Fractal and Fractional

Research

16 pages, 1504 KiB

Open AccessArticle

A Novel and Effective Scheme for Solving the Fractional Telegraph Problem via the Spectral Element Method

by Tao Liu, Runqi Xue, Bolin Ding, Davron A. Juraev, Behzad Nemati Saray and Fazlollah Soleymani

Fractal Fract. 2024, 8(12), 711; https://doi.org/10.3390/fractalfract8120711 - 29 Nov 2024

Viewed by 582

Abstract

The combination of fractional derivatives (due to their global behavior) and the challenges related to hyperbolic PDEs pose formidable obstacles in solving fractional hyperbolic equations. Due to the importance and applications of the fractional telegraph equation, solving it and presenting accurate solutions via [...] Read more.

The combination of fractional derivatives (due to their global behavior) and the challenges related to hyperbolic PDEs pose formidable obstacles in solving fractional hyperbolic equations. Due to the importance and applications of the fractional telegraph equation, solving it and presenting accurate solutions via a novel and effective method can be useful. This work introduces and implements a method based on the spectral element method (SEM) that relies on interpolating scaling functions (ISFs). Through the use of an orthonormal projection, the method maps the equation to scaling spaces raised from multi-resolution analysis (MRA). To achieve this, the Caputo fractional derivative (CFD) is represented by ISFs as a square matrix. Remarkable efficiency, ease of implementation, and precision are the distinguishing features of the presented method. An analysis is provided to demonstrate the convergence of the scheme, and illustrative examples validate our method. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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16 pages, 301 KiB

Open AccessArticle

On Higher-Order Nonlinear Fractional Elastic Equations with Dependence on Lower Order Derivatives in Nonlinearity

by Yujun Cui, Chunyu Liang and Yumei Zou

Fractal Fract. 2024, 8(7), 398; https://doi.org/10.3390/fractalfract8070398 - 2 Jul 2024

Viewed by 573

Abstract

The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given [...] Read more.

The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given to illustrate the key results. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

21 pages, 369 KiB

Open AccessArticle

Study on a Nonlocal Fractional Coupled System Involving (k,ψ)-Hilfer Derivatives and (k,ψ)-Riemann–Liouville Integral Operators

by Ayub Samadi, Sotiris K. Ntouyas and Jessada Tariboon

Fractal Fract. 2024, 8(4), 211; https://doi.org/10.3390/fractalfract8040211 - 4 Apr 2024

Viewed by 1010

Abstract

This paper deals with a nonlocal fractional coupled system of

(k, ψ)

-Hilfer fractional differential equations, which involve, in boundary conditions,

(k, ψ)

-Hilfer fractional derivatives and

(k, ψ)

-Riemann–Liouville fractional integrals. The existence [...] Read more.

This paper deals with a nonlocal fractional coupled system of

(k, ψ)

-Hilfer fractional differential equations, which involve, in boundary conditions,

(k, ψ)

-Hilfer fractional derivatives and

(k, ψ)

-Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel’skiĭ’s fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

30 pages, 402 KiB

Open AccessArticle

Existence of Solutions for the Initial Value Problem with Hadamard Fractional Derivatives in Locally Convex Spaces

by Weiwei Liu and Lishan Liu

Fractal Fract. 2024, 8(4), 191; https://doi.org/10.3390/fractalfract8040191 - 27 Mar 2024

Cited by 1 | Viewed by 1091

Abstract

In this paper, we investigate an initial value problem for a nonlinear fractional differential equation on an infinite interval. The differential operator is taken in the Hadamard sense and the nonlinear term involves two lower-order fractional derivatives of the unknown function. In order [...] Read more.

In this paper, we investigate an initial value problem for a nonlinear fractional differential equation on an infinite interval. The differential operator is taken in the Hadamard sense and the nonlinear term involves two lower-order fractional derivatives of the unknown function. In order to establish the global existence criteria, we first verify that there exists a unique positive solution to an integral equation based on a class of new integral inequality. Next, we construct a locally convex space, which is metrizable and complete. On this space, applying Schäuder’s fixed point theorem, we obtain the existence of at least one solution to the initial value problem. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

16 pages, 1292 KiB

Open AccessArticle

A Novel Hybrid Crossover Dynamics of Monkeypox Disease Mathematical Model with Time Delay: Numerical Treatments

by Nasser H. Sweilam, Seham M. Al-Mekhlafi, Saleh M. Hassan, Nehaya R. Alsenaideh and Abdelaziz E. Radwan

Fractal Fract. 2024, 8(4), 185; https://doi.org/10.3390/fractalfract8040185 - 24 Mar 2024

Cited by 3 | Viewed by 1213

Abstract

In this paper, we improved a mathematical model of monkeypox disease with a time delay to a crossover model by incorporating variable-order and fractional differential equations, along with stochastic fractional derivatives, in three different time intervals. The stability and positivity of the solutions [...] Read more.

In this paper, we improved a mathematical model of monkeypox disease with a time delay to a crossover model by incorporating variable-order and fractional differential equations, along with stochastic fractional derivatives, in three different time intervals. The stability and positivity of the solutions for the proposed model are discussed. Two numerical methods are constructed to study the behavior of the proposed models. These methods are the nonstandard modified Euler Maruyama technique and the nonstandard Caputo proportional constant Adams-Bashfourth fifth step method. Many numerical experiments were conducted to verify the efficiency of the methods and support the theoretical results. This study’s originality is the use of fresh data simulation techniques and different solution methodologies. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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23 pages, 437 KiB

Open AccessArticle

Monotone Positive Radial Solution of Double Index Logarithm Parabolic Equations

by Mengru Liu and Lihong Zhang

Fractal Fract. 2024, 8(3), 173; https://doi.org/10.3390/fractalfract8030173 - 16 Mar 2024

Viewed by 1195

Abstract

This article mainly studies the double index logarithmic nonlinear fractional

g -

Laplacian parabolic equations with the Marchaud fractional time derivatives

\partial_{t}^{α}

. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double [...] Read more.

This article mainly studies the double index logarithmic nonlinear fractional

g -

Laplacian parabolic equations with the Marchaud fractional time derivatives

\partial_{t}^{α}

. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double non-locality of space-time and the nonlinearity of the fractional

g -

Laplacian, we establish the unbounded narrow domain principle, which provides a starting point for the moving plane method. Meanwhile, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of a non-local operator are established; then, these averaging effects are applied twice to ensure that the plane can be continuously moved toward infinity. Based on the above, the monotonicity of a positive solution for the above fractional

g -

Laplacian parabolic equations is studied. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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19 pages, 342 KiB

Open AccessArticle

On a Faster Iterative Method for Solving Fractional Delay Differential Equations in Banach Spaces

by James Abah Ugboh, Joseph Oboyi, Mfon Okon Udo, Hossam A. Nabwey, Austine Efut Ofem and Ojen Kumar Narain

Fractal Fract. 2024, 8(3), 166; https://doi.org/10.3390/fractalfract8030166 - 14 Mar 2024

Cited by 2 | Viewed by 1221

Abstract

In this paper, we consider a faster iterative method for approximating the fixed points of generalized

α

-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our [...] Read more.

In this paper, we consider a faster iterative method for approximating the fixed points of generalized

α

-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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33 pages, 1380 KiB

Open AccessArticle

Proposal for Use of the Fractional Derivative of Radial Functions in Interpolation Problems

by Anthony Torres-Hernandez, Fernando Brambila-Paz and Rafael Ramirez-Melendez

Fractal Fract. 2024, 8(1), 16; https://doi.org/10.3390/fractalfract8010016 - 23 Dec 2023

Viewed by 5405

Abstract

This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions [...] Read more.

This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions for use in interpolation problems. Furthermore, a technique is employed to precondition the matrices generated in the presented problems through

Q R

decomposition. Similarly, a method is introduced to define two different types of abelian groups for any fractional operator defined in the interval

[0, 1)

, among which the Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative, and Caputo fractional derivative are worth mentioning. Finally, a form of radial interpolant is suggested for application in solving fractional differential equations using the asymmetric collocation method, and examples of its implementation in differential operators utilizing the aforementioned fractional operators are shown. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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24 pages, 724 KiB

Open AccessArticle

Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications

by Junjie Wang, Shoucheng Yuan and Xiao Liu

Fractal Fract. 2023, 7(12), 868; https://doi.org/10.3390/fractalfract7120868 - 6 Dec 2023

Cited by 1 | Viewed by 1421

Abstract

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and [...] Read more.

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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14 pages, 312 KiB

Open AccessArticle

Maximum Principle for Variable-Order Fractional Conformable Differential Equation with a Generalized Tempered Fractional Laplace Operator

by Tingting Guan and Lihong Zhang

Fractal Fract. 2023, 7(11), 798; https://doi.org/10.3390/fractalfract7110798 - 1 Nov 2023

Cited by 1 | Viewed by 1488

Abstract

In this paper, we investigate properties of solutions to a space-time fractional variable-order conformable nonlinear differential equation with a generalized tempered fractional Laplace operatorby using the maximum principle. We first establish some new important fractional various-order conformable inequalities. With these inequalities, we prove [...] Read more.

In this paper, we investigate properties of solutions to a space-time fractional variable-order conformable nonlinear differential equation with a generalized tempered fractional Laplace operatorby using the maximum principle. We first establish some new important fractional various-order conformable inequalities. With these inequalities, we prove a new maximum principle with space-time fractional variable-order conformable derivatives and a generalized tempered fractional Laplace operator. Moreover, we discuss some results about comparison principles and properties of solutions for a family of space-time fractional variable-order conformable nonlinear differential equations with a generalized tempered fractional Laplace operator by maximum principle. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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