 )
2 (60)
𝑛
Now applying the inverse triple Laplace transform on the × ( − (4𝑠2 ) Hypergeometric2𝐹1
previous equation, we obtain the following solution:
1+𝑛 2+𝑛 1
𝑢 (𝑥, 𝑦, 𝑡) ×( , , 1 − 𝑛, − 2 )
2 2 𝑠
2 2 2
𝐺 (𝑝, 𝑠, 𝑘) 𝑘𝑠𝑝/ (1 + 𝑝 ) (1 + 𝑠 ) (−1 + 𝑘 ) + cos (𝑛𝜋) Hypergeometric2𝐹1
= 𝐿−1
𝑥𝑦𝑡 [ +
1 + 𝑝𝑠𝑘 1 + 𝑝𝑠𝑘
1+𝑛 2+𝑛 1
1/ (1 + 𝑝2 ) (1 + 𝑠2 ) (−1 + 𝑘2 ) ×( , , 1 + 𝑛, − 2 ))
2 2 𝑠
− ]
1 + 𝑝𝑠𝑘 𝑛
× ( − (4𝑝2 ) Hypergeometric2𝐹1
= cos (𝑥) cos (𝑦) cos (−𝑡) .
(61) 1+𝑛 2+𝑛 1
×( , , 1 − 𝑛, − 2 )
2 2 𝑝
This is the exact solution for nonhomogeneous Mboctara
equation. + cos (𝑛𝜋) Hypergeometric2𝐹1
Example 4. consider the following nonlinear nonhomoge- 1+𝑛 2+𝑛 1
neous with variable coefficient Mboctara equation: ×( , , 1 + 𝑛, − 2 )) .
2 2 𝑝
(64)
𝑒𝑥+𝑦+𝑡 𝜕𝑥𝑦𝑡 𝑢 (𝑥, 𝑦, 𝑡) − 3𝑢2 (𝑥, 𝑦, 𝑡) + 𝑒𝑥+𝑦+𝑡 𝑢 (𝑥, 𝑦, 𝑡)
= 𝑒2𝑥+2𝑦+2𝑡 , 7. Conclusion
(62)
𝑢𝑥 (𝑥, 𝑦, 0) = 𝑒𝑥+𝑦 , 𝑢 (0, 0, 0) = 1, This work presents the definition of the triple Laplace trans-
form. Some triple Laplace transform is presented in Table 1.
𝑢 (1, 0, 0) = 𝑒, 𝜕𝑥𝑦𝑡 𝑢 (0, 0, 0) = 1. Some theorems and properties of this new relatively new
operator are presented. Applications of the new operator, for
Now applying the triple Laplace transform on both sides solving some kind of third-order partial differential equations
of (62) and then using the properties of the triple Laplace called Mboctara equation, are presented. Numerical solutions
transform and after factorising as in the previous examples of the Mboctara equation are given.
and taking the inverse triple Laplace transform, we obtain
the following as an exact solution of this type of Mboctara
equation: References
[1] G. L. Lamb Jr., Introductory Applications of Partial Differential
𝑢 (𝑥, 𝑦, 𝑡) = 𝑒𝑥+𝑦+𝑡 . (63) Equations with Emphasis on Wave Propagation and Diffusion,
John Wiley & Sons, New York, NY, USA, 1995.
The numerical simulations of the exact solutions of the Mboc-
[2] U. T. Myint, Differential Equations of Mathematical Physics,
tara equation are depicted in Figure 1(a) (4.1), Figure 1(b)
American Elsevier, New York, NY, USA, 1980.
(4.6), Figure 1(c) (4.6) and Figure 1(d) (4.11), respectively.
[3] C. Constanda, Solution Techniques for Elementary Partial Differ-
ential Equations, Chapman & Hall/CRC, New York, NY, USA,
6. Triple Laplace Transform of Some 2002.
Functions of Three Variables [4] D. G. Duffy, Transform Methods for Solving Partial Differential
Equations, CRC Press, New York, NY, USA, 2004.
In this section, we examine the triple Laplace transform of
[5] A. Babakhani and R. S. Dahiya, “Systems of multi-dimensional
some functions in Table 1:
Laplace transforms and a heat equation,” in Proceedings of the
16th Conference on Applied Mathematics, vol. 7 of ElectronicJour-
𝐿 𝑥𝑦𝑡 (𝑌𝑛 (𝑥) 𝑌𝑛 (𝑦) 𝑌𝑛 (𝑡)) nal of Differential Equations, pp. 25–36, University of Central
−1−𝑛
Oklahoma, Edmond, Okla, USA, 2001.
= 8−𝑛 (𝑘𝑠𝑝) 𝐶𝑠𝑐3 [𝑛𝜋]
[6] Y. A. Brychkov, H. -J. Glaeske, A. P. Prudnikov, and V. K. Tuan,
Multidimensional Integral Transformations, Gordon and Breach
2 𝑛
× ( − (4𝑘 ) Hypergeometric2𝐹1 Science Publishers, Philadelphia, Pa, USA, 1992.
[7] A. Atangana and A. Kilicman, “A possible generalization of
1+𝑛 2+𝑛 1 acoustic wave equation using the concept of perturbed deriva-
×( , , 1 − 𝑛, − 2 )
2 2 𝑘 tive order,” Mathematical Problems in Engineering, vol. 2013,
+ cos (𝑛𝜋) Hypergeometric2𝐹1
[8] A. Kılıçman and H. Eltayeb, “A note on the classifications of
1+𝑛 2+𝑛 1 hyperbolic and elliptic equations with polynomial coefficients,”
×( , , 1 + 𝑛, − 2 )) Applied Mathematics Letters, vol. 21, no. 11, pp. 1124–1128, 2008.
2 2 𝑘
[9] H. Eltayeb, A. Kılıçman, and P. Ravi Agarwal, “An analysis on

classifications of hyperbolic and elliptic PDEs,” Mathematical
Sciences, vol. 6, article 47, 2012.
[10] H. Eltayeb and A. Kılıçman, “A note on double laplace transform
and telegraphic equations,” Abstract and Applied Analysis, vol.
2013, Article ID 932578, 6 pages, 2013.
[11] A. Kılıçman and H. E. Gadain, “On the applications of Laplace
and Sumudu transforms,” Journal of the Franklin Institute, vol.
347, no. 5, pp. 848–862, 2010.
[12] R. P. Kanwal, Generalized Functions Theory and Applications,
Birkhäauser, Boston, Mass, USA, 2004.
http://dx.doi.org/10.1155/2013/759801
Research Article
Approximate Solution of Tuberculosis Disease Population
Dynamics Model
Abdon Atangana1 and Necdet Bildik2

1
Institute for Groundwater Studies, University of the Free State, P.O. Box 9300, Bloemfontein, South Africa
2
Department of Mathematics, Faculty of Art & Sciences, Celal Bayar University, Muradiye Campus, 45047 Manisa, Turkey
Correspondence should be addressed to Abdon Atangana; abdonatangana@yahoo.fr
Received 22 March 2013; Accepted 2 June 2013
Copyright © 2013 A. Atangana and N. Bildik. This is an open access article distributed under the Creative Commons Attribution
cited.
We examine possible approximate solutions of both integer and noninteger systems of nonlinear differential equations describing
tuberculosis disease population dynamics. The approximate solutions are obtained via the relatively new analytical technique, the
homotopy decomposition method (HDM). The technique is described and illustrated with numerical example. The numerical
simulations show that the approximate solutions are continuous functions of the noninteger-order derivative. The technique used
for solving these problems is friendly, very easy, and less time consuming.
1. Introduction 𝐼𝐴(𝑡) as captured in the model system of ordinary differential

equations that follows.
Tuberculosis, MTB, or TB (short for tubercle bacillus) is a
common and in many cases lethal, infectious disease caused 𝑑𝑆 (𝑡)
by various strains of Mycobacterium, usually Mycobacterium = V𝑓𝑁 − 𝛼𝐼𝐴𝑆 (𝑡) + 𝛿𝑆 (𝑡) + 𝑇𝐴𝐼𝐴(𝑡) + 𝑇𝐿 𝐼𝐿 (𝑡) ,
𝑑𝑡
Tuberculosis [1]. Tuberculosis typically attacks the lungs, but
can also affect other parts of the body. It is spread through 𝑑𝐼𝐿 (𝑡)
= (1 − 𝑃) 𝛼𝐼𝐴𝑆 (𝑡) − 𝛽𝐴 𝐼𝐿 (𝑡) − 𝑇𝐿 𝐼𝐿 (𝑡) − 𝛿𝐼𝐿 (𝑡) ,
the air when people who have an active TB infection cough, 𝑑𝑡
sneeze, or otherwise transmit their saliva through the air 𝑑𝐼𝐴 (𝑡)
[2]. Most infections are asymptomatic and latent, but about = 𝑃𝛼𝐼𝐴𝑆 (𝑡) + 𝛽𝐴 𝐼𝐿 (𝑡) − 𝑇𝐴𝐼𝐴 (𝑡) − 𝛿𝐼𝐴 (𝑡) − 𝜀𝐼𝐴 (𝑡)
𝑑𝑡
one in ten latent infections eventually progresses to active (1)
disease which, if left untreated, kills more than 50% of those
so infected. Interested reader can find more about this model subject to the initial conditions
in [3–7].
Based on the standard SIRS model, the model population 𝑆 (0) = 𝑁, 𝐼𝐿 (0) ≥ 0, 𝐼𝐴 (0) ≥ 0, (2)
was compartmentalised into the susceptible (𝑆) and the
infected (𝐼) which is further broken down into latently where 𝑁 is the total number of new people in the location of
infected (𝐼𝐿 ) and actively infected (𝐼𝐴) while the recovered interest; 𝑆 is the number of susceptible people in the location;
subpopulation is ploughed back into the susceptible group 𝐼𝐿 is the number of TB latently infected people; 𝐼𝐴 is the
due to the possibility of reinfection after successful treatment number of TB actively infected people; V is the probability
of the earlier infection. The model monitors the temporary that a susceptible person is not vaccinated; 𝑓 is the efficient
dynamics in the population of susceptible people (𝑡), TB rate of vaccines; 𝑇𝐿 is the success rate of latent 𝑇𝐵 therapy; 𝑇𝐴
latently infected people 𝐼𝐿 (𝑡), and TB actively infected people is the active TB treatment cure rate; 𝛼 is the TB instantaneous
incidence rate per susceptible; 𝛿 is humans natural death rate; The multi-integral in (3) can be transformed to
𝑃 is the proportion of infection instantaneously degenerating 𝑡 𝑡1 𝑡𝑚−1
into active TB; 𝜀 is the TB-induced death rate; and 𝛽𝐴 is the ∫ ∫ ⋅⋅⋅∫ 𝐿 (𝑈 (𝑥, 𝜏))
breakdown rate from latent to active TB. The equilibrium 0 0 0
analysis of the model was studied in [8]. Equation (1) together + 𝑁 (𝑈 (𝑥, 𝜏)) + 𝑓 (𝑥, 𝜏) 𝑑𝜏 ⋅ ⋅ ⋅ 𝑑𝑡
with (2) does not have an exact solution and is usually solved (6)
𝑡
numerically. 1
The purpose of this paper is to derive approximate analyt- = ∫ (𝑡 − 𝜏)𝑚−1 𝐿 (𝑈 (𝑥, 𝜏))
(𝑚 − 1)! 0
ical solutions for the standard form as well as the fractional
version of (1) together with (2) using the relatively new + 𝑁 (𝑈 (𝑥, 𝜏)) + 𝑓 (𝑥, 𝜏) 𝑑𝜏
analytical technique, the homotopy decomposition method so that (3) can be reformulated as
(HDM).
𝑚−1 𝑘
The paper is structured as follows. In Section 2, we present 𝑡
𝑈 (𝑥, 𝑡) = ∑ 𝑦𝑖 (𝑥)
the basic ideal of the homotopy decomposition method for 𝑘=0
𝑘!
solving partial differential equations. We present the applica-
𝑡
tion of the HDM for system Tuberculosis disease population 1
dynamics model in Section 3. In Section 4, we present the + ∫ (𝑡 − 𝜏)𝑚−1 𝐿 (𝑈 (𝑥, 𝜏))
(𝑚 − 1)! 0
application of the HDM for system of fractional Tuberculosis
disease population dynamics model. The conclusions are then + 𝑁 (𝑈 (𝑥, 𝜏)) + 𝑓 (𝑥, 𝜏) 𝑑𝜏.
given finally in Section 5. (7)
Using the homotopy scheme, the solution of the aforemen-
tioned integral equation is given in series form as
2. Fundamental Information about Homotopy
∞
Decomposition Method 𝑈 (𝑥, 𝑡, 𝑝) = ∑ 𝑝𝑛 𝑈𝑛 (𝑥, 𝑡) ,
𝑛=0 (8)
To demonstrate the elementary notion of this technique,
we consider a universal nonlinear nonhomogeneous partial 𝑈 (𝑥, 𝑡) = lim 𝑈 (𝑥, 𝑡, 𝑝) ,
differential equation with initial conditions of the following 𝑝→1
form [9–13]. and the nonlinear term can be decomposed as
∞
𝑚
𝜕 𝑈 (𝑥, 𝑡) 𝑁𝑈 (𝑟, 𝑡) = ∑ 𝑝𝑛 H𝑛 (𝑈) , (9)
= 𝐿 (𝑈 (𝑥, 𝑡)) + 𝑁 (𝑈 (𝑥, 𝑡)) + 𝑓 (𝑥, 𝑡) , 𝑛=1
𝜕𝑡𝑚 (3)
where 𝑝 ∈ (0, 1] is an implanting parameter. H𝑛 (𝑈) is the
𝑚 = 1, 2, 3 . . . , polynomials that can be engendered by
𝑛
1 𝜕𝑛 [
focused on the primary condition H𝑛 (𝑈0 , . . . , 𝑈𝑛 ) = 𝑛
𝑁 ( ∑𝑝𝑗 𝑈𝑗 (𝑥, 𝑡))] ,
𝑛! 𝜕𝑝 𝑗=0 (10)
[ ]
𝜕𝑖 𝑈 (𝑥, 0) 𝜕𝑚−1 𝑈 (𝑥, 0) 𝑛 = 0, 1, 2 . . . .
= 𝑦𝑖 (𝑥) , = 0,
𝜕𝑡𝑖 𝜕𝑡𝑚−1 (4) The homotopy decomposition method is obtained by the
𝑖 = 0, 1, 2, . . . , 𝑚 − 2, combination of decomposition method with Abel integral
and is given by
∞
where 𝑚 is the order of the derivative, where 𝑓 is an identified ∑ 𝑝𝑛 𝑈𝑛 (𝑥, 𝑡)
function, 𝑁 is the common nonlinear differential operator, 𝐿 𝑛=0
denotes a linear differential operator, and 𝑚 is the order of 1
the derivative. The procedures first stage here is to apply the = 𝑇 (𝑥, 𝑡) + 𝑝
(𝑚 − 1)!
inverse operator 𝜕𝑚 /𝜕𝑡𝑚 on both sides of (3) to obtain
𝑡 ∞
(11)
𝑚−1 𝑛
× ∫ (𝑡 − 𝜏) [𝑓 (𝑥, 𝜏) + 𝐿 ( ∑ 𝑝 𝑈𝑛 (𝑥, 𝜏))
𝑚−1 𝑘 0
𝑡 𝑑𝑘 𝑢 (𝑥, 0) 𝑛=0
𝑈 (𝑥, 𝑡) = ∑
𝑘=0
𝑘! 𝑑𝑡𝑘 ∞
+ ∑ 𝑝𝑛 H𝑛 (𝑈)] 𝑑𝜏
𝑡 𝑡1 𝑡𝑚−1 𝑛=0
+ ∫ ∫ ⋅⋅⋅∫ 𝐿 (𝑈 (𝑥, 𝜏))
0 0 0 with
𝑚−1 𝑘
+ 𝑁 (𝑈 (𝑥, 𝜏)) + 𝑓 (𝑥, 𝜏) 𝑑𝜏 ⋅ ⋅ ⋅ 𝑑𝑡. 𝑡
𝑇 (𝑥, 𝑡) = ∑ 𝑦𝑖 (𝑥) . (12)
(5) 𝑘=0
𝑘!
Relating the terms of same powers of 𝑝, this gives solutions

of various orders. The initial guess of the approximation is −𝑇𝐿 𝐼𝐿(𝑛−1) (𝜏) − 𝛿𝐼𝐿(𝑛−1) (𝜏) ) 𝑑𝜏,
𝑇(𝑥, 𝑡) that is actually the Taylor series of the exact solution
of order 𝑚. Note that this initial guess insures the uniqueness 𝐼𝐿𝑛 (0) = 0
of the series decompositions [9].
𝑝𝑛 : 𝐼𝐴𝑛 (𝑡)
3. Application of the HDM to the Model with 𝑡 𝑛−1
Integer-Order Derivative = ∫ (𝑃𝛼 ∑ 𝐼𝐴𝑗 𝑆𝑛−𝑗−1 (𝜏) + 𝛽𝐴 𝐼𝐿(𝑛−1) (𝜏) − 𝑇𝐴 𝐼𝐴(𝑛−1) (𝜏)
0 𝑗=0
In this section, we employ this method for deriving the set
of the mathematical equations describing the tuberculosis
disease population dynamics model. −𝛿𝐼𝐴(𝑛−1) (𝜏) − 𝜀𝐼𝐴(𝑛−1) (𝜏) ) 𝑑𝜏, 𝐼𝐴𝑛 (0) = 0.
Resulting from the steps involved in the HDM method,
we reach at the following integral equations that are very (13)
simple to solve:
Integrating the previous, we obtain the following compo-
𝑝0 : 𝑆0 (𝑡) = 𝑆 (0) , nents:
𝑆0 (𝑡) = 𝑆 (0) ; 𝐼𝐿0 (𝑡) = 𝐼𝐿 (0) ;
𝑝0 : 𝐼𝐿0 (𝑡) = 𝐼𝐿 (0) ,
𝐼𝐴0 (𝑡) = 𝐼𝐴 (0) ,
𝑝0 : 𝐼𝐴0 (𝑡) = 𝐼𝐴 (0) ,
𝑆1 (𝑡) = (V𝑓𝑁 − 𝛼𝐼𝐴0 𝑆0 + 𝛿𝑆0 + 𝑇𝐴𝐼𝐴0 + 𝑇𝐿 𝐼𝐿0 ) 𝑡,
𝑝1 : 𝑆1 (𝑡)
𝐼𝐿1 (𝑡) = ((1 − 𝑃) 𝛼𝐼𝐴0 𝑆0 − 𝛽𝐴𝐼𝐿0 − 𝑇𝐿 𝐼𝐿 0 − 𝛿𝐼𝐿0 ) 𝑡,
𝑡
= ∫ (V𝑓𝑁 − 𝛼𝐼𝐴0 𝑆0 (𝜏) + 𝛿𝑆0 (𝜏) 𝐼𝐴1 (𝑡) = (𝑃𝛼𝐼𝐴0 𝑆0 + 𝛽𝐴 𝐼𝐿0 − 𝑇𝐴𝐼𝐴0 − 𝛿𝐼𝐴0 − 𝜀𝐼𝐴0 ) 𝑡.
0
(14)
+𝑇𝐴𝐼𝐴0 (𝜏) + 𝑇𝐿 𝐼𝐿0 (𝜏)) 𝑑𝜏, 𝑆1 (0) = 0,
For simplicity, let us put
𝑝1 : 𝐼𝐿1 (𝑡)
𝑎 = (V𝑓𝑁 − 𝛼𝐼𝐴0 𝑆0 + 𝛿𝑆0 + 𝑇𝐴𝐼𝐴0 + 𝑇𝐿 𝐼𝐿0 ) ,
𝑡
= ∫ ((1 − 𝑃) 𝛼𝐼𝐴0 𝑆0 (𝜏) − 𝛽𝐴 𝐼𝐿0 (𝜏) 𝑏 = ((1 − 𝑃) 𝛼𝐼𝐴0 𝑆0 − 𝛽𝐴 𝐼𝐿0 − 𝑇𝐿 𝐼𝐿 0 − 𝛿𝐼𝐿0 ) ,
0
−𝑇𝐿 𝐼𝐿 0 (𝜏) − 𝛿𝐼𝐿0 (𝜏)) 𝑑𝜏, 𝐼𝐿1 (0) = 0, 𝑐 = (𝑃𝛼𝐼𝐴0 𝑆0 + 𝛽𝐴 𝐼𝐿0 − 𝑇𝐴𝐼𝐴0 − 𝛿𝐼𝐴0 − 𝜀𝐼𝐴0 ) ,
1
𝑝1 : 𝐼𝐴1 (𝑡) 𝑆2 (𝑡) = 𝑡2 (𝑏𝑇𝐴 + 𝑐𝑇𝐿 − 𝑎𝐼𝐴0 𝛼 − 𝑏𝑆0 𝛼 + 𝑎𝛿)
2
𝑡
= ∫ (𝑃𝛼𝐼𝐴0 𝑆0 (𝜏) + 𝛽𝐴 𝐼𝐿0 (𝜏) 𝑡2
= 𝑎,
0 2 1
(15)
−𝑇𝐴𝐼𝐴0 (𝜏) − 𝛿𝐼𝐴0 (𝜏) − 𝜀𝐼𝐴0 (𝜏)) 𝑑𝜏, 𝐼𝐴1 (0) = 0, 1
𝐼𝐿2 (𝑡) = 𝑡2 (−𝑐𝑇𝐿 + 𝑎𝐼𝐴0 𝛼 − 𝑎𝐼𝐴0 𝑃𝛼
2
..
. 𝑡2
+𝑏𝑆0 𝛼 − 𝑐𝛽𝐴 − 𝑐𝛿) = 𝑏,
𝑝𝑛 : 𝑆𝑛 (𝑡) 2 1
1 2
𝑡 𝑛−1 𝐼𝐴2 (𝑡) = 𝑡 (𝑎𝐼𝐴0 𝑃𝛼 + 𝑏𝑃𝛼𝑆0 − 𝑏𝑇𝐴
= ∫ (V𝑓𝑁 − 𝛼 ∑ 𝐼𝐴𝑗 𝑆𝑛−𝑗−1 (𝜏) + 𝛿𝑆𝑛−1 (𝜏) 2
0 𝑗=0 𝑡2
+𝑐𝛽𝐴 − 𝑏𝛿 − 𝑏𝜀) = 𝑐1 .
2
+𝑇𝐴 𝐼𝐴(𝑛−1) (𝜏) + 𝑇𝐿 𝐼𝐿(𝑛−1) (𝜏) ) 𝑑𝜏, In general, we obtain the following recursive formulas:
𝑡𝑛
𝑆𝑛−1 (0) = 0, 𝑆𝑛 (𝑡) = 𝑎,
𝑛! 𝑛
𝑛
𝑝 : 𝐼𝐿𝑛 (𝑡) 𝑡𝑛
𝐼𝐿𝑛 (𝑡) = 𝑏, (16)
𝑛! 𝑛
𝑡 𝑛−1
= ∫ ((1 − 𝑃) 𝛼 ∑ 𝐼𝐴𝑗 𝑆𝑛−𝑗−1 (𝜏) − 𝛽𝐴 𝐼𝐿(𝑛−1) (𝜏) 𝑡𝑛
0 𝐼𝐴𝑛 (𝑡) = 𝑐𝑛 ,
𝑗=0 𝑛!
where 𝑎𝑛 , 𝑏𝑛 , and 𝑐𝑛 depend on the fixed set of empirical Susceptible people

parameters. It therefore follows that the approximate solution 100
of the system (1) is given as 90
𝑁 80
𝑡𝑛
𝑆𝑁 (𝑡) = ∑ 𝑎𝑛 , 70
𝑛=0 𝑛!
60
𝑁 𝑛
𝑡 50
𝐼𝐿𝑁 (𝑡) = ∑ 𝑏𝑛 , (17)
𝑛=0 𝑛! 40
𝑁 30
𝑡𝑛
𝐼𝐴𝑁 (𝑡) = ∑ 𝑐𝑛 .
𝑛=0 𝑛!
20
10
If for instance one supposes that the total number of new
people in the location of interest is 𝑁 = 100; the initial 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
number of susceptible people in the location is 𝑆(0) = 96; Time
the initial number of TB latently infected people is 𝐼𝐿 (0) = 3;
the initial number of TB actively infected people is 𝐼𝐴(0) = 1; Figure 1: Approximate solution for the number of susceptible
the probability that a susceptible person is not vaccinated is people in the location.
V = 0.5; the efficient rate of vaccines is 𝑓 = 0.5; the success
rate of latent TB therapy is 𝑇𝐿 = 0.8; the active TB treatment
Latent people
cure rate is 𝑇𝐴 = 0.74; the TB instantaneous incidence rate 9
per susceptible is 𝛼 = 0.41; humans natural death rate is
𝛿 = 1/(366 × 70); the proportion of infection instantaneously 8
degenerating into active TB is 𝑃 = 0.0197; the TB-induced 7
death rate is 𝜀 = 0.0735; and the breakdown rate from latent
to active TB is 𝛽𝐴 = 0.01, then the following approximate 6
solution is obtained as a result of the first 8 terms of the series
5
decomposition:
4
𝑆 (𝑡) = 96 − 11.2162𝑡 + 62.1069𝑡2 − 29.5924𝑡3 − 149.2𝑡4
3
+ 48.3455𝑡5 − 20.6378𝑡6 + 15.5857𝑡7 + ⋅ ⋅ ⋅ 2
𝐼𝐿 (𝑡) = 3 + 36.8527𝑡 − 62.9161𝑡2 − 797.302𝑡3 + 151.174𝑡4 1
0
− 48.8926𝑡5 + 20.7629𝑡6 − 15.6036𝑡7 + ⋅ ⋅ ⋅ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2 3 Time
𝐼𝐴 (𝑡) = 1 − 0.706394𝑡 + 0.252053𝑡 − 0.252832𝑡
Figure 2: Approximate solution for the number of TB latently
− 1.96203𝑡4 + 0.573666𝑡5 − 0.131459𝑡6 infected people.
+ 0.0190148𝑡7 + ⋅ ⋅ ⋅ .
(18) people will also vanish because of the natural death rate and
the death due to TB.
If in addition we assume that no new person migrates or
is born in this area, we obtain the following figures. The
approximate solutions of the main problem are depicted in 4. Application of the HDM to the Model with
Figures 1, 2, and 3, respectively. Noninteger-Order Derivative
Figure 1 shows that, if there is migration or newborn in
the location of interest, the number of susceptible people will Fractional calculus has been used to model physical and
vanish as time goes, because of the natural death rate and due engineering processes, which are found to be best described
to TB. Note that any person that is latently infected is removed by fractional differential equations. It is worth noting that the
from the set of susceptible. Figure 2 indicates that the number standard mathematical models of integer-order derivatives,
of people that are latently infected will increase up to a including nonlinear models, do not work adequately in
certain time and then vanish as time goes. The number of many cases. In the recent years, fractional calculus has
susceptible people, will become latently infected since some played a very important role in various fields such as
are not vaccinated against the TB and finally will vanish due mechanics, electricity, chemistry, biology, economics, notably
to. Figure 3 indicates that the number of TB actively infected control theory, and signal and image processing. Major topics
Active people Definition 4 (partial derivatives of fractional order). Assume

4 now that 𝑓(x) is a function of 𝑛 variables 𝑥𝑖 𝑖 = 1, . . . , 𝑛 also
3.5
of class 𝐶 on 𝐷 ∈ R𝑛 . We define partial derivative of order 𝛼
for 𝑓 respect to 𝑥𝑖 the function
3 1 𝑥𝑖
󵄨
𝜕𝑥𝑖 𝑓 (𝑥𝑗 )󵄨󵄨󵄨󵄨𝑥 =𝑡 𝑑𝑡. (22)
𝑚−𝛼−1 𝑚
𝑎𝜕𝛼x 𝑓 = ∫ (𝑥𝑖 − 𝑡)
2.5 Γ (𝑚 − 𝛼) 𝑎 𝑗
where 𝜕𝑥𝑚𝑖 is the usual partial derivative of integer-order 𝑚.

2
1.5 4.2. Approximate Solution of Fractional Version. The system

of equations under investigation here is given as
1
𝑑𝜇 𝑆 (𝑡)
0.5
= V𝑓𝑁 − 𝛼𝐼𝐴𝑆 (𝑡) + 𝛿𝑆 (𝑡)
𝑑𝑡𝜇
0 + 𝑇𝐴 𝐼𝐴 (𝑡) + 𝑇𝐿 𝐼𝐿 (𝑡) , 0 < 𝜇 ≤ 1,
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time 𝑑𝜂 𝐼𝐿 (𝑡)
= (1 − 𝑃) 𝛼𝐼𝐴𝑆 (𝑡) − 𝛽𝐴𝐼𝐿 (𝑡)
Figure 3: Approximate solution for the number of TB actively
𝑑𝑡𝜂 (23)
infected people. − 𝑇𝐿 𝐼𝐿 (𝑡) − 𝛿𝐼𝐿 (𝑡) , 0 < 𝜂 ≤ 1,
𝑑𝜐 𝐼𝐴 (𝑡)
include anomalous diffusion; vibration and control; continu- = 𝑃𝛼𝐼𝐴𝑆 (𝑡) + 𝛽𝐴𝐼𝐿 (𝑡) − 𝑇𝐴 𝐼𝐴 (𝑡)
𝑑𝑡𝜐
ous time random walk; Levy statistics, fractional Brownian
motion; fractional neutron point kinetic model; power law; − 𝛿𝐼𝐴 (𝑡) − 𝜀𝐼𝐴 (𝑡) , 0 < 𝜐 ≤ 1.
Riesz potential; fractional derivative and fractals; computa-
Following the discussion presented earlier, we arrive at the
tional fractional derivative equations; nonlocal phenomena;
following equations:
history-dependent process; porous media; fractional filters;
biomedical engineering; fractional phase-locked loops, and 𝑝0 : 𝑆0 (𝑡) = 𝑆 (0) ,
groundwater problem (see [14–21]).
𝑝0 : 𝐼𝐿0 (𝑡) = 𝐼𝐿 (0) ,
4.1. Properties and Definitions
𝑝0 : 𝐼𝐴0 (𝑡) = 𝐼𝐴 (0) ,
Definition 1. A real function 𝑓(𝑥), 𝑥 > 0, is said to be in the
space 𝐶𝜇 , 𝜇 ∈ R, if there exists a real number 𝑝 > 𝜇, such that 𝑝1 : 𝑆1 (𝑡)
𝑓(𝑥) = 𝑥𝑝 ℎ(𝑥), where ℎ(𝑥) ∈ 𝐶[0, ∞), and it is said to be in 1 𝑡
space 𝐶𝜇𝑚 if 𝑓(𝑚) ∈ 𝐶𝜇 , 𝑚 ∈ N. = ∫ (𝑡 − 𝜏)𝜇−1
Γ (𝜇) 0
Definition 2. The Riemann-Liouville fractional integral oper- × (V𝑓𝑁 − 𝛼𝐼𝐴0 𝑆0 (𝜏) + 𝛿𝑆0 (𝜏)
ator of order 𝛼 ≥ 0, of a function 𝑓 ∈ 𝐶𝜇 , 𝜇 ≥ −1, is defined
as +𝑇𝐴𝐼𝐴0 (𝜏) + 𝑇𝐿 𝐼𝐿0 (𝜏)) 𝑑𝜏,
𝑥
1
𝐽𝛼 𝑓 (𝑥) = ∫ (𝑥 − 𝑡)𝛼−1 𝑓 (𝑡) 𝑑𝑡, 𝛼 > 0, 𝑥 > 0 𝑆1 (0) = 0,
Γ (𝛼) 0 (19)
𝐽0 𝑓 (𝑥) = 𝑓 (𝑥) . 𝑝1 : 𝐼𝐿1 (𝑡)
𝑡
1
Properties of the operator can be found in [14–16]. We = ∫ (𝑡 − 𝜏)𝜂−1
mention only the following: for 𝑓 ∈ 𝐶𝜇 , 𝜇 ≥ −1, 𝛼, 𝛽 ≥ 0, Γ (𝜂) 0
and 𝛾 > −1, × ((1 − 𝑃) 𝛼𝐼𝐴0 𝑆0 (𝜏) − 𝛽𝐴𝐼𝐿0 (𝜏)
𝛼 𝛽 𝛼+𝛽
𝐽 𝐽 𝑓 (𝑥) = 𝐽 𝑓 (𝑥) , −𝑇𝐿 𝐼𝐿 0 (𝜏) − 𝛿𝐼𝐿0 (𝜏)) 𝑑𝜏,
𝛾 Γ (𝛾 + 1) 𝛼+𝛾 (20) 𝐼𝐿1 (0) = 0,
𝐽𝛼 𝐽𝛽 𝑓 (𝑥) = 𝐽𝛽 𝐽𝛼 𝑓 (𝑥) 𝐽𝛼 𝑥 = 𝑥 .
Γ (𝛼 + 𝛾 + 1) 𝑝1 : 𝐼𝐴1 (𝑡)
Lemma 3. If 𝑚 − 1 < 𝛼 ≤ 𝑚, 𝑚 ∈ N, and 𝑓 ∈ 𝐶𝜇𝑚 , 𝜇 ≥ −1, 1 𝑡
then = ∫ (𝑡 − 𝜏)𝜐−1
Γ (𝜐) 0
𝐷𝛼 𝐽𝛼 𝑓 (𝑥) = 𝑓 (𝑥) , × (𝑃𝛼𝐼𝐴0 𝑆0 (𝜏) + 𝛽𝐴 𝐼𝐿0 (𝜏)
𝑚−1
𝑥𝑘 (21) −𝑇𝐴𝐼𝐴0 (𝜏) − 𝛿𝐼𝐴0 (𝜏) − 𝜀𝐼𝐴0 (𝜏)) 𝑑𝜏,
𝐽𝛼 𝐷0𝛼 𝑓 (𝑥) = 𝑓 (𝑥) − ∑ 𝑓(𝑘) (0+ ) , 𝑥 > 0.
𝑘=0
𝑘! 𝐼𝐴1 (0) = 0,
𝑝𝑛 : 𝑆𝑛 (𝑡) 29.8522𝑡𝜂 4.58965𝑡𝜇

𝐼𝐿2 (𝑡) = − 𝑡𝜂 ( +
𝑡 Γ (1 + 2𝜂) Γ (1 + 𝜂 + 𝜇)
1
= ∫ (𝑡 − 𝜏)𝜇−1
Γ (𝜇) 0 27.7492𝑡𝜐
+ ),
𝑛−1 Γ (1 + 𝜐 + 𝜇)
× (V𝑓𝑁 − 𝛼 ∑ 𝐼𝐴𝑗 𝑆𝑛−𝑗−1 (𝜏) + 𝛿𝑆𝑛−1 (𝜏)
𝑗=0 0.368527𝑡𝜂 0.00901337𝑡𝜇
𝐼𝐴2 (𝑡) = 𝑡𝜐 ( −
Γ (1 + 𝜂 + 𝜐) Γ (1 + 𝜐 + 𝜇)
+𝑇𝐴𝐼𝐴(𝑛−1) (𝜏) + 𝑇𝐿 𝐼𝐿(𝑛−1) (𝜏) ) 𝑑𝜏,
0.520184𝑡𝜐
+ ),
𝑆𝑛−1 (0) = 0, Γ (1 + 2𝜐)
𝑝𝑛 : 𝐼𝐿𝑛 (𝑡) 3.24846𝑡𝜇+𝜐 Γ (1 + 𝜇 + 𝜐)

𝑆3 (𝑡) = 𝑡𝜇 (−
𝑡 Γ (1 + 𝜇) Γ (1 + 𝜐) Γ (1 + 2𝜇 + 𝜐)
1
= ∫ (𝑡 − 𝜏)𝜂−1
Γ (𝜂) 0 0.298522𝑡2𝜂 15.7583𝑡𝜂+𝜇
− −
𝑛−1 Γ (1 + 2𝜂 + 𝜐) Γ (1 + 𝜂 + 2𝜇)
× ((1 − 𝑃) 𝛼 ∑ 𝐼𝐴𝑗 𝑆𝑛−𝑗−1 (𝜏) − 𝛽𝐴𝐼𝐿(𝑛−1) (𝜏)
𝑗=0 1.88509𝑡2𝜇 36.4319𝑡𝜂+𝜐
− −
Γ (1 + 3𝜇) Γ (1 + 𝜇 + 𝜐 + 𝜂)
−𝑇𝐿 𝐼𝐿(𝑛−1) (𝜏) − 𝛿𝐼𝐿(𝑛−1) (𝜏) ) 𝑑𝜏,
10.836𝑡𝜇+𝜐 20.0895𝑡2𝜐
− − ),
𝐼𝐿𝑛 (0) = 0, Γ (1 + 2𝜇 + 𝜐) Γ (1 + 𝜇 + 2𝜐)
𝑝𝑛 : 𝐼𝐴𝑛 (𝑡) 1148.5𝑡2𝜂 164.513𝑡𝜂+𝜇 1.88158𝑡2𝜇

𝐼𝐿3 (𝑡) = 𝑡𝜂 (− − +
𝑡
Γ (1 + 3𝜂) Γ (1 + 2𝜂 + 𝜇) Γ (1 + 𝜂 + 2𝜇)
1
= ∫ (𝑡 − 𝜏)𝜐−1
Γ (𝜐) 0 1067.59𝑡𝜂 11.1633𝑡𝜇+𝜐
− +
𝑛−1 Γ (1 + 2𝜂 + 𝜐) Γ (1 + 𝜂 + 𝜇 + 𝜐)
× (𝑃𝛼 ∑ 𝐼𝐴𝑗 𝑆𝑛−𝑗−1 (𝜏) + 𝛽𝐴 𝐼𝐿(𝑛−1) (𝜏)
𝑗=0 3.2421𝑡𝜇+𝜐 Γ (1 + 𝜇 + 𝜐)
+ ),
− 𝑇𝐴 𝐼𝐴(𝑛−1) (𝜏) − 𝛿𝐼𝐴(𝑛−1) (𝜏) Γ (1 + 𝜇) Γ (1 + 𝜐) Γ (1 + 𝜂 + 𝜇 + 𝜐)
0.00636699𝑡𝜇+𝜐 Γ (1 + 𝜇 + 𝜐)
−𝜀𝐼𝐴(𝑛−1) (𝜏) ) 𝑑𝜏, 𝐼𝐴3 (𝑡) = 𝑡𝜐 (
Γ (1 + 𝜇) Γ (1 + 𝜐) Γ (1 + 2𝜐 + 𝜇)
𝐼𝐴𝑛 (0) = 0. 0.298522𝑡2𝜂 0.0222046𝑡𝜂+𝜇
− −
(24) Γ (1 + 2𝜂 + 𝜐) Γ (1 + 𝜂 + 2𝜇)
Integrating the previous, we obtain the following compo- 0.00369513𝑡2𝜇 0.548873𝑡𝜂+𝜐

+ −
nents: Γ (1 + 3𝜇) Γ (1 + 𝜇 + 2𝜐)
𝑆0 (𝑡) = 𝑆 (0) ; 𝐼𝐿0 (𝑡) = 𝐼𝐿 (0) ; 0.0285603𝑡𝜇+𝜐 0.38306𝑡2𝜐

+ − ).
Γ (1 + 𝜇 + 2𝜐) Γ (1 + 3𝜐)
𝐼𝐴0 (𝑡) = 𝐼𝐴 (0) ,
(25)
11.2162𝑡𝜇
𝑆1 (𝑡) = − ;
Γ (1 + 𝜇)
The remaining terms can be obtained in the same manner.
36.8527𝑡𝜂 But here only few terms of the series solutions are considered,
𝐼𝐿1 (𝑡) = , and the asymptotic solution is given as
Γ (1 + 𝜂)
𝑆 (𝑡) = 𝑆0 (𝑡) + 𝑆1 (𝑡) + 𝑆2 (𝑡) + 𝑆3 (𝑡) + ⋅ ⋅ ⋅ ,
0.706394𝑡𝜐
𝐼𝐴1 (𝑡) = − ,
Γ (1 + 𝜐) 𝐼𝐿 (𝑡) = 𝐼𝐿0 (𝑡) + 𝐼𝐿1 (𝑥, 𝑡) + 𝐼𝐿2 (𝑥, 𝑡) + 𝐼𝐿3 (𝑥, 𝑡) + ⋅ ⋅ ⋅ ,
29.4822𝑡𝜂 4.59822𝑡𝜇
𝑆2 (𝑡) = 𝑡𝜇 ( + 𝐼𝐴 (𝑡) = 𝐼𝐴0 (𝑡) + 𝐼𝐴1 (𝑥, 𝑡) + 𝐼𝐴2 (𝑥, 𝑡) + 𝐼𝐴3 (𝑥, 𝑡) + ⋅ ⋅ ⋅ .
Γ (1 + 𝜂 + 𝜇) Γ (1 + 2𝜇) (26)
27.2809𝑡𝜐 The following figures show the simulated solutions for differ-
+ ),
Γ (1 + 𝜐 + 𝜇) ent values of the fractional order derivatives. The approximate
Susceptible people Susceptible people

100 100
90 90
80
80
70
70
60
60
50
50
40
30 40
20 30
10 20
0 10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Figure 4: Approximate for 𝜇 = 0.45, 𝜂 = 0.7, and 𝜐 = 0.85. Time
Latent people Figure 7: Approximate for 𝜇 = 0.045, 𝜂 = 0.5, and 𝜐 = 0.085.

9
8 Latent people
4
7
3.5
6
5 3
4 2.5
3 2
2 1.5
1
1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5
Time
0
Figure 5: Approximate for 𝜇 = 0.45, 𝜂 = 0.7, and 𝜐 = 0.85. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time
Active people
4 Figure 8: Approximate for 𝜇 = 0.045, 𝜂 = 0.5, and 𝜐 = 0.085.
3.5
3
solutions of the main problem are depicted in Figures 4, 5, 6,
7, 8, and 9, respectively.
2.5 The numerical simulations show that the approximate
solutions are continuous functions of the noninteger-order
2 derivative. It is worth noting that the standard mathemati-
cal models of integer-order derivatives, including nonlinear
1.5 models, do not work adequately in many cases. It is therefore
advisable to use the fractional model for describing this
1 problem.
0.5
5. Conclusion
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 The tuberculosis model was examined for the case of integer-
Time and noninteger-order derivatives. Both systems of nonlinear
equations were solved with an iterative analytical model
Figure 6: Approximate for 𝜇 = 0.45, 𝜂 = 0.7, and 𝜐 = 0.85.
Active people isotropic strain-hardening plate supported by a rigid substrate

4 using the concept of non-integer derivatives,” Abstract Applied
Analysis, vol. 2013, Article ID 671321, 9 pages, 2013.
3.5
[11] A. Atangana and J. F. Botha, “Analytical solution of the ground-
3 water flow equation obtained via homotopy decomposition
method,” Journal of Earth Science & Climatic Change, vol. 3, p.
2.5 115, 2012.
[12] A. Atangana and E. Alabaraoye, “Solving a system of fractional
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1.5 Segel equations,” Advances in Difference Equations, vol. 2013, p.
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[13] A. Atangana and A. Secer, “Time-fractional coupled- the
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http://dx.doi.org/10.1155/2013/580461
Research Article
A Rectangular Mixed Finite Element Method with a Continuous
Flux for an Elliptic Equation Modelling Darcy Flow
Xindong Li and Hongxing Rui

School of Mathematics, Shandong University, Jinan 250100, China
Correspondence should be addressed to Hongxing Rui; hxrui@sdu.edu.cn
Received 26 March 2013; Accepted 29 May 2013
Copyright © 2013 X. Li and H. Rui. This is an open access article distributed under the Creative Commons Attribution License,
We introduce a mixed finite element method for an elliptic equation modelling Darcy flow in porous media. We use a staggered mesh
where the two components of the velocity and the pressure are defined on three different sets of grid nodes. In the present mixed
finite element, the approximate velocity is continuous and the conservation law still holds locally. The LBB consistent condition is
established, while the L2 error estimates are obtained for both the velocity and the pressure. Numerical examples are presented to
confirm the theoretical analysis.
1. Introduction we track the characteristic segment using the approximate

velocity, the discontinuities of the velocity may introduce
We consider the discretization technique for the elliptic some difficulties when the characteristic line cross the edges
problem modelling the flow in saturated porous media, or of element. While applying mass-conservative characteristic
the classical Darcy flow problem, including a system of mass finite element method to the coupled system of compressible
conservation law and Darcy’s law [1, 2]. The most popular miscible displacement in porous media, the continuous flux
numerical methods for this elliptic equation focus on mixed is crucial [10]. A brief description of this point will be found
finite element methods, since by this kind of methods the at the last part of this paper.
original scalar variable, called pressure, and its vector flux, To overcome this disadvantage, Arbogast and Wheeler
named Darcy velocity, can be approximated simultaneously [11] introduced a mixed finite element method with an
and maintain the local conservation. The classical theory for approximate velocity continuous in both the normal direc-
the mixed finite element, which includes the LBB consistent tion and the tangential direction, which was got by adding
condition, the existence and uniqueness of the approximate some freedom to the RT mixed finite element. In this
solution, and the error estimate, has been established. Some paper, we introduced a mixed finite element method with an
mixed finite element methods such as RT mixed finite ele- approximate velocity continuous in all directions. It is based
ment and BDM mixed finite element are introduced (as in [3– on rectangular mesh and uses continuous piecewise bilinear
6]), which satisfy the consistent condition and have optimal functions to approximate the velocity components and uses
order error estimate [7, 8]. Give some stabilized mixed finite piecewise constant functions to approximate the pressure. We
methods by adding to the classical mixed formulation some obtain the element by improving a kind of element for Stokes
least squares residual forms of the governing equations. equation and Navier-Stokes equation given by Han [12], Han
By using the abovementioned mixed finite element meth- and Wu [13], and Han and Yan [14]. By using this mixed finite
ods, the approximate velocity is continuous in the normal element, we can get continuous velocity vector and maintain
direction and discontinuous in the tangential direction on the local conservation. Comparing to the mixed finite ele-
the edges of the element. This is reasonable for the case of ment method in [11], we need less degrees of freedom for the
heterogenous permeability, yet it is desirable that the flux same convergence rate. The LBB consistent condition and 𝐿2
be continuous in some applications [9]. In particular, when error estimates of velocity and pressure are also established.
The outline of the rest of this paper is organized as follows. The following discussion and discrete analysis are related
In Sections 2 and 3, we recall the model problem and weak to the weak form (6). Let 𝑉0 be a closed subspace of 𝑉 via
formulation for the mixed finite element method and then
establish the discrete inf-sup consistent condition and 𝐿2 𝑉0 = {V ∈ 𝑉 : 𝑏 (V, 𝑞) = 0, ∀𝑞 ∈ 𝑆} . (8)
error estimates for the velocity and the pressure in Section 4.
In Section 5, we present some numerical examples which For the bilinear forms 𝑎(𝑢, V) and 𝑏(V, 𝑞), we have the standard
verify the efficiency of the proposed mixed finite element result.
method. A valuable application of this method to mass-con-
servative characteristic (MCC) scheme for the coupled com- Lemma 1. The bilinear form 𝑎(𝑢, V) is bounded on 𝑉 × 𝑉 and
pressible miscible displacement in porous media closes the coercive on 𝑉0 , and the bilinear form 𝑏(V, 𝑞) is bound on 𝑉 × 𝑆.
paper in Section 6. Namely,
(1) there exist two constants 𝐶1 > 0 and 𝛼 > 0 such that
2. The Mixed Finite Formulation
for Darcy Equation |𝑎 (𝑢, V)| ≤ 𝐶1 ‖𝑢‖ 𝐻(div ,Ω) ‖V‖ 𝐻(div ,Ω) ∀𝑢, V ∈ 𝑉,
(9)
The mathematical model for viscous flow in porous media 𝑎 (𝑢, 𝑢) ≥ 𝛼‖𝑢‖ 2𝐻(div ,Ω) ∀𝑢 ∈ 𝑉0 ,
includes Darcy’s law and conservation law of mass, written as
follows: (2) there is a constant 𝐶2 > 0 such that
𝜅 󵄨󵄨 󵄨 󵄩 󵄩
𝑢 = − ∇𝑝 on Ω (Darcy’s law) 󵄨󵄨𝑏 (V, 𝑞)󵄨󵄨󵄨 ≤ 𝐶2 󵄩󵄩󵄩𝑞󵄩󵄩󵄩 0,Ω ‖V‖ 𝐻(div ,Ω) ∀𝑞 ∈ 𝑆, V ∈ 𝑉. (10)
𝜇
div 𝑢 = 𝜙 on Ω (mass conservation) (1)
For the space V and S, the Ladyzhenskaya-Babuska- ̆
Brezzi(L-B-B) condition holds; see [15, 16], for example.
𝑢⋅𝑛=0 on Γ,
where 𝜅 > 0 is the permeability, 𝜇 > 0 is the viscosity, and 𝜙 is Lemma 2. There is a constant 𝛽 > 0 such that
the volumetric flow rate source or sink. Γ is the boundary of
Ω, and 𝑛 is the unit outward normal vector to Γ. The variable 𝑏 (V, 𝑞) 󵄩 󵄩
sup ≥ 𝛽󵄩󵄩󵄩𝑞󵄩󵄩󵄩 0,Ω , ∀𝑞 ∈ 𝑆. (11)
𝑢 = (𝑢1 , 𝑢2 ) is the Darcy velocity vector, and p is the pressure. V∈𝑉 ‖V‖ 𝐻(div ,Ω)
The source 𝜙 must satisfy the consistency constraint
It is clear that there exists a unique solution (𝑢, 𝑝) ∈ 𝑉×𝑆 to
∫ 𝜙𝑑Ω = 0. (2) the Problem (6).
Ω
2
Let 𝐿 (Ω) be the space of square integrable function in Ω 3. Finite Element Discretization
with inner product (⋅, ⋅) and norm ‖ ⋅ ‖. We use the notation
of the Hilbert space In this section, we present the mixed finite element based on
rectangular mesh for the Darcy flow problem.
2
𝐻 (div , Ω) = {𝑢 ∈ [𝐿2 (Ω)] ; div 𝑢 ∈ 𝐿2 (Ω)} , (3) In [13], Han and Wu introduced a mixed finite element for
Stokes problem and then extended to solve the Navier-Stokes
with norm problem [14]. Based on this element, we introduced the new
1/2 mixed finite element with a continuous flux approximation
‖𝑢‖ 𝐻(div ,Ω) = {‖𝑢‖ 2 + ‖ div 𝑢‖ 2 } . (4) for Darcy flow problem.
For simplicity, we suppose that the domain Ω is a unit
Define the following subspaces of 𝐻(div , Ω) and 𝐿2 (Ω): square, and the mixed finite element discussed here can be
𝑉 = 𝐻0 (div , Ω) = {𝑢 ∈ 𝐻 (div , Ω) : 𝑢 ⋅ 𝑛 = 0 on Γ} , easily generalized to the case when the domain Ω is a rectan-
gular.
Let 𝑁 be a given integer and ℎ = 1/𝑁. We construct the
𝑆 = {𝑞 | 𝑞 ∈ 𝐿2 (Ω) : ∫ 𝑞𝑑Ω = 0} .
Ω finite-dimensional subspaces of 𝑆 and 𝑉 by introducing three
(5) different quadrangulations 𝜏ℎ , 𝜏1ℎ , 𝜏2ℎ of Ω.
First, we divide Ω into uniform squares
The classical weak variational formulation of Problem (1) is
as follows: find (𝑢, 𝑝) ∈ 𝑉 × 𝑆, such that
𝑇𝑖,𝑗 = {(𝑥, 𝑦) : 𝑥𝑖−1 ≤ 𝑥 ≤ 𝑥𝑖 , 𝑦𝑗−1 ≤ 𝑦 ≤ 𝑦𝑗 } ,
𝑎 (𝑢, V) − 𝑏 (V, 𝑝) = 0 ∀V ∈ 𝑉, (12)
(6) 𝑖, 𝑗 = 1, . . . , 𝑁,
𝑏 (𝑢, 𝑞) = (𝜙, 𝑞) ∀𝑞 ∈ 𝑆.
where 𝑥𝑖 = 𝑖ℎ and 𝑦𝑗 = 𝑗ℎ. The corresponding quadrangula-
Here, tion is denoted by 𝜏ℎ . See Figure 1(a).
𝜇
𝑎 (𝑢, V) = ∫ 𝑢 ⋅ V𝑑𝑥 𝑏 (V, 𝑞) = ∫ 𝑞 div V𝑑𝑥. (7) 𝜏𝑖,𝑗 = {𝑇𝑖,𝑗 : 𝑖, 𝑗 = 1, . . . , 𝑁} . (13)
Ω𝜅 Ω
Γ3
Γ4 Γ2
Γ1
(a) (b)
(c)
Figure 1: Quadrangulations: (a)𝜏ℎ , (b)𝜏1ℎ , and (c) 𝜏2ℎ .
Then, for all 𝑇𝑖,𝑗 ∈ 𝜏ℎ , we connect all the neighbor mid- where 𝑄1,1 denotes the piecewise bilinear polynomial space
points of the vertical sides of 𝑇𝑖,𝑗 by straight segments if the with respect to the variables 𝑥 and 𝑦. Let
neighbor midpoints have the same vertical coordinate. Then,
Ω is divided into squares and rectangles. The corresponding
quadrangulation is denoted by 𝜏1ℎ (see Figure 1(b)). Similarly, 𝑉ℎ = 𝑉ℎ1 × 𝑉ℎ2 . (16)
for all 𝑇𝑖,𝑗 ∈ 𝜏ℎ , we connect all the neighbor midpoints of the
horizontal sides of 𝑇𝑖,𝑗 by straight line segments if the neigh-
bor midpoints have the same horizontal coordinate. Then, we Obviously, 𝑉ℎ is a subspace of 𝑉.
obtained the third quadrangulation of Ω, which is denoted by Using the subspaces 𝑉ℎ and 𝑆ℎ instead of 𝑉 and 𝑆 in the
𝜏2ℎ (see Figure 1(c)). variational Problem (6), we obtain the discrete problem: find
Based on the quadrangulation 𝜏ℎ , we define the piecewise (𝑢ℎ , 𝑝ℎ ) ∈ 𝑉ℎ × 𝑆ℎ , such that
constant functional space used to approximate the pressure
𝑆ℎ := {𝑞ℎ : 𝑞ℎ | 𝑇 = constant, ∀𝑇 ∈ 𝜏ℎ ; ∫ 𝑞ℎ 𝑑𝑥 = 0} . 𝑎 (𝑢ℎ , Vℎ ) − 𝑏 (Vℎ , 𝑝ℎ ) = 0 ∀Vℎ ∈ 𝑉ℎ ,

Ω
(14) (17)
𝑏 (𝑢ℎ , 𝑞ℎ ) = (𝜙, 𝑞ℎ ) ∀𝑞ℎ ∈ 𝑆ℎ .
𝑆ℎ is a subspace of 𝑆.
Furthermore, using the quadrangulations 𝜏1ℎ and 𝜏2ℎ , we
construct a subspace of 𝑉. Denote by Γ1 , Γ2 , Γ3 , and Γ4 the 4. Convergence Analysis and Error Estimate
south, right, north, and left sides on the boundary of Ω. Set
In this section, we give the corresponding convergence anal-
𝑉ℎ1 = {Vℎ ∈ 𝐶(0) (Ω) : Vℎ | 𝑇1 ∈ 𝑄1,1 (𝑇1 ) ∀𝑇1 ∈ 𝜏1ℎ , ysis and error estimate. Firstly, we define an interpolating for
Vℎ = 0 on Γ2 ∪ Γ4 } , the following analysis.
(15) For the quadrangulation 𝜏ℎ , we divided the edges of all
𝑉ℎ2 = {Vℎ ∈ 𝐶(0) (Ω) : Vℎ | 𝑇2 ∈ 𝑄1,1 (𝑇2 ) ∀𝑇2 ∈ 𝜏2ℎ , squares into two sets. The first one denoted by 𝐿 𝑉 contains
all vertical edges, and the second one denoted by 𝐿 𝐻 contains
Vℎ = 0 on Γ1 ∪ Γ3 } , all horizontal edges. We define the interpolation operator
Π : 𝑉 → 𝑉ℎ by Π𝑢 = (Π1ℎ 𝑢1 , Π2ℎ 𝑢2 ) ∈ 𝑉ℎ1 × 𝑉ℎ2 , which satisfy

the following:
∫ Π1ℎ 𝑢1 𝑑𝑠 = ∫ 𝑢1 𝑑𝑠 ∀𝑙 ∈ 𝐿 𝑉󸀠 ,
𝑙 𝑙
(18)
∫ Π2ℎ 𝑢2 𝑑𝑠 = ∫ 𝑢2 𝑑𝑠 ∀𝑙 ∈ 𝐿 𝐻󸀠 ,
𝑙 𝑙
where 𝐿 𝑉󸀠 is a set of edges of elements got by bisecting the

most bottom element edges and the most top element edges
of 𝐿 𝑉 and 𝐿 𝐻󸀠 are got by bisecting the most left element edges
and the most right element edges of 𝐿 𝐻. See Figures 2 and 3.
Lemma 3. For any 𝑢 ∈ 𝑉, the interpolating Π𝑢 ∈ 𝑉ℎ is unique-

ly determined by (18).
Proof. It is easy to see that (18) is equivalent to an equation of Some edges on L V Corresponding edges on L V󳰀
𝐴𝑋 = 𝐵, where A is a matrix and X, B are vectors. Direct (a) (b)
calculation shows that
Figure 2: Some edges on 𝐿 𝑉 and corresponding edges on 𝐿 𝑉󸀠 .
𝐴 = ℎ ∗ diag {𝐴 1 , 𝐴 1 , . . . } , (19)
and the form of submatrix 𝐴 1 is as follows
1 1 (iii) For any 𝑢 ∈ 𝑉, we have that
0 0 0 ⋅⋅⋅ 0 0 0 0
4 4
3 1 ∫ 𝑞ℎ div (𝑢 − Π𝑢) 𝑑𝑥 = 0, ∀𝑞ℎ ∈ 𝑆ℎ . (25)
(0 0 0 ⋅⋅⋅ 0 0 0 0) Ω
( 8 8 )
( )
( 1 3 1 )
( ) Proof. The estimates (22), (23), and (24) follow from Defini-
(0 0 ⋅⋅⋅ 0 0 0 0)
( 8 4 8 ) tion (18) and the approximation theory; see [1], for example.
( )
( )
( 0 0 1 3 1 ⋅⋅⋅ 0 0 0 0)
( 8 4 8 ). (20) For (25), based on Green formulation, we know that
( )
(⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ )
( 1 3 1 )
( ) ∫ 𝑞ℎ div (𝑢 − Π𝑢) 𝑑𝑥 = ∑ ∫ 𝑞ℎ div (𝑢 − Π𝑢) 𝑑𝑥
( 0 0 0 0 0 ⋅⋅⋅ 0)
( 8 4 8 ) Ω 𝑇∈𝜏ℎ 𝑇
( )
( 1 3 )
( 0 0 0 0 0 ⋅⋅⋅ 0 0)
8 8 = ∑ ∫ ⃗
𝑞ℎ (𝑢 − Π𝑢) ⋅ 𝑛𝑑𝑠
𝑇∈𝜏ℎ 𝜕𝑇
1 1
0 0 0 0 0 ⋅⋅⋅ 0 0
( 4 4)
− ∑ ∫ ∇𝑞ℎ ⋅ (𝑢 − Π𝑢) 𝑑𝑥
We can see that the matrix is invertible and the equation is 𝑇∈𝜏ℎ 𝑇
solvable, and therefore X can be uniquely determined.
Assume that the solution (𝑢, 𝑝) of Problem (6) has the = ∑ ∫ 𝑞ℎ (𝑢1 − Π1ℎ 𝑢1 ) 𝑛1 𝑑𝑠
𝑙∈𝐿 𝑉 𝑙
following smoothness properties:
𝑢 ∈ 𝑉󸀠 := 𝑉 ⋂ 𝐻2 ((Ω))2 , 𝑝 ∈ 𝑆 ⋂ 𝐻1 (Ω) . (21) + ∑ ∫ 𝑞ℎ (𝑢2 − Π2ℎ 𝑢2 ) 𝑛2 𝑑𝑠

𝑙∈𝐿 𝐻 𝑙
Then, we should give the following lemma about the proper-
ties of the interpolations defined in (18). = ∑ ∫ 𝑞ℎ (𝑢1 − Π1ℎ 𝑢1 ) 𝑛1 𝑑𝑠
𝑙∈𝐿 𝑉󸀠 𝑙
Lemma 4. (i) There exist two constants 𝐶3 and 𝐶4 indepen-
dent of h, such that
+ ∑ ∫ 𝑞ℎ (𝑢2 − Π2ℎ 𝑢2 ) 𝑛2 𝑑𝑠
𝑗−𝑖 𝑙
|𝑢 − Π𝑢|𝑖,2,Ω ≤ 𝐶3 ℎ |𝑢| 𝑗,2,Ω , 𝑖 = 0, 1, 𝑖 ≤ 𝑗 ≤ 2, (22) 𝑙∈𝐿 𝐻󸀠
󵄩 󵄩
inf 󵄩󵄩󵄩𝑝 − 𝑞ℎ 󵄩󵄩󵄩 ≤ 𝐶4 ℎ|𝑝|1,Ω . (23)
= 0.
𝑞ℎ ∈𝑆ℎ
(26)
(ii) There exists a constant 𝐶5 independent of h such that
Here, 𝑛⃗
= (𝑛1 , 𝑛2 ), and we use (18) for the last step. The proof
‖Π𝑢‖ 𝐻(div ,Ω) ≤ 𝐶5 ‖𝑢‖ 1,Ω ∀𝑢 ∈ 𝑉. (24) is completed.
Some edges on L H Corresponding edges on L H󳰀
(a) (b)
Figure 3: Some edges on 𝐿 𝐻 and corresponding edges on 𝐿 𝐻󸀠 .
Theorem 5. The discrete Inf-sup condition is valid; namely, Lemma 6. For any 𝑢 ∈ 𝑄1,1 (𝜏1ℎ ) × 𝑄1,1 (𝜏2ℎ ), there holds the
there is a constant 𝛽 ≥ 0, such that equivalence Π󸀠 𝑢 = 𝑃𝑉󸀠 𝑢; namely,
𝑏 (V , 𝑞 ) 󵄩 󵄩 (𝑢 − Π󸀠 𝑢, V) = 0, ∀V ∈ 𝑉ℎ󸀠 . (34)
sup 󵄩󵄩 󵄩󵄩 ℎ ℎ ≥ 𝛽 󵄩󵄩󵄩𝑞ℎ 󵄩󵄩󵄩 , ∀𝑞ℎ ∈ 𝑆ℎ . (27)
Vℎ ∈𝑉ℎ 󵄩 V 󵄩
󵄩 ℎ 󵄩 𝐻(div ,Ω) 𝑉ℎ󸀠
Proof. As the definition of is based on each element 𝑇, we
Proof. From the process above, we obtain that 𝑏(V, 𝑞ℎ ) = focus our discussion on arbitrary element 𝑒 ⊂ 𝜏ℎ , 𝑒 = [𝑥0 , 𝑥0 +
𝑏(ΠV, 𝑞ℎ ), any V ∈ 𝑉, 𝑞ℎ ∈ 𝑆ℎ . For any 𝑝ℎ ∈ 𝑆ℎ , there exists ℎ] × [𝑦0 , 𝑦0 + ℎ]. Firstly, we consider the x-component (see
V ∈ (𝐻01 (Ω))2 , such that Figure 4). The analysis for y-component is similar.
For a function 𝑈1 ∈ 𝑉ℎ1 , on an element 𝑒, it is uniquely
󵄩 󵄩
∇ ⋅ V = 𝑞ℎ , ‖V‖ 1,Ω ≤ 𝐶6 󵄩󵄩󵄩𝑞ℎ 󵄩󵄩󵄩 , (28) given by its node values 𝑢𝑖 , 𝑖 = 1, . . . , 6. As 𝑈1 is a continuous
bilinear function on each of the two parts as shown in
where 𝐶6 is a constant independent of 𝑞ℎ ; then we obtain Figure 4. Then, from (32), we know that Π󸀠 𝑢1 = 𝑎 + 𝑏𝑥 is
𝑏 (V , 𝑞 ) 𝑏 (ΠV, 𝑞ℎ ) given by
sup 󵄩󵄩 󵄩󵄩 ℎ ℎ ≥
ℎ
󵄩Vℎ 󵄩󵄩 𝐻(div ,Ω) ‖ΠV‖ 𝐻(div ,Ω)
Vℎ ∈𝑉ℎ 󵄩 ∫ (𝑎 + 𝑏𝑥) 𝑑𝑠 = (𝑎 + 𝑏𝑥0 ) ∗ ℎ = (𝑢1 + 2𝑢3 + 𝑢5 ) ∗
𝑙1 4
𝑏 (V, 𝑞ℎ )
= (29) ℎ
‖ΠV‖ 𝐻(div ,Ω) ∫ (𝑎 + 𝑏𝑥) 𝑑𝑠 = (𝑎 + 𝑏 (𝑥0 + ℎ)) ∗ ℎ = (𝑢2 + 2𝑢4 + 𝑢6 ) ∗ .
𝑙2 4
󵄩󵄩 󵄩󵄩 2 (35)
󵄩󵄩𝑞ℎ 󵄩󵄩 0
= .
‖ΠV‖ 𝐻(div ,Ω) We deduce that
𝑢1 + 2𝑢3 + 𝑢5 − 4𝑏𝑥0
Using Lemma 4, we have that 𝑎= ,
4
󵄩 󵄩2 (36)
𝑏 (V , 𝑞 ) 1 󵄩󵄩󵄩𝑞ℎ 󵄩󵄩󵄩 0 1 󵄩󵄩 󵄩󵄩 1
sup 󵄩󵄩 󵄩󵄩 ℎ ℎ ≥ ≥ 󵄩󵄩𝑞ℎ 󵄩󵄩 . (30) 𝑏 = ((𝑢2 − 𝑢1 ) + 2 (𝑢4 − 𝑢3 ) + (𝑢6 − 𝑢5 )) ∗ .
Vℎ ∈𝑉ℎ 󵄩 V 󵄩
󵄩 ℎ 󵄩 𝐻(div ,Ω) 𝐶5 ‖V‖ 1,Ω 𝐶5 𝐶6 4ℎ
It is clear that we just need to verify (34) for both V = 1 and
Taking 𝛽 = 1/𝐶5 𝐶6 , we complete the proof of (27). V = 𝑥.
With the analysis technique presented by Arbogast and We first consider V = 1. Denote by 𝜑𝑖 the node basis
Wheeler [11], we consider the numerical analysis of the mixed function at the point 𝑖, which implies that 𝜑𝑖 (𝑥𝑗 ) = 𝛿𝑖,𝑗 , which
finite element presented in this paper. Recall the 𝑅𝑇0 mixed has the value 1 if and only if 𝑖 = 𝑗; otherwise, it is zero. By
element spaces 𝑉ℎ󸀠 × 𝑆ℎ󸀠 [3, 5, 6] based on the partition 𝜏ℎ direct calculation, we can get the basis, for example,
1 2 2 4 4
𝑉ℎ󸀠 = 𝑄1,0 (𝜏ℎ ) × 𝑄0,1 (𝜏ℎ ) , 𝑆ℎ󸀠 = 𝑆ℎ . (31) 𝜑1 = (2 − 𝑥 + 𝑥0 ) (2 − 𝑦 + 𝑦0 ) , (37)
4 ℎ ℎ ℎ ℎ
Define the interpolation operator Π󸀠 : 𝑉 → 𝑉ℎ󸀠 by the so
following equations: ∫ 𝑈1 𝑑𝑥 𝑑𝑦
𝑒
󸀠
∫ Π 𝑢1 𝑑𝑠 = ∫ 𝑢1 𝑑𝑠 ∀𝑙 ∈ 𝐿 𝑉, ℎ
𝑙 𝑙 𝑥0 +ℎ 𝑦0 +
(32)
=∫ ∫ 2 (𝑢 𝜑 + 𝑢 𝜑 + 𝑢 𝜑 + 𝑢 𝜑 ) 𝑑𝑥 𝑑𝑦
1 1 2 2 3 3 4 4
∫ Π󸀠 𝑢2 𝑑𝑠 = ∫ 𝑢2 𝑑𝑠 ∀𝑙 ∈ 𝐿 𝐻. 𝑥0 𝑦0
𝑙 𝑙
𝑥0 +ℎ 𝑦0 +ℎ
Denote by 𝑃𝑆 : 𝑆 → 𝑆ℎ the 𝐿2 projection operator and by +∫ ∫ (𝑢3 𝜑3 + 𝑢4 𝜑4 + 𝑢5 𝜑5 + 𝑢6 𝜑6 ) 𝑑𝑥 𝑑𝑦
𝑥0 𝑦0 +ℎ/2
𝑃𝑉󸀠 : 𝑉 → 𝑉ℎ󸀠 the (𝐿2 (Ω))2 vector projection operator. The
following properties of the projections hold: ℎ2 ℎ2
󵄩󵄩 󵄩 = (𝑢1 + 𝑢2 + 𝑢3 + 𝑢4 ) + (𝑢 + 𝑢4 + 𝑢5 + 𝑢6 )
󵄩󵄩𝑝 − 𝑃𝑆 𝑝󵄩󵄩󵄩 0 ≤ 𝐶ℎ|𝑝|1 8 8 3
󵄩󵄩 󵄩󵄩 (33) ℎ2
󵄩󵄩𝑢 − 𝑃𝑉󸀠 𝑢󵄩󵄩 0 ≤ 𝐶ℎ‖𝑢‖ 1 . = (𝑢 + 𝑢2 + 2𝑢3 + 2𝑢4 + 𝑢5 + 𝑢6 ) .
8 1
Then, we have an important property about the operator Π󸀠 . (38)
u5 u6
Q1,1
l1 l2
Q1,0 u3 u4
Q1,1
u1 u2
One element on 𝜏h Its corresponding portion on 𝜏h1
(a) (b)
Figure 4: An element on 𝜏ℎ and its corresponding portion on 𝜏1ℎ .
By direct computation, we can easily see that Theorem 7. If (𝑢, 𝑝) satisfy (6) and (𝑢ℎ , 𝑝ℎ ) satisfy (17), then
∫ 𝑒 Π󸀠 𝑈1 𝑑𝑥 𝑑𝑦 has the same value, so there exists a positive constant 𝐶 independent of ℎ such that the
following error estimates hold:
∫ Π󸀠 𝑈1 𝑑𝑥 𝑑𝑦 = ∫ 𝑈1 𝑑𝑥 𝑑𝑦. (39) 󵄩󵄩 󵄩
𝑒 𝑒 󵄩󵄩𝑢 − 𝑢ℎ 󵄩󵄩󵄩 0 ≤ 𝐶ℎ‖𝑢‖ 1 ,
(43)
When V = 𝑥, we have that 󵄩󵄩 󵄩 󵄩 󵄩
󵄩󵄩𝑝 − 𝑝ℎ 󵄩󵄩󵄩 0 ≤ 𝐶ℎ (‖𝑢‖ 1 + 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 1 ) .
𝑥0 +ℎ 𝑦0 +ℎ
∫ Π󸀠 𝑈1 ∗ 𝑥𝑑𝑥 𝑑𝑦 = ∫ ∫ 𝑎𝑥 + 𝑏𝑥2 𝑑𝑥 𝑑𝑦 Proof. First, we focus on the error 𝑢 − 𝑢ℎ . From (6), (17), (18),
𝑒 𝑥0 𝑦0 and (32), we derive that
ℎ3 𝑏 (𝑢, 𝑞ℎ ) = 𝑏 (𝑢ℎ , 𝑞ℎ ) = 𝑏 (Π𝑢, 𝑞ℎ ) = 𝑏 (Π󸀠 𝑢, 𝑞ℎ ) , ∀𝑞ℎ ∈ 𝑆ℎ .
= 𝑎 (𝑥0 ℎ2 + ) (40)
2 (44)
1
+ 𝑏 (𝑥02 ℎ2 + 𝑥0 ℎ3 + ℎ4 ) , Let V = Π󸀠 Vℎ in (6); then
3
where 𝑎, 𝑏 are defined in (36). Next, we compare the coeffi- 𝑎 (𝑢, Π󸀠 Vℎ ) − 𝑏 (Π󸀠 Vℎ , 𝑝) = 0. (45)
cients of 𝑢𝑖 in (40) with the coefficients in ∫ 𝑒 𝑈1 ∗ 𝑥𝑑𝑥 𝑑𝑦,
Namely,
𝑦0 +ℎ/2 𝑥0 +ℎ
∫ 𝜑1 ∗ 𝑥𝑑𝑥 𝑑𝑦 = ∫ ∫ 𝜑 ∗ 𝑥𝑑𝑥 𝑑𝑦 𝑎 (𝑃V󸀠 𝑢, Vℎ ) − (𝑃𝑠 ∇ ⋅ Vℎ , 𝑝) = 𝑎 (𝑃V󸀠 𝑢, Vℎ ) − 𝑏 (Vℎ , 𝑃𝑠 𝑝) = 0.
𝑒 𝑦0 𝑥0
(46)
ℎ 𝑥0 +ℎ 1 2 2
= ∫ (2𝑥 − 𝑥2 + 𝑥0 𝑥) (41) Here, we used the property ∇ ⋅ Π󸀠 V = 𝑃𝑠 ∇ ⋅ V. Subtracting from
2 𝑥0 4 ℎ ℎ
(17), we get that
1 3 1
= ℎ + 𝑥0 ℎ2 = 𝑘1 , 𝑎 (𝑃V󸀠 𝑢 − 𝑢ℎ , Vℎ ) − 𝑏 (Vℎ , 𝑃𝑠 𝑝 − 𝑝ℎ ) = 0. (47)
24 8
which determine 𝑘1 as the coefficient of 𝑢1 . With similar com- Take
putation, we obtain that
Vℎ = Π𝑢 − 𝑢ℎ , 𝑞ℎ = 𝑃𝑠 𝑝 − 𝑝ℎ . (48)
2 3
𝑥ℎ ℎ
𝑘5 = 𝑘1 , 𝑘2 = 𝑘6 = 0 + , Then
8 12
(42)
𝑥0 ℎ2 ℎ3 𝑥0 ℎ2 ℎ3 𝑎 (𝑃V󸀠 𝑢 − 𝑢ℎ , Π𝑢 − 𝑢ℎ ) − 𝑏 (Π𝑢 − 𝑢ℎ , 𝑃𝑠 𝑝 − 𝑝ℎ ) = 0. (49)
𝑘3 = + , 𝑘4 = + .
4 12 4 6 Due to (44), we find that
Comparing with (40), we can find that (34) is true with V = 𝑥.
So, we certify the lemma. 𝑏 (Π𝑢 − 𝑢ℎ , 𝑃𝑠 𝑝 − 𝑝ℎ ) = 0. (50)
Now, we analyze the error 𝑢 − 𝑢ℎ based on the equations Now, we estimate the right hand terms of the above ine-
above quality. From (33), (22), and (52), we have
𝑎 (𝑃V󸀠 𝑢 − 𝑢ℎ , Π∇𝜙 − 𝑃V󸀠 ∇𝜙) = 𝑎 (𝑃V󸀠 𝑢 − 𝑢, Π∇𝜙 − ∇𝜙)

𝑎 (𝑢 − 𝑢ℎ , 𝑢 − 𝑢ℎ )
+ 𝑎 (𝑢 − 𝑢ℎ , Π∇𝜙 − ∇𝜙)
= 𝑎 (𝑢 − 𝑢ℎ , 𝑢 − Π𝑢) + 𝑎 (𝑢 − 𝑢ℎ , Π𝑢 − 𝑢ℎ )
+ 𝑎 (𝑃V󸀠 𝑢 − 𝑢, ∇𝜙 − 𝑃V󸀠 ∇𝜙)
= 𝑎 (𝑢 − 𝑢ℎ , 𝑢 − Π𝑢) + 𝑎 (𝑢 − 𝑃V󸀠 𝑢, Π𝑢 − 𝑢ℎ )
+ 𝑎 (𝑢 − 𝑢ℎ , ∇𝜙 − 𝑃V󸀠 ∇𝜙)
+ 𝑎 (𝑃V󸀠 𝑢 − 𝑢ℎ , Π𝑢 − 𝑢ℎ )
≤ 𝐶ℎ2 ‖𝑢‖ 1 | 𝜙|𝐻2
󵄩 󵄩2 1 (51)
≤ 𝜖1 󵄩󵄩󵄩𝑢 − 𝑢ℎ 󵄩󵄩󵄩 0 + ‖𝑢 − Π𝑢‖ 20 󵄩 󵄩
≤ 𝐶ℎ2 ‖𝑢‖ 1 | 󵄩󵄩󵄩𝑃𝑠 𝑝 − 𝑝ℎ 󵄩󵄩󵄩 0 .
𝜖1
(55)
1󵄩 󵄩2
+ 𝜖2 ‖Π𝑢 − 𝑢‖ 20 + 󵄩󵄩󵄩𝑢 − 𝑃V󸀠 𝑢󵄩󵄩󵄩 0 It is easy to see that
𝜖2
󵄩 󵄩2 1 󵄩 󵄩2 𝑎 (𝑢 − 𝑢ℎ , 𝑃V󸀠 ∇𝜙 − ∇𝜙) ≤ 𝐶ℎ2 ‖𝑢‖ 1 |𝜙|𝐻2

+ 𝜖3 󵄩󵄩󵄩𝑢 − 𝑢ℎ 󵄩󵄩󵄩 0 + 󵄩󵄩󵄩𝑢 − 𝑃V󸀠 𝑢󵄩󵄩󵄩 0 ,
𝜖3 󵄩 󵄩
≤ 𝐶ℎ2 ‖𝑢‖ 1 󵄩󵄩󵄩𝑃𝑠 𝑝 − 𝑝ℎ 󵄩󵄩󵄩 0 ,
where 𝜖𝑖 > 0, 𝑖 = 1, 2, 3 are positive constants. Take the value 𝑎 (𝑢 − 𝑢ℎ , ∇𝜙)

of 𝜖1 = 𝜖3 = 𝜇/4𝜅, 𝜖2 = 1, and combining with (22) and (33),
= 𝑎 (𝑢 − Π𝑢, ∇𝜙) + 𝑎 (Π𝑢 − 𝑢ℎ , ∇𝜙 − 𝑃V󸀠 ∇𝜙) (56)
we conclude that
+ 𝑎 (Π𝑢 − 𝑢ℎ , 𝑃V󸀠 ∇𝜙)
󵄩󵄩 󵄩
󵄩󵄩𝑢 − 𝑢ℎ 󵄩󵄩󵄩 0 ≤ 𝐶ℎ‖𝑢‖ 1 . (52) 󵄩 󵄩 󵄩 󵄩
≤ 𝐶 (ℎ2 |𝑢| 2 󵄩󵄩󵄩𝜙󵄩󵄩󵄩 2 + ℎ2 |𝑢| 1 󵄩󵄩󵄩𝜙󵄩󵄩󵄩 2 )
󵄩 󵄩
≤ 𝐶ℎ2 ‖𝑢‖ 2 󵄩󵄩󵄩𝑃𝑠 𝑝 − 𝑝ℎ 󵄩󵄩󵄩 0 .
We also can obtain a higher order error estimate for
‖𝑃𝑠 𝑝 − 𝑝ℎ ‖. Consider the classical duality argument. Let 𝜙 be Here, we used the fact that 𝑎(Π𝑢 − 𝑢ℎ , 𝑃V󸀠 ∇𝜙) = 0 which is got
the solution of the following elliptical problem: from the Green formulation and (44).
Combining the above inequalities, we conclude that
󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩
𝜕𝜙 󵄩󵄩𝑝 − 𝑝ℎ 󵄩󵄩󵄩 0 ≤ 󵄩󵄩󵄩𝑝 − 𝑃𝑠 𝑝󵄩󵄩󵄩 0 + 󵄩󵄩󵄩𝑃𝑠 𝑝 − 𝑝ℎ 󵄩󵄩󵄩 0
Δ𝜙 = 𝑃𝑠 𝑝 − 𝑝ℎ , = 0. (53)
(57)
𝜕𝑛 󵄩 󵄩
≤ 𝐶ℎ (‖𝑢‖ 1 + 󵄩󵄩󵄩𝑝󵄩󵄩󵄩 0 ) .
By the elliptic regularity, the estimate holds: |𝜙|𝐻2 ≤ We complete the proof.
𝐶‖𝑃𝑠 𝑝 − 𝑝ℎ ‖0 . So
It is worth mentioning that we analyze this mixed finite
element method in a direct way as it is not straightforward to
󵄩󵄩 󵄩2 apply the classical inf-sup theory. We just have the coercivity
󵄩󵄩𝑃𝑠 𝑝 − 𝑝ℎ 󵄩󵄩󵄩 0 property for 𝑎(𝑢ℎ , Vℎ ) on the normal 𝐿 2 space, not in the
subspace of V0ℎ = {Vℎ ∈ 𝑉ℎ : 𝑏(Vℎ , 𝑞ℎ ) = 0, ∀𝑞ℎ ∈ 𝑆ℎ }, and
= (𝑃𝑠 𝑝 − 𝑝ℎ , ∇ ⋅ ∇𝜙) the same issue also occurs in [11]. The problem is that testing
(∇ ⋅ V, 𝑤) by 𝑤 ∈ 𝑊ℎ does not control the full divergence
= (𝑃𝑠 𝑝 − 𝑝ℎ , ∇ ⋅ Π∇𝜙)
of 𝑉, and it does not occur when this method is applied to
= 𝑎 (𝑃V󸀠 𝑢 − 𝑢ℎ , Π∇𝜙) Stokes or Navier-Stokes equations (as in [13, 14]). As a result,
we just obtain a convergence rate of ‖𝑢 − 𝑢ℎ ‖0 . Failing to
= 𝑎 (𝑃V󸀠 𝑢 − 𝑢ℎ , Π∇𝜙 − 𝑃V󸀠 ∇𝜙) + 𝑎 (𝑃V󸀠 𝑢 − 𝑢ℎ , 𝑃V󸀠 ∇𝜙) obtain convergence rate of ‖𝑢 − 𝑢𝑢 ‖𝐻(div ,Ω) is a weak point
of this proposed mixed formulation compared to the clas-
= 𝑎 (𝑃V󸀠 𝑢 − 𝑢ℎ , Π∇𝜙 − 𝑃V󸀠 ∇𝜙) + 𝑎 (𝑃V󸀠 𝑢 − 𝑢, 𝑃V󸀠 ∇𝜙) sical Raviart-Thomas mixed method. But the significance of
continuous flux applied to mass conservation can be found in
+ 𝑎 (𝑢 − 𝑢ℎ , 𝑃V󸀠 ∇𝜙) Section 6.
= 𝑎 (𝑃V󸀠 𝑢 − 𝑢ℎ , Π∇𝜙 − 𝑃V󸀠 ∇𝜙) + 𝑎 (𝑢 − 𝑢ℎ , 𝑃V󸀠 ∇𝜙 − ∇𝜙)

5. Numerical Examples
+ 𝑎 (𝑢 − 𝑢ℎ , ∇𝜙) . In this section, we present some numerical results for the
(54) model Problem (1). For simplicity, we assume that the domain
Table 1: Three numerical test cases. where 𝛾(𝐶) and 𝑑 are the gravity coefficient and vertical
coordinate, 𝜙(𝑥) is the porosity of the rock, and 𝐶𝑞 ̃ rep-
Case Coefficient 𝜇/𝜅 True solution 𝑢 True solution 𝑝
resents a known source. 𝐷(𝑥, 𝑢) is the molecular diffusion
𝑥2 𝑦 − 𝑥4 𝑦
1 1 ( 4 ) (𝑥 − 1/2)(𝑦 − 1/2) and mechanical dispersion coefficient. For convenience, we
𝑥𝑦 − 𝑥𝑦2 ̃ and 𝑎(𝐶) = 𝜇(𝐶)𝐾−1 . Let 𝜒 : (0, 𝑇] → 𝑅2
𝑒2𝑥𝑦
2
0 denote that 𝑓 = 𝐶𝑞
𝑥2 𝑦 − 𝑥4 𝑦
2 ( 1 ) ( 4 ) (𝑥 − 1/2)(𝑦 − 1/2) be the solution of the ordinary differential equation
0 𝑥𝑦 − 𝑥𝑦2
𝑥+𝑦 𝑑𝜒 u (𝜒 (𝑥, 𝑡; 𝜏) , 𝜏)
𝑒2𝑥𝑦
2
0 = ,
𝑒−𝑥𝑦 𝑥 𝑑𝜏 𝜙 (𝑥)
3 ( 1 ) ( 2 ) 𝑦𝑒 (60)
0 𝑥 cos 𝑦
𝑥+𝑦 𝜒 (𝑥, 𝑡; 𝑡) = 𝑥.
Let 𝑉 = 𝐻(div , Ω), 𝑆 = 𝐿20 (Ω), 𝑀 = 𝐻1 (Ω); then, we derive
is a unit square Ω = [0, 1] × [0, 1] and the test cases are sum- the entire weak formulation for the model: find (u, 𝑝, 𝐶) ∈
marized in Table 1. We can choose the boundary conditions 𝑉 × 𝑆 × 𝑀, such that
and the right hand terms according to the analytical solutions.
We compare our method to the formulation constructed (𝑎 (𝐶) 𝑢, V) − (𝑝, ∇ ⋅ V) = (𝛾 (𝐶) ∇𝑑, k) , ∀k ∈ 𝑉,
by Arbogast and Wheeler [11]. Its corresponding discrete 𝑑𝐶 (𝜒, 𝜏)
finite element spaces are (𝜙 (𝑥) + 𝑔𝐶, 𝑤) + (𝐷∇𝐶, ∇𝑤) = (𝑓, 𝑤) ,
𝑑𝜏 (61)
2
𝑉ℎ = {Vℎ ∈ (𝐶(0) (Ω)) : Vℎ | 𝑇 ∈ 𝑄1,2 (𝑇)
∀𝑤 ∈ 𝑀,
× 𝑄2,1 (𝑇) , ∀ 𝑇 ∈ 𝜏ℎ } , (∇ ⋅ 𝑢, 𝜑) = (𝑔, 𝜑) , ∀𝜑 ∈ 𝑆.
(58)
𝑆ℎ = {𝑞ℎ : 𝑞ℎ | 𝑇 = constant, Let Δ𝑡 be the time step for both concentration and pressure;
define
∀𝑇 ∈ 𝜏ℎ ; ∫ 𝑞ℎ 𝑑𝑥 = 0} . 𝑀ℎ = {Vℎ ∈ 𝐶(0) (Ω) : Vℎ | 𝑇 ∈ 𝑄1,1 (𝑇) , ∀𝑇 ∈ 𝜏ℎ } . (62)
Ω
The results of the error estimate with various norms are listed Combing with the new characteristic finite element method
in Table 2, while the corresponding convergence rates of the which preserves the mass balance proposed by Rui and Tabata
presented method are shown in Table 3. [10], the approximate characteristic line of 𝜒 is defined as
Close results of numerical errors for both formulations 𝑢ℎ𝑛
are shown in Table 2. From Table 3, we can see that 𝑝 con- 𝜒𝑛 (𝑥) = 𝑥 − Δ𝑡. (63)
𝜙 (𝑥)
verges to 𝑝ℎ as 𝑂(ℎ) and 𝑃𝑠 𝑝−𝑝ℎ as 𝑂(ℎ2 ) for our formulation,
which both agree with the theorem. From the examples, We obtain the corresponding full-discrete mass-conservative
we can observe that 𝑢ℎ converges to 𝑢 somewhat better characteristic (MCC) scheme: find (𝑢ℎ , 𝑝ℎ , 𝐶ℎ ) ∈ 𝑉ℎ ×𝑆ℎ ×𝑀ℎ ,
than expected, and it appears that on the uniform grid we such that
attain 𝑂(ℎ3/2 ) superconvergence in the 𝐿2 norm which is (𝑎 (𝐶ℎ𝑛 ) 𝑢ℎ𝑛 , Vℎ ) − (𝑝ℎ , ∇ ⋅ Vℎ )
similar to the tests of Arbogast’s formulation [11]. Yet, the
degrees of freedom of our method are less than Arbogast’s = (𝛾 (𝐶ℎ𝑛 ) ∇𝑑, Vℎ ) , ∀Vℎ ∈ 𝑉ℎ
scheme. As in the case of 64 ∗ 64, the degrees of freedom of
Arbogast’s scheme are 20866 and 12676 for our formulation. 𝜙𝐶ℎ𝑛 − (𝜙𝐶ℎ𝑛−1 ) ∘ 𝜒𝑛 𝛾𝑛
( , 𝜑ℎ ) + (𝐷 (𝑢ℎ𝑛 ) ∇𝐶ℎ𝑛 , ∇𝜑ℎ )
The convergence rate of ‖𝑢 − 𝑢ℎ ‖𝐻(div ,Ω) is first order, but here Δ𝑡
we cannot give the corresponding analysis.
= (𝑓, 𝜑ℎ ) , ∀𝜑ℎ ∈ 𝑀ℎ
6. A Valuable Application (∇ ⋅ 𝑢ℎ𝑛 , 𝑞ℎ ) = (𝑔, 𝑞ℎ ) , ∀𝑞ℎ ∈ 𝑆ℎ
In this section, we briefly show an application of the proposed
̃0 ,
𝐶ℎ0 = 𝐶
mixed finite element method to the miscible displacement
of one incompressible fluid by another in porous media. The (64)
model is as follows: where
𝜇 (𝐶) 𝐾−1 𝑢 + ∇𝑝 = 𝛾 (𝐶) ∇𝑑, (𝑥, 𝑡) ∈ Ω × 𝐽, 𝜕𝜒𝑛
𝛾𝑛 = det ( )
𝜕𝐶 𝜕𝑥
𝜙 ̃
+ ∇ ⋅ (𝑢𝐶) − ∇ ⋅ (𝐷 (𝑢) ∇𝐶) = 𝐶𝑞, (𝑥, 𝑡) ∈ Ω × 𝐽,
𝜕𝑡 ∇ ⋅ 𝑢ℎ𝑛 ∇𝜙
∇ ⋅ 𝑢 = 𝑔, (𝑥, 𝑡) ∈ Ω × 𝐽, =1− Δ𝑡 + 𝑢ℎ𝑛 2 Δ𝑡 (65)
𝜙 𝜙
𝑢 ⋅ 𝑛 = 𝑔1 , (𝑥, 𝑡) ∈ 𝜕Ω × 𝐽,
𝑛 𝑛
𝐶 (𝑥, 0) = 𝐶0 (𝑥) , 𝑥 ∈ Ω, 𝑢ℎ,1 𝑢ℎ,2
+ ∇( ) ⋅ curl ( ) Δ𝑡2 .
(59) 𝜙 𝜙
Table 2: The numerical error for fm. 1 (our formulation) and fm. 2 (Arbogast’s formulation).
‖𝑢 − 𝑢ℎ ‖ ‖∇ ⋅ (𝑢 − 𝑢ℎ )‖ ‖𝑝 − 𝑝ℎ ‖ ‖𝑃𝑠 𝑝 − 𝑝ℎ ‖
Case Mesh
fm. 1 fm. 2 fm. 1 fm. 2 fm. 1 fm. 2 fm. 1 fm. 2
4 4.90𝑒 − 2 5.67𝑒 − 2 3.06𝑒 − 1 3.24𝑒 − 1 2.93𝑒 − 2 2.93𝑒 − 2 4.53𝑒 − 3 4.17𝑒 − 3
8 1.78𝑒 − 2 2.05𝑒 − 2 1.53𝑒 − 1 1.62𝑒 − 1 1.47𝑒 − 2 1.47𝑒 − 2 1.24𝑒 − 3 1.20𝑒 − 3
1 16 6.45𝑒 − 3 7.37𝑒 − 3 7.67𝑒 − 2 8.13𝑒 − 2 7.37𝑒 − 3 7.37𝑒 − 3 3.18𝑒 − 4 3.15𝑒 − 4
32 2.31𝑒 − 3 2.64𝑒 − 3 3.84𝑒 − 2 4.07𝑒 − 2 3.68𝑒 − 3 3.68𝑒 − 3 8.01𝑒 − 5 7.98𝑒 − 5
64 8.25𝑒 − 4 9.38𝑒 − 4 1.92𝑒 − 2 2.03𝑒 − 2 1.84𝑒 − 3 1.84𝑒 − 3 2.01𝑒 − 5 2.01𝑒 − 5
4 4.70𝑒 − 2 5.47𝑒 − 2 2.99𝑒 − 1 3.22𝑒 − 1 2.95𝑒 − 2 2.94𝑒 − 2 5.42𝑒 − 3 4.97𝑒 − 3
8 1.72𝑒 − 2 1.99𝑒 − 2 1.53𝑒 − 1 1.63𝑒 − 1 1.47𝑒 − 2 1.47𝑒 − 2 1.54𝑒 − 3 1.48𝑒 − 3
2 16 6.25𝑒 − 3 7.19𝑒 − 3 7.75𝑒 − 2 8.27𝑒 − 2 7.37𝑒 − 3 7.37𝑒 − 3 4.04𝑒 − 4 3.98𝑒 − 4
32 2.25𝑒 − 3 2.58𝑒 − 3 3.89𝑒 − 2 4.15𝑒 − 2 3.68𝑒 − 3 3.68𝑒 − 3 1.03𝑒 − 4 1.02𝑒 − 4
64 8.08𝑒 − 4 9.21𝑒 − 4 1.95𝑒 − 2 2.08𝑒 − 2 1.84𝑒 − 3 1.84𝑒 − 3 2.59𝑒 − 5 2.58𝑒 − 5
4 9.65𝑒 − 2 1.09𝑒 − 1 4.14𝑒 − 1 4.67𝑒 − 1 1.49𝑒 − 1 1.49𝑒 − 1 7.39𝑒 − 3 6.21𝑒 − 3
8 3.79𝑒 − 2 4.31𝑒 − 2 2.16𝑒 − 1 2.46𝑒 − 1 7.44𝑒 − 2 7.44𝑒 − 2 2.14𝑒 − 3 1.89𝑒 − 3
3 16 1.42𝑒 − 2 1.62𝑒 − 2 1.11𝑒 − 1 1.28𝑒 − 1 3.72𝑒 − 2 3.72𝑒 − 2 5.72𝑒 − 4 5.19𝑒 − 4
32 5.19𝑒 − 3 5.91𝑒 − 3 5.63𝑒 − 2 6.51𝑒 − 2 1.86𝑒 − 2 1.86𝑒 − 2 1.47𝑒 − 4 1.35𝑒 − 4
64 1.87𝑒 − 3 2.13𝑒 − 3 2.84𝑒 − 2 3.28𝑒 − 2 9.31𝑒 − 3 9.31𝑒 − 3 3.72𝑒 − 5 3.44𝑒 − 5
Table 3: The corresponding convergence rates of fm. 1 and fm. 2.
‖𝑢 − 𝑢ℎ ‖ ‖∇ ⋅ (𝑢 − 𝑢ℎ )‖ ‖𝑝 − 𝑝ℎ ‖ ‖𝑃𝑠 𝑝 − 𝑝ℎ ‖
Case Mesh
fm. 1 fm. 2 fm. 1 fm. 2 fm. 1 fm. 2 fm. 1 fm. 2
8 1.459 1.468 0.997 1.001 0.995 0.993 1.875 1.795
16 1.468 1.476 0.998 0.995 0.999 0.999 1.961 1.934
1 32 1.479 1.484 1.000 0.998 1.000 1.000 1.987 1.978
64 1.486 1.489 1.000 1.000 1.000 1.000 1.996 1.993
8 1.449 1.457 0.968 0.978 0.999 0.996 1.817 1.742
16 1.462 1.471 0.983 0.984 1.001 1.001 1.930 1.901
2 32 1.471 1.479 0.993 0.993 1.001 1.000 1.976 1.960
64 1.480 1.485 0.997 0.997 1.000 1.000 1.989 1.984
8 1.347 1.340 0.942 0.924 0.998 0.997 1.787 1.708
16 1.416 1.414 0.957 0.945 0.999 0.999 1.906 1.870
3 32 1.452 1.452 0.979 0.973 1.000 1.000 1.959 1.942
64 1.472 1.473 0.990 0.986 1.000 1.000 1.983 1.975
We can see that the continuous flux is indispensable for 𝛾𝑛 . Now, we select 𝜇(𝐶) = 𝐶, and the following analytical solu-
Let 𝜑ℎ = 1 in (64), and summing it up from 𝑛 = 1 to 𝑁, we tion of the problem is
get the mass balance
𝑁 𝑢 (𝑥, 𝑦, 𝑡) = (𝑒𝑥 + 𝑡, 𝑒𝑦 + 𝑡) ,
∫ 𝜙𝐶ℎ𝑁𝑑𝑥 =∫ 𝜙𝐶ℎ0 𝑑𝑥 + Δ𝑡 ∑ ∫ 𝑓 𝑑𝑥. 𝑛
(66)
Ω Ω 𝑛=1 Ω 𝑝 (𝑥, 𝑦, 𝑡) = 𝑒−𝑡 (𝑥2 + 𝑦2 ) ,
(68)
Here, we just give numerical example to show the feasibility −𝑡 1 2 1 2
𝐶 (𝑥, 𝑦, 𝑡) = 𝑒 ((𝑥 − ) + (𝑦 − ) ) .
of this application, and the theoretical analysis of stability, 2 2
mass balance, and convergence of this discrete scheme will be
discussed in the future. Firstly, we define compute mass error The error results with different norms of this numerical simu-
and relative mass error as follows: lation can be listed in Tables 4 and 5, and at last we give a mass
compute mass error : ∫ 𝜙𝐶ℎ𝑁𝑑𝑥 error to check the mass conservation in Table 6.
Ω As can be seen from Tables 4 and 5, we conjecture that
𝑁 almost all the convergence rates are true in general. From
− (∫ 𝜙𝐶ℎ0 𝑑𝑥 + Δ𝑡 ∑ ∫ 𝑓𝑛 𝑑𝑥) , Table 6 we find that mass balance is right as computational
Ω 𝑛=1 Ω
mass error resulting from computer is inevitable and nearly
∫Ω 𝜙𝐶ℎ𝑁𝑑𝑥 − ∫Ω 𝜙𝐶 𝑑𝑥𝑁
invariable for different meshes, while the relative mass error
relative mass error : . decreases as was expected. The corresponding theoretical
∫Ω 𝜙𝐶𝑁𝑑𝑥
analysis about this system will be considered in the future
(67) work.
Table 4: Numerical error and convergence rate (Δ𝑡 = 𝐶ℎ).
Mesh 5×5 10 × 10 20 × 20 40 × 40
Norm type Error Rate Error Rate Error Rate Error Rate
‖𝑢‖ 𝑙2 (𝐿2 ) 1.83𝑒 − 4 — 7.10𝑒 − 5 1.36 3.38𝑒 − 5 1.07 1.65𝑒 − 5 1.03
‖𝑢‖ 𝑙∞ (𝐿2 ) 1.29𝑒 − 2 — 5.19𝑒 − 3 1.31 2.64𝑒 − 3 0.97 1.37𝑒 − 3 0.95
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑝󵄩󵄩 𝑙2 (𝐿2 ) 1.33𝑒 − 3 — 6.67𝑒 − 4 1.00 3.33𝑒 − 4 1.00 1.67𝑒 − 4 1.00
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑝󵄩󵄩 𝑙∞ (𝐿2 ) 9.43𝑒 − 2 — 4.71𝑒 − 2 1.00 2.35𝑒 − 2 1.00 1.18𝑒 − 2 1.00
‖𝐶‖ 𝑙2 (𝐻1 ) 2.32𝑒 − 3 — 1.16𝑒 − 3 1.01 5.78𝑒 − 4 1.00 2.88𝑒 − 4 1.00
‖𝐶‖ 𝑙∞ (𝐻1 ) 1.63𝑒 − 1 — 8.18𝑒 − 2 1.00 4.11𝑒 − 2 0.99 2.05𝑒 − 2 0.99
Table 5: Numerical error and convergence rate (Δ𝑡 = 𝐶ℎ2 ).
Mesh 5×5 10 × 10 20 × 20 40 × 40
Norm type Error Rate Error Rate Error Rate Error Rate
‖𝐶‖ 𝑙2 (𝐿2 ) 8.48𝑒 − 5 — 2.13𝑒 − 5 1.995 5.37𝑒 − 6 1.986 1.36𝑒 − 6 1.971
‖𝐶‖ 𝑙∞ (𝐿2 ) 1.34𝑒 − 2 — 3.37𝑒 − 3 1.989 8.56𝑒 − 4 1.978 2.21𝑒 − 4 1.952
Table 6: Mass error for concentration 𝐶 (Δ𝑡 = 𝐶ℎ). [9] J. Bear, Dynamics of Fluids in Porous Media, Dover, New York,
NY, USA, 1972.
Mesh 5×5 10 × 10 20 × 20 40 × 40
[10] H. Rui and M. Tabata, “A mass-conservative characteristic finite
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Relative mass error 2.068𝑒 − 2 5.427𝑒 − 3 1.487𝑒 − 3 4.371𝑒 − 4 Scientific Computing, vol. 43, no. 3, pp. 416–432, 2010.
[11] T. Arbogast and M. F. Wheeler, “A family of rectangular mixed
elements with a continuous flux for second order elliptic prob-
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1914–1931, 2005.
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no. 2, pp. 135–143, 1987.
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and the MAC method for the Stokes equations,” SIAM Journal
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the Navier-Stokes Equations, vol. 749, Springer, New York, NY, putational Mathematics, vol. 26, no. 6, pp. 816–824, 2008.
USA, 1979. [15] I. Babuška, “Error-bounds for finite element method,” Numer-
[3] F. Brezzi, J. Douglas, Jr., R. Durán, and M. Fortin, “Mixed finite ische Mathematik, vol. 16, pp. 322–333, 1971.
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Analytical and Numerical Methods For Solving Partial Differential Equations and Integral Equations Arising in Physical Models

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Abstract and Applied Analysis

Analytical and Numerical Methods

Analytical and Numerical Methods for Solving

Guest Editors: Santanu Saha Ray, Om P. Agrawal, R. K. Bera,

Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional

Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse

Decoupling the Stationary Navier-Stokes-Darcy System with the Beavers-Joseph-Saffman Interface

New Exact Solutions for a Generalized Double Sinh-Gordon Equation,

A Pressure-Stabilized Lagrange-Galerkin Method in a Parallel Domain Decomposition System,

Approximate Solution of Tuberculosis Disease Population Dynamics Model,

Correspondence should be addressed to Santanu Saha Ray; santanusaharay@yahoo.com

Received 15 December 2013; Accepted 15 December 2013; Published 9 January 2014

in the parameters space. According to the operator theory,

S. Saha Ray and P. K. Sahu

Correspondence should be addressed to S. Saha Ray; santanusaharay@yahoo.com

Received 3 September 2013; Accepted 3 October 2013

Academic Editor: Rasajit Bera

1. Introduction functions only, then several approximate methods of solution

3.1. B-Spline Wavelet Method

{ 𝐵 (2𝑗−𝑗0 𝑥) 𝑖 = −𝑚 + 1, . . . , −1, + ∑ ∑ 𝑑𝑗,𝑘 𝜓𝑗,𝑘 (𝑥) ,

where 𝑞𝑖,𝑘 = 𝑞𝑘−2𝑖 . 1

Suppose that Φ ̃ are the dual functions of Φ and Ψ,

𝑦 (𝑥) = 𝐶𝑇 Ψ (𝑥) , (25) ⋃ 𝑉𝑗 is dense in 𝐿2 (𝑅) , (35)

By applying (25)–(28) in (7) we have 𝑉𝑗1 ∩ 𝑉𝑗2 = 𝑉𝑗2 , 𝑗1 > 𝑗2 ,

⟨𝜑, 𝜑⟩ − ⟨𝐾𝜑, 𝜑⟩ We apply variation iteration method for (46). According to

where 𝐻(1) = [1], 𝐻(2) = [ 11 −1

The integration of the Haar function vector ℎ(𝑚) (𝑡) is

𝑇 (93) Block-Pulse coefficients of 𝑦𝑗 (𝑥), 𝑗 = 1, 2, . . . , 𝑛 in the interval

which is a linear system From (110)–(112), we get

If 𝑢 is a solution of (121), then it is also solution of (126). By

where 𝐽(𝑥, 𝑡) = 𝜕𝐾(𝑥, 𝑡)/𝜕𝑡.

𝑢󸀠 (𝑥𝑛 ) = 𝑓󸀠 (𝑥𝑛 ) where

By solving this system with (𝑛 + 3) nonlinear equations and

𝑦 (𝑥) ≈ 𝑌𝑇 Ψ (𝑥) , For solving (148) by Homotopy perturbation method (HPM)

1 ∞ In the view of ADM, the nonlinear term 𝑁𝑢 can be repre-

[16] S. Abbasbandy, “Numerical solutions of the integral equations:

Correspondence should be addressed to Hasan Bulut; hbulut@firat.edu.tr

Received 24 May 2013; Revised 1 August 2013; Accepted 18 August 2013

Academic Editor: Santanu Saha Ray

1. Introduction of the KdV equation, where, in particular, the case 𝑙 = 𝑚 =

nonlinearity. In Section 3, as applications, we procure some

(𝑞𝑙 )𝑡 + 𝑎𝑞𝑚 𝑞𝑥 + 𝑏(𝑞𝑛 )𝑥𝑥𝑥 = 0, (1) 𝑁 (𝑢, 𝑢󸀠 , 𝑢󸀠󸀠 , . . .) = 0. (4)

where Let 𝑙 = 𝑛, applying balance and using the following transfor-

Φ󸀠 (Γ) Ψ (Γ) − Φ (Γ) Ψ󸀠 (Γ) 𝛿 − 𝑐 (𝑚 + 1) (𝑚 + 1 − 𝑛)2 V2 + 𝑎(𝑚 + 1 − 𝑛)2 V3

According to the balance principle, we can find a relation of

where 𝑢 (𝑥, 𝑡) = [𝜏0 + 𝜏1 𝛼1 + 𝜏1 (𝛼2 − 𝛼1 )

Also 𝛼1 , 𝛼2 , and 𝛼3 are the roots of the polynomial equation √𝛼2 − 𝛼1

1-soliton wave solution where 𝐴 1 = √−6𝑏𝑛𝜉0 (1+𝑚)(1+𝑚+𝑛)/𝑎𝜉1 𝜏1 (1+𝑚−𝑛)2. Inte-

Remark 1. If we choose the corresponding values for some 𝛼1 > 𝛼2 > 𝛼3 ,

Substituting solutions (31) into (5) and (13), we have

2𝑎𝑛𝜉0 𝜏1 = [𝜏0 + 𝜏1 𝛼1 + ((𝛼1 − 𝛼2 ) 𝜏1 )

= [𝜏0 + 𝜏1 𝛼2 + ((𝛼2 − 𝛼1 ) 𝜏1 ) [ √(𝛼1 − 𝛼2 ) (𝛼1 − 𝛼3 )

2𝑎𝑛𝜉0 𝜏1 (𝛼2 − 𝛼3 ) (𝛼1 − 𝛼4 )

+ 𝜏2 (𝛼2 + ((𝛼1 − 𝛼2 ) (𝛼4 − 𝛼2 ) 𝜏1 ) 2

× ((𝛼1 − 𝛼4 ) 𝑠𝑛2 (𝜑, 𝑙)

Substituting these results into (6) and (10), we get ± (𝜂 − 𝜂0 )

= 3𝐴 2 ∫ ( (𝑑Γ) 𝛼1 > 𝛼2 > 𝛼3 > 𝛼4 ,

2𝑎(1 + 𝑚 − 𝑛)2 𝜁1 𝜏1 𝜁0 + 𝜁1 Γ (46)

+𝜁0 𝜏1 (𝑀 + 2𝑎(1 + 𝑚 − 𝑛)2 𝜁0 𝜏1 )) +𝑖√𝛼1 −𝛼2 (𝐸 (𝜑, 𝑙)−𝐹 (𝜑, 𝑙)) ) ,