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4 pages, 183 KiB

Open AccessEditorial

Preface to “Applications of Partial Differential Equations in Engineering”

by Francisco Ureña, Ángel García and Antonio M. Vargas

Mathematics 2023, 11(1), 199; https://doi.org/10.3390/math11010199 - 30 Dec 2022

Cited by 1 | Viewed by 1889

Abstract

Many problems in the broad spectrum of science require the solution of a partial differential equation [...] Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

Research

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13 pages, 976 KiB

Open AccessArticle

Using a Separable Mathematical Entropy to Construct Entropy-Stable Schemes for a Reduced Blood Flow Model

by Sonia Valbuena and Carlos A. Vega

Mathematics 2022, 10(18), 3314; https://doi.org/10.3390/math10183314 - 13 Sep 2022

Cited by 1 | Viewed by 1203

Abstract

The aim of this paper is to derive a separable entropy for a one-dimensional reduced blood flow model, which will be used to treat the symmetrizability of the model in full generality and for constructing entropy conservative fluxes, which are one of the [...] Read more.

The aim of this paper is to derive a separable entropy for a one-dimensional reduced blood flow model, which will be used to treat the symmetrizability of the model in full generality and for constructing entropy conservative fluxes, which are one of the essential building blocks of designing entropy-stable schemes. Time discretization is conducted by implicit–explicit (IMEX) Runge–Kutta schemes, but solutions for nonlinear systems will not be required due to the particular form of the source term. To validate the numerical schemes obtained, some numerical tests are presented. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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Example-1: Numerical solutions of the ideal tourniquet problem at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.005</mn> </mrow> </semantics></math> s on a mesh with 200 cells: area (left) and velocity (right). Full article ">Figure 2
Example-2: Numerical solutions (for radius) of the wave equation at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.004</mn> </mrow> </semantics></math> s on a mesh with 200 cells with enlarged views in selected regions. Full article ">Figure 3
Example-2: Numerical solutions (for velocity) of the wave equation at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.004</mn> </mrow> </semantics></math> s on a mesh with 200 cells with enlarged views in selected regions. Full article ">Figure 4
Example-3: Numerical solutions of the wave damping at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>25</mn> </mrow> </semantics></math> s with <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (top) and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.005053</mn> </mrow> </semantics></math> (bottom) and the corresponding enlarged views. The friction term has been treated with either the H-LDIRK3(2,2,2) method or the SI method. Full article ">

9 pages, 418 KiB

Open AccessArticle

Investigation and Analysis of Acoustojets by Spectral Element Method

by Ibrahim Mahariq, Ibrahim H. Giden, Shadi Alboon, Wael Hosny Fouad Aly, Ahmed Youssef and Hamza Kurt

Mathematics 2022, 10(17), 3145; https://doi.org/10.3390/math10173145 - 1 Sep 2022

Cited by 14 | Viewed by 1843

Abstract

In this study, acoustic wave scattering in a homogeneous media by an obstacle is examined in the case of plane wave excitation and the formation of acoustic jets is explored. Spectral element method (SEM) is employed for the approximate solution of scattered acoustic [...] Read more.

In this study, acoustic wave scattering in a homogeneous media by an obstacle is examined in the case of plane wave excitation and the formation of acoustic jets is explored. Spectral element method (SEM) is employed for the approximate solution of scattered acoustic waves’ calculations. An important finding of the study is the concurrence of whispering gallery modes and acoustic jet in the case of proper adjustment of structural parameters, which has not been reported before in the literature. Furthermore, numerical findings based on SEM calculations show that the main characteristics of acoustic jet can be explored and controlled by changing the targeted parameters. Microscopy and imaging applications utilizing acoustic wave can benefit from the conducted study presented in this manuscript. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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13 pages, 10917 KiB

Open AccessArticle

Some Important Points of the Josephson Effect via Two Superconductors in Complex Bases

by Fernando S. Vidal Causanilles, Haci Mehmet Baskonus, Juan Luis García Guirao and Germán Rodríguez Bermúdez

Mathematics 2022, 10(15), 2591; https://doi.org/10.3390/math10152591 - 25 Jul 2022

Cited by 12 | Viewed by 1665

Abstract

In this paper, we study the extraction of some analytical solutions to the nonlinear perturbed sine-Gordon equation with the long Josephson junction properties. The model studied was formed to observe the long Josephson junction properties separated by two superconductors. Moreover, it is also [...] Read more.

In this paper, we study the extraction of some analytical solutions to the nonlinear perturbed sine-Gordon equation with the long Josephson junction properties. The model studied was formed to observe the long Josephson junction properties separated by two superconductors. Moreover, it is also used to explain the Josephson effect arising in the highly nonlinear nature of the Josephson junctions. This provides the shunt inductances to realize a Josephson left-handed transmission line. A powerful scheme is used to extract the complex function solutions. These complex results are used to explain deeper properties of Josephson effects in the frame of impedance. Various simulations of solutions obtained in this paper are also reported. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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25 pages, 7977 KiB

Open AccessArticle

Magnetorheological Fluid of High-Speed Unsteady Flow in a Narrow-Long Gap: An Unsteady Numerical Model and Analysis

by Pengfei Zheng, Baolin Hou and Mingsong Zou

Mathematics 2022, 10(14), 2493; https://doi.org/10.3390/math10142493 - 18 Jul 2022

Cited by 2 | Viewed by 1325

Abstract

To investigate the unsteady flow field generated by magnetorheological (MR) fluid of a high-speed unsteady laminar boundary layer flow in a narrow-long gap of the magnetorheological absorber (MRA), a new unsteady numerical model is proposed. The gap has magnetic-field-activated and inactivated regions, with [...] Read more.

To investigate the unsteady flow field generated by magnetorheological (MR) fluid of a high-speed unsteady laminar boundary layer flow in a narrow-long gap of the magnetorheological absorber (MRA), a new unsteady numerical model is proposed. The gap has magnetic-field-activated and inactivated regions, with MR fluid flowing as bi-viscous (non-Newtonian) and Newtonian fluid. The unsteady flow field is described by the unsteady incompressible governing partial differential equation (PDE) and initial-boundary conditions with the moving boundary. The space-time solution domain is discretized using the finite difference method, and the governing PDE is transformed into implicit partial difference equations. The volume flow rate function is constructed to solve numerical solutions of pressure gradient and fluid velocity based on mass conservation, the continuity equation, and the bisection method. The accuracy of unsteady numerical model is validated by the experiment data. The results show that the fluid acceleration profiles along the gap’s height are non-uniform distribution. Further, the volume flow rate and excitation current has a significant impact on the dynamic distribution of fluid velocity profiles, and the moving boundary makes the flow field asymmetric about the central plane. Furthermore, as the transition stress increases, the thickness of the pre-yield region in the activated region increases. There is also a transition flow phenomenon in the activated region as the volume flow rate increases. Finally, the unsteady numerical model has good stability and convergence. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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12 pages, 370 KiB

Open AccessArticle

A Novel Spatio-Temporal Fully Meshless Method for Parabolic PDEs

by Juan José Benito, Ángel García, Mihaela Negreanu, Francisco Ureña and Antonio M. Vargas

Mathematics 2022, 10(11), 1870; https://doi.org/10.3390/math10111870 - 30 May 2022

Cited by 7 | Viewed by 1676

Abstract

We introduce a meshless method derived by considering the time variable as a spatial variable without the need to extend further conditions to the solution of linear and non-linear parabolic PDEs. The method is based on a moving least squares method, more precisely, [...] Read more.

We introduce a meshless method derived by considering the time variable as a spatial variable without the need to extend further conditions to the solution of linear and non-linear parabolic PDEs. The method is based on a moving least squares method, more precisely, the generalized finite difference method (GFDM), which allows us to select well-conditioned stars. Several 2D and 3D examples, including the time variable, are shown for both regular and irregular node distributions. The results are compared with explicit GFDM both in terms of errors and execution time. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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10 pages, 269 KiB

Open AccessArticle

Multiple Comparison Procedures for Exponential Mean Lifetimes Compared with Several Controls

by Shu-Fei Wu

Mathematics 2022, 10(4), 609; https://doi.org/10.3390/math10040609 - 16 Feb 2022

Cited by 2 | Viewed by 1704

Abstract

Under heteroscedasticity, we propose one-stage multiple comparison procedures for several treatment groups compared with several control groups in terms of exponential mean lifetimes. The simultaneous confidence intervals including one-sided and two-sided confidence intervals for the difference between the mean lifetime from the i [...] Read more.

Under heteroscedasticity, we propose one-stage multiple comparison procedures for several treatment groups compared with several control groups in terms of exponential mean lifetimes. The simultaneous confidence intervals including one-sided and two-sided confidence intervals for the difference between the mean lifetime from the i-th treatment group and the mean lifetime from the j-th control group are developed in this research. The required critical values are obtained and tabulated for the practical use of users. The experimenters can use these simultaneous confidence intervals to determine whether the treatment mean lifetimes are better than several controls or worse than several controls under a specified confidence level. Finally, one example of comparing the mean duration of remission using four drugs for treating leukemia is used for the aims of illustrations. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

15 pages, 5036 KiB

Open AccessArticle

A Study on the Experimental Design for the Lifetime Performance Index of Rayleigh Lifetime Distribution under Progressive Type I Interval Censoring

by Shu-Fei Wu, Tzu-Hsuan Liu, Yu-Hua Lai and Wei-Tsung Chang

Mathematics 2022, 10(3), 517; https://doi.org/10.3390/math10030517 - 5 Feb 2022

Cited by 6 | Viewed by 1787

Abstract

With the rapid development of technology, improving product life performance has become a very important issue in recent decades. The lifetime performance index is used in this research for the assessment of the lifetime performance of products following the Rayleigh distribution. Based on [...] Read more.

With the rapid development of technology, improving product life performance has become a very important issue in recent decades. The lifetime performance index is used in this research for the assessment of the lifetime performance of products following the Rayleigh distribution. Based on the hypothesis testing procedure with this index, using the maximum likelihood estimator as a testing statistic, the sampling design is determined and the related values are tabulated for practical use to reach the given power level and minimize the total experimental cost under progressive type I interval censoring. When the inspection interval length is fixed and the number of inspection intervals is not fixed, the required number of inspection intervals and sample size with the minimum total cost are determined and tabulated. When the termination time is not fixed, the required number of inspection intervals, sample size and equal interval length reaching the minimum total cost are determined and tabulated. Lastly, a practical example is given to illustrate the use of this sampling design for the testing procedure to determine whether the process is capable. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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9 pages, 643 KiB

Open AccessArticle

A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

by Ángel García, Mihaela Negreanu, Francisco Ureña and Antonio M. Vargas

Mathematics 2021, 9(22), 2843; https://doi.org/10.3390/math9222843 - 10 Nov 2021

Cited by 5 | Viewed by 1676

Abstract

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using [...] Read more.

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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Clouds of nodes with irregular distribution. Full article ">Figure 2
Solutions given by the method for <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (left) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> (centre) and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> (right) using the first cloud of <a href="#mathematics-09-02843-f001" class="html-fig">Figure 1</a>. Full article ">Figure 3
Solutions given by the method for <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (left) <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (centre) and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> (right) using the second cloud of <a href="#mathematics-09-02843-f001" class="html-fig">Figure 1</a>. Full article ">Figure 4
Solutions given by the method for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> </mrow> </semantics></math> and 10 in the second example. Full article ">

18 pages, 679 KiB

Open AccessArticle

Mixed Mesh Finite Volume Method for 1D Hyperbolic Systems with Application to Plug-Flow Heat Exchangers

by Jiří Dostál and Vladimír Havlena

Mathematics 2021, 9(20), 2609; https://doi.org/10.3390/math9202609 - 16 Oct 2021

Cited by 2 | Viewed by 2017

Abstract

We present a finite volume method formulated on a mixed Eulerian-Lagrangian mesh for highly advective 1D hyperbolic systems altogether with its application to plug-flow heat exchanger modeling/simulation. Advection of sharp moving fronts is an important problem in fluid dynamics, and even a simple [...] Read more.

We present a finite volume method formulated on a mixed Eulerian-Lagrangian mesh for highly advective 1D hyperbolic systems altogether with its application to plug-flow heat exchanger modeling/simulation. Advection of sharp moving fronts is an important problem in fluid dynamics, and even a simple transport equation cannot be solved precisely by having a finite number of nodes/elements/volumes. Finite volume methods are known to introduce numerical diffusion, and there exist a wide variety of schemes to minimize its occurrence; the most recent being adaptive grid methods such as moving mesh methods or adaptive mesh refinement methods. We present a solution method for a class of hyperbolic systems with one nonzero time-dependent characteristic velocity. This property allows us to rigorously define a finite volume method on a grid that is continuously moving by the characteristic velocity (Lagrangian grid) along a static Eulerian grid. The advective flux of the flowing field is, by this approach, removed from cell-to-cell interactions, and the ability to advect sharp fronts is therefore enhanced. The price to pay is a fixed velocity-dependent time sampling and a time delay in the solution. For these reasons, the method is best suited for systems with a dominating advection component. We illustrate the method’s properties on an illustrative advection-decay equation example and a 1D plug flow heat exchanger. Such heat exchanger model can then serve as a convection-accurate dynamic model in estimation and control algorithms for which it was developed. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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Illustration of the advected grid movement against the static grid. Full article ">Figure 2
Illustration of a discrete state jump in otherwise continuous state evolution. Full article ">Figure 3
Method comparison for the advection-decay test at x = 1; number of volumes for all methods is N = 5. Full article ">Figure 4
Time sequence of inlet water temperature Tw,in, inlet air temperature Ta,in, water mass flow <math display="inline"><semantics> <mover accent="true"> <mi>m</mi> <mo>˙</mo> </mover> </semantics></math> and air volumetric flow <math display="inline"><semantics> <mover accent="true"> <mi>V</mi> <mo>˙</mo> </mover> </semantics></math>. Full article ">Figure 5
Time evolution of the body and water temperatures Tb, Tw at x = [0.125, 0.375, 0.625, 0.875] (from top to bottom). The fine FVM solution was spatially averaged to the cell size of the latter two solutions. Full article ">Figure 6
Time evolution of the heat output Q and the water outlet temperature Tw,out (at x = 1). Full article ">

17 pages, 1525 KiB

Open AccessArticle

Adaptive Boundary Control for a Certain Class of Reaction–Advection–Diffusion System

by Oscar F. Murillo-García and Francisco Jurado

Mathematics 2021, 9(18), 2224; https://doi.org/10.3390/math9182224 - 10 Sep 2021

Cited by 3 | Viewed by 2610

Abstract

Several phenomena in nature are subjected to the interaction of various physical parameters, which, if these latter are well known, allow us to predict the behavior of such phenomena. In most cases, these physical parameters are not exactly known, or even more these [...] Read more.

Several phenomena in nature are subjected to the interaction of various physical parameters, which, if these latter are well known, allow us to predict the behavior of such phenomena. In most cases, these physical parameters are not exactly known, or even more these are unknown, so identification techniques could be employed in order to estimate their values. Systems for which their inputs and outputs vary both temporally and spatially are the so-called distributed parameter systems (DPSs) modeled through partial differential equations (PDEs). The way in which their parameters evolve with respect to time may not always be known and may also induce undesired behavior of the dynamics of the system. To reverse the above, the well-known adaptive boundary control technique can be used to estimate the unknown parameters assuring a stable behavior of the dynamics of the system. In this work, we focus our attention on the design of an adaptive boundary control for a parabolic type reaction–advection–diffusion PDE under the assumption of unknown parameters for both advection and reaction terms and Robin and Neumann boundary conditions. An identifier PDE system is established and parameter update laws are designed following the certainty equivalence approach with a passive identifier. The performance of the adaptive Neumann stabilizing controller is validated via numerical simulation. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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11 pages, 497 KiB

Open AccessArticle

Convective Heat Transfer of a Hybrid Nanofluid over a Nonlinearly Stretching Surface with Radiation Effect

by Emad H. Aly, Alin V. Roşca, Natalia C. Roşca and Ioan Pop

Mathematics 2021, 9(18), 2220; https://doi.org/10.3390/math9182220 - 10 Sep 2021

Cited by 26 | Viewed by 1809

Abstract

The flow of the hybrid nanofluid (copper–titanium dioxide/water) over a nonlinearly stretching surface was studied with suction and radiation effect. The governing partial differential equations were then converted into non-linear ordinary differential equations by using proper similarity transformations. Therefore, these equations were solved [...] Read more.

The flow of the hybrid nanofluid (copper–titanium dioxide/water) over a nonlinearly stretching surface was studied with suction and radiation effect. The governing partial differential equations were then converted into non-linear ordinary differential equations by using proper similarity transformations. Therefore, these equations were solved by applying a numerical technique, namely Chebyshev pseudo spectral differentiation matrix. The results of the flow field, temperature distribution, reduced skin friction coefficient and reduced Nusselt number were deduced. It was found that the rising of the mass flux parameter slows down the velocity and, hence, decreases the temperature. Further, on enlarging the stretching parameter, the velocity and temperature increases and decreases, respectively. In addition, it was mentioned that the radiation parameter can effectively control the thermal boundary layer. Finally, the temperature decreases when the values of the temperature parameter increases. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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18 pages, 3437 KiB

Open AccessArticle

Reliability Sampling Design for the Lifetime Performance Index of Gompertz Lifetime Distribution under Progressive Type I Interval Censoring

by Shu-Fei Wu, Yi-Jun Xie, Mao-Feng Liao and Wei-Tsung Chang

Mathematics 2021, 9(17), 2109; https://doi.org/10.3390/math9172109 - 31 Aug 2021

Cited by 5 | Viewed by 1682

Abstract

In this artificial intelligence era, the constantly changing technology makes production techniques become sophisticated and complicated. Therefore, manufacturers are dedicated to improving the quality of products by increasing the lifetime in order to achieve the quality standards demanded by consumers. For products with [...] Read more.

In this artificial intelligence era, the constantly changing technology makes production techniques become sophisticated and complicated. Therefore, manufacturers are dedicated to improving the quality of products by increasing the lifetime in order to achieve the quality standards demanded by consumers. For products with lifetime following a Gompertz distribution, the lifetime performance index was used to measure the performance of manufacturing process under progressive type I interval censoring. The sampling design is investigated to reach the given level of significance and power level. When inspection interval length is fixed and the number of inspection intervals is not fixed, the required number of inspection intervals and sample size with minimum total cost are determined and tabulated. When the termination time is not fixed, the required number of inspection intervals, sample size, and equal interval length reaching the minimum total cost are determined and tabulated. The optimal parameter values are tabulated for the practical use of users. Finally, one practical example is given for the illustrative aim to show the implementation of this sampling design to collect data and the collected data are used to conduct the testing procedure to see if the process is capable. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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Minimum sample size for the test at <math display="inline"><semantics> <mi>α</mi> </semantics></math> = 0.05, p = 0.05, and m = 5. Full article ">Figure 2
Minimum sample size for the test at <math display="inline"><semantics> <mi>α</mi> </semantics></math> = 0.05, <math display="inline"><semantics> <mi>β</mi> </semantics></math> = 0.2, and p = 0.05. Full article ">Figure 3
Minimum sample size for the test at <math display="inline"><semantics> <mi>α</mi> </semantics></math> = 0.05, <math display="inline"><semantics> <mi>β</mi> </semantics></math> = 0.2, and m = 5. Full article ">Figure 4
Minimum sample size for the test at <math display="inline"><semantics> <mi>β</mi> </semantics></math> = 0.2 under p = 0.05 and m = 5. Full article ">Figure 5
(a) Total cost curve at m = 2(1) m0; (b) total cost curve at m = 2(1) m0. Full article ">Figure 6
(a) Total cost curve at m = 2(1)m0; (b) total cost curve at m = 1(1)m0. Full article ">Figure 7
The p-values for Gini test versus k values. Full article ">

14 pages, 828 KiB

Open AccessArticle

On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

by Junjiang Lai and Hongyu Liu

Mathematics 2021, 9(16), 2014; https://doi.org/10.3390/math9162014 - 23 Aug 2021

Cited by 5 | Viewed by 1822

Abstract

In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability [...] Read more.

In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability estimates as well as optimal a priori error estimates. Extensive numerical experiments are conducted to verify the promising features of the newly proposed method. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

9 pages, 241 KiB

Open AccessArticle

On a Class of Second-Order PDE&PDI Constrained Robust Modified Optimization Problems

by Savin Treanţă

Mathematics 2021, 9(13), 1473; https://doi.org/10.3390/math9131473 - 23 Jun 2021

Cited by 8 | Viewed by 1797

Abstract

In this paper, by using scalar multiple integral cost functionals and the notion of convexity associated with a multiple integral functional driven by an uncertain multi-time controlled second-order Lagrangian, we develop a new mathematical framework on multi-dimensional scalar variational control problems with mixed [...] Read more.

In this paper, by using scalar multiple integral cost functionals and the notion of convexity associated with a multiple integral functional driven by an uncertain multi-time controlled second-order Lagrangian, we develop a new mathematical framework on multi-dimensional scalar variational control problems with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Concretely, we introduce and investigate an auxiliary (modified) variational control problem, which is much easier to study, and provide some equivalence results by using the notion of a normal weak robust optimal solution. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

15 pages, 2779 KiB

Open AccessArticle

Convergence and Numerical Solution of a Model for Tumor Growth

by Juan J. Benito, Ángel García, María Lucía Gavete, Mihaela Negreanu, Francisco Ureña and Antonio M. Vargas

Mathematics 2021, 9(12), 1355; https://doi.org/10.3390/math9121355 - 11 Jun 2021

Cited by 6 | Viewed by 2035

Abstract

In this paper, we show the application of the meshless numerical method called “Generalized Finite Diference Method” (GFDM) for solving a model for tumor growth with nutrient density, extracellular matrix and matrix degrading enzymes, [recently proposed by Li and Hu]. We derive the [...] Read more.

In this paper, we show the application of the meshless numerical method called “Generalized Finite Diference Method” (GFDM) for solving a model for tumor growth with nutrient density, extracellular matrix and matrix degrading enzymes, [recently proposed by Li and Hu]. We derive the discretization of the parabolic–hyperbolic–parabolic–elliptic system by means of the explicit formulae of the GFDM. We provide a theoretical proof of the convergence of the spatial–temporal scheme to the continuous solution and we show several examples over regular and irregular distribution of points. This shows the feasibility of the method for solving this nonlinear model appearing in Biology and Medicine in complicated and realistic domains. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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25 pages, 5928 KiB

Open AccessArticle

Four-Quadrant Riemann Problem for a 2×2 System II

by Jinah Hwang, Suyeon Shin, Myoungin Shin and Woonjae Hwang

Mathematics 2021, 9(6), 592; https://doi.org/10.3390/math9060592 - 10 Mar 2021

Cited by 2 | Viewed by 2225

Abstract

In previous work, we considered a four-quadrant Riemann problem for a

2 \times 2

hyperbolic system in which delta shock appears at the initial discontinuity without assuming that each jump of the initial data projects exactly one plane elementary wave. In this paper, [...] Read more.

In previous work, we considered a four-quadrant Riemann problem for a

2 \times 2

hyperbolic system in which delta shock appears at the initial discontinuity without assuming that each jump of the initial data projects exactly one plane elementary wave. In this paper, we consider the case that does not involve a delta shock at the initial discontinuity. We classified 18 topologically distinct solutions and constructed analytic and numerical solutions for each case. The constructed analytic solutions show the rich structure of wave interactions in the Riemann problem, which coincide with the computed numerical solutions. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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23 pages, 8862 KiB

Open AccessArticle

Conventional Partial and Complete Solutions of the Fundamental Equations of Fluid Mechanics in the Problem of Periodic Internal Waves with Accompanying Ligaments Generation

by Yuli D. Chashechkin

Mathematics 2021, 9(6), 586; https://doi.org/10.3390/math9060586 - 10 Mar 2021

Cited by 22 | Viewed by 1963

Abstract

The problem of generating beams of periodic internal waves in a viscous, exponentially stratified fluid by a band oscillating along an inclined plane is considered by the methods of the theory of singular perturbations in the linear and weakly nonlinear approximations. The complete [...] Read more.

The problem of generating beams of periodic internal waves in a viscous, exponentially stratified fluid by a band oscillating along an inclined plane is considered by the methods of the theory of singular perturbations in the linear and weakly nonlinear approximations. The complete solution to the linear problem, which satisfies the boundary conditions on the emitting surface, is constructed taking into account the previously proposed classification of flow structural components described by complete solutions of the linearized system of fundamental equations without involving additional force or mass sources. Analyses includes all components satisfying the dispersion relation that are periodic waves and thin accompanying ligaments, the transverse scale of which is determined by the kinematic viscosity and the buoyancy frequency. Ligaments are located both near the emitting surface and in the bulk of the liquid in the form of wave beam envelopes. Calculations show that in a nonlinear description of all components, both waves and ligaments interact directly with each other in all combinations: waves-waves, waves-ligaments, and ligaments-ligaments. Direct interactions of the components that generate new harmonics of internal waves occur despite the differences in their scales. Additionally, the problem of generating internal waves by a rapidly bi-harmonically oscillating vertical band is considered. If the difference in the frequencies of the spectral components of the band movement is less than the buoyancy frequency, the nonlinear interacting ligaments generate periodic waves as well. The estimates made show that the amplitudes of such waves are large enough to be observed under laboratory conditions. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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Periodic internal waves beams and a family of ligaments (envelopes of beams and thin flows on the emitting surface), generated by vertical oscillations of a horizontal disk D = 8 cm in diameter with a period of To = 6.3 s and a velocity amplitude of Uo = 0.25 cm/s in a liquid with a buoyancy period of Tb = 4.2 s: (a–c)—fields of the vertical component of velocity νz, the first <math display="inline"><semantics> <mrow> <mfrac> <mrow> <msub> <mrow> <mo>∂</mo> <mi>ν</mi> </mrow> <mi>z</mi> </msub> </mrow> <mrow> <mo>∂</mo> <mi>r</mi> </mrow> </mfrac> </mrow> </semantics></math> and second derivatives <math display="inline"><semantics> <mrow> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <msub> <mo>ν</mo> <mi>z</mi> </msub> </mrow> <mrow> <msup> <mrow> <mo>∂</mo> <mi>r</mi> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </semantics></math>. Full article ">Figure 2
Schlieren images of the harmonic internal waves beams and ligaments (horizontal interfaces) generated by the sloping band placed under angle φ = 35° band oscillating along its plane: (a)—period of buoyancy Tb = 5.5 s and band oscillation T = 6.5 s, width a = 1 cm and amplitude of oscillations A = 0.15 cm, beam slope angle θ = 57° ≠ φ —general case; (b) φ = 45°, Tb = 5.5 s, T = 7.8 s, a = 6 cm, A = 0.15 cm, θ = φ = 45°—critical case: (c)—φ = 35°, Tb = 5.5 s, T = 13 s, a = 1 cm A = 0.15 cm, slope of main beam θ1 = 25°, second harmonics beam θ2 = 60°. Full article ">Figure 3
Schlieren images of periodic flows induced by oscillating sphere (D = 4.5 cm), (а–с)—Tb = 11.2, 7.3, 11.2 s; A = 1, 2.8, 2.8 cm, ω/N = 0.73, 0.8,0.8. Full article ">Figure 4
Basic coordinate frames. Full article ">

22 pages, 4363 KiB

Open AccessArticle

Four-Quadrant Riemann Problem for a 2 × 2 System Involving Delta Shock

by Jinah Hwang, Suyeon Shin, Myoungin Shin and Woonjae Hwang

Mathematics 2021, 9(2), 138; https://doi.org/10.3390/math9020138 - 10 Jan 2021

Cited by 2 | Viewed by 2517

Abstract

In this paper, a four-quadrant Riemann problem for a

2 \times 2

system of hyperbolic conservation laws is considered in the case of delta shock appearing at the initial discontinuity. We also remove the restriction in that there is only one planar wave [...] Read more.

In this paper, a four-quadrant Riemann problem for a

2 \times 2

system of hyperbolic conservation laws is considered in the case of delta shock appearing at the initial discontinuity. We also remove the restriction in that there is only one planar wave at each initial discontinuity. We include the existence of two elementary waves at each initial discontinuity and classify 14 topologically distinct solutions. For each case, we construct an analytic solution and compute the numerical solution. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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8 pages, 256 KiB

Open AccessArticle

Multiple Comparisons for Exponential Median Lifetimes with the Control Based on Doubly Censored Samples

by Shu-Fei Wu

Mathematics 2021, 9(1), 76; https://doi.org/10.3390/math9010076 - 31 Dec 2020

Cited by 1 | Viewed by 1586

Abstract

Under doubly censoring, the one-stage multiple comparison procedures with the control in terms of exponential median lifetimes are presented. The uniformly minimum variance unbiased estimator for median lifetime is found. The upper bounds, lower bounds and two-sided confidence intervals for the difference between [...] Read more.

Under doubly censoring, the one-stage multiple comparison procedures with the control in terms of exponential median lifetimes are presented. The uniformly minimum variance unbiased estimator for median lifetime is found. The upper bounds, lower bounds and two-sided confidence intervals for the difference between each median lifetimes and the median lifetime of the control population are developed. Statistical tables of critical values are constructed for the practical use of our proposed procedures. Users can use these simultaneous confidence intervals to determine whether the performance of treatment populations is better than or worse than the control population in agriculture and pharmaceutical industries. At last, one practical example is provided to illustrate the proposed procedures. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

14 pages, 321 KiB

Open AccessArticle

Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)

by Antonio Falcó, Lucía Hilario, Nicolás Montés, Marta C. Mora and Enrique Nadal

Mathematics 2021, 9(1), 34; https://doi.org/10.3390/math9010034 - 25 Dec 2020

Cited by 1 | Viewed by 1890

Abstract

A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems. However, it is well-known that the bottleneck of its practical implementation focuses on the computation of the so-called best [...] Read more.

A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems. However, it is well-known that the bottleneck of its practical implementation focuses on the computation of the so-called best rank-one approximation. Motivated by this fact, we are going to discuss some of the geometrical aspects of the best rank-one approximation procedure. More precisely, our main result is to construct explicitly a vector field over a low-dimensional vector space and to prove that we can identify its stationary points with the critical points of the best rank-one optimization problem. To obtain this result, we endow the set of tensors with fixed rank-one with an explicit geometric structure. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

13 pages, 1212 KiB

Open AccessArticle

Complex Ginzburg–Landau Equation with Generalized Finite Differences

by Eduardo Salete, Antonio M. Vargas, Ángel García, Mihaela Negreanu, Juan J. Benito and Francisco Ureña

Mathematics 2020, 8(12), 2248; https://doi.org/10.3390/math8122248 - 20 Dec 2020

Cited by 4 | Viewed by 2783

Abstract

In this paper we obtain a novel implementation for irregular clouds of nodes of the meshless method called Generalized Finite Difference Method for solving the complex Ginzburg–Landau equation. We derive the explicit formulae for the spatial derivative and an explicit scheme by splitting [...] Read more.

In this paper we obtain a novel implementation for irregular clouds of nodes of the meshless method called Generalized Finite Difference Method for solving the complex Ginzburg–Landau equation. We derive the explicit formulae for the spatial derivative and an explicit scheme by splitting the equation into a system of two parabolic PDEs. We prove the conditional convergence of the numerical scheme towards the continuous solution under certain assumptions. We obtain a second order approximation as it is clear from the numerical results. Finally, we provide several examples of its application over irregular domains in order to test the accuracy of the explicit scheme, as well as comparison with other numerical methods. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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Figure 1
Irregular clouds of points with 55, 197 and 743 nodes respectively. Full article ">Figure 2
Analytical solutions of (<a href="#FD1-mathematics-08-02248" class="html-disp-formula">1</a>). Full article ">Figure 3
Approximate solutions in the Example 1. Full article ">Figure 4
Approximate solutions in the Example 2. Full article ">Figure 5
Approximate solutions in the Example 3. Full article ">Figure 6
<math display="inline"><semantics> <msup> <mi>l</mi> <mn>2</mn> </msup> </semantics></math> and <math display="inline"><semantics> <msup> <mi>l</mi> <mo>∞</mo> </msup> </semantics></math> norms of the errors of the real and imaginary parts for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, respectively versus nodes. Full article ">Figure 7
<math display="inline"><semantics> <msup> <mi>l</mi> <mn>2</mn> </msup> </semantics></math> and <math display="inline"><semantics> <msup> <mi>l</mi> <mo>∞</mo> </msup> </semantics></math> norms of the errors of the real and imaginary parts Example 3, respectively versus time. Full article ">

16 pages, 1065 KiB

Open AccessArticle

State Feedback Regulation Problem to the Reaction-Diffusion Equation

by Francisco Jurado and Andrés A. Ramírez

Mathematics 2020, 8(11), 1983; https://doi.org/10.3390/math8111983 - 6 Nov 2020

Cited by 2 | Viewed by 2642

Abstract

In this work, we explore the state feedback regulator problem (SFRP) in order to achieve the goal for trajectory tracking with harmonic disturbance rejection to one-dimensional (1-D) reaction-diffusion (R-D) equation, namely, a partial differential equation of parabolic type, while taking into account bounded [...] Read more.

In this work, we explore the state feedback regulator problem (SFRP) in order to achieve the goal for trajectory tracking with harmonic disturbance rejection to one-dimensional (1-D) reaction-diffusion (R-D) equation, namely, a partial differential equation of parabolic type, while taking into account bounded input, output, and disturbance operators, a finite-dimensional exosystem (exogenous system), and the state of the exosystem as the state to the feedback law. As is well-known, the SFRP can be solved only if the so-called Francis (regulator) equations have solution. In our work, we try with the solution of the Francis equations from the 1-D R-D equation following given criteria to the eigenvalues from the exosystem and transfer function of the system, but the state operator is here defined in terms of the Sturm–Liouville differential operator (SLDO). Within this framework, the SFRP is then solved for the 1-D R-D equation. The numerical simulation results validate the performance of the regulator. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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14 pages, 950 KiB

Open AccessArticle

A Meshless Method Based on the Laplace Transform for the 2D Multi-Term Time Fractional Partial Integro-Differential Equation

by Kamran Kamran, Zahir Shah, Poom Kumam and Nasser Aedh Alreshidi

Mathematics 2020, 8(11), 1972; https://doi.org/10.3390/math8111972 - 6 Nov 2020

Cited by 11 | Viewed by 2310

Abstract

In this article, we propose a localized transform based meshless method for approximating the solution of the 2D multi-term partial integro-differential equation involving the time fractional derivative in Caputo’s sense with a weakly singular kernel. The purpose of coupling the localized meshless method [...] Read more.

In this article, we propose a localized transform based meshless method for approximating the solution of the 2D multi-term partial integro-differential equation involving the time fractional derivative in Caputo’s sense with a weakly singular kernel. The purpose of coupling the localized meshless method with the Laplace transform is to avoid the time stepping procedure by eliminating the time variable. Then, we utilize the local meshless method for spatial discretization. The solution of the original problem is obtained as a contour integral in the complex plane. In the literature, numerous contours are available; in our work, we will use the recently introduced improved Talbot contour. We approximate the contour integral using the midpoint rule. The bounds of stability for the differentiation matrix of the scheme are derived, and the convergence is discussed. The accuracy, efficiency, and stability of the scheme are validated by numerical experiments. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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Figure 1
Node distribution in the square domain. Full article ">Figure 2
(a) The exact solution in the square domain; (b) the numerical solution in the square domain. Full article ">Figure 3
(a) Node distribution in the nut-shaped domain; (b) the exact and numerical solutions in the nut-shaped domain. Full article ">Figure 4
(a) Nodes in the L-shaped domain; (b) the exact and numerical solutions in the L-shaped domain. Full article ">Figure 5
Absolute error using Robin boundary conditions in the L-shaped domain, with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mrow/> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.75</mn> <mo>,</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.95</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>N</mi> <mo>=</mo> <mn>736</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>n</mi> <mo>=</mo> <mn>73</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>. Full article ">

22 pages, 1024 KiB

Open AccessArticle

Boundary Control for a Certain Class of Reaction-Advection-Diffusion System

by Eduardo Cruz-Quintero and Francisco Jurado

Mathematics 2020, 8(11), 1854; https://doi.org/10.3390/math8111854 - 22 Oct 2020

Cited by 4 | Viewed by 2863

Abstract

There are physical phenomena, involving diffusion and structural vibrations, modeled by partial differential equations (PDEs) whose solution reflects their spatial distribution. Systems whose dynamics evolve on an infinite-dimensional Hilbert space, i.e., infinite-dimensional systems, are modeled by PDEs. The aim when designing a controller [...] Read more.

There are physical phenomena, involving diffusion and structural vibrations, modeled by partial differential equations (PDEs) whose solution reflects their spatial distribution. Systems whose dynamics evolve on an infinite-dimensional Hilbert space, i.e., infinite-dimensional systems, are modeled by PDEs. The aim when designing a controller for infinite-dimensional systems is similar to that for finite-dimensional systems, i.e., the control system must be stable. Another common goal is to design the controller in such a way that the response of the system does not be affected by external disturbances. The controller design for finite-dimensional systems is not an easy task, so, the controller design for infinite-dimensional systems is even more challenging. The backstepping control approach is a dominant methodology for boundary feedback design. In this work, we try with the backstepping design for the boundary control of a reaction-advection-diffusion (R-A-D) equation, namely, a type parabolic PDE, but with constant coefficients and Neumann boundary conditions, with actuation in one of these latter. The heat equation with Neumann boundary conditions is considered as the target system. Dynamics of the open- and closed-loop solution of the PDE system are validated via numerical simulation. The MATLAB

^{®}

-based numerical algorithm related with the implementation of the control scheme is here included. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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13 pages, 3612 KiB

Open AccessArticle

The Application of Accurate Exponential Solution of a Differential Equation in Optimizing Stability Control of One Class of Chaotic System

by Hao Jia and Chen Guo

Mathematics 2020, 8(10), 1740; https://doi.org/10.3390/math8101740 - 10 Oct 2020

Cited by 2 | Viewed by 1865

Abstract

For many nonlinear systems in our life, the chaos phenomenon generated under certain conditions in special cases will split the system and result in a crash-down of the system. This paper discusses the stable control of one class of chaotic systems and a [...] Read more.

For many nonlinear systems in our life, the chaos phenomenon generated under certain conditions in special cases will split the system and result in a crash-down of the system. This paper discusses the stable control of one class of chaotic systems and a control method based on the accurate exponential solution of a differential equation is used. Compared with other methods, the advantages are: this method determines that the system can exponentially converge at the origin and the convergence rate can be easily regulated. The chaotic system with unknown parameters is also deduced and validated by using this method. In practical application, it is found that the ship’s electric system also has the same model, so it has certain practical significance. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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30 pages, 4450 KiB

Open AccessArticle

Closed-Form Solutions and Conserved Vectors of a Generalized (3+1)-Dimensional Breaking Soliton Equation of Engineering and Nonlinear Science

by Chaudry Masood Khalique and Oke Davies Adeyemo

Mathematics 2020, 8(10), 1692; https://doi.org/10.3390/math8101692 - 1 Oct 2020

Cited by 18 | Viewed by 2028

Abstract

In this article, we examine a (3+1)-dimensional generalized breaking soliton equation which is highly applicable in the fields of engineering and nonlinear sciences. Closed-form solutions in the form of Jacobi elliptic functions of the underlying equation are derived by the method of Lie [...] Read more.

In this article, we examine a (3+1)-dimensional generalized breaking soliton equation which is highly applicable in the fields of engineering and nonlinear sciences. Closed-form solutions in the form of Jacobi elliptic functions of the underlying equation are derived by the method of Lie symmetry reductions together with direct integration. Moreover, the

(G^{'} / G)

-expansion technique is engaged, which consequently guarantees closed-form solutions of the equation structured in the form of trigonometric and hyperbolic functions. In addition, we secure a power series analytical solution of the underlying equation. Finally, we construct local conserved vectors of the aforementioned equation by employing two approaches: the general multiplier method and Ibragimov’s theorem. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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9 pages, 208 KiB

Open AccessArticle

One-Stage Multiple Comparisons with the Control for Exponential Median Lifetimes under Heteroscedasticity

by Shu-Fei Wu

Mathematics 2020, 8(9), 1405; https://doi.org/10.3390/math8091405 - 21 Aug 2020

Cited by 2 | Viewed by 1545

Abstract

When the additional sample for the second stage may not be available, one-stage multiple comparisons for exponential median lifetimes with the control under heteroscedasticity including one-sided and two-sided confidence intervals are proposed in this paper since the median is a more robust measure [...] Read more.

When the additional sample for the second stage may not be available, one-stage multiple comparisons for exponential median lifetimes with the control under heteroscedasticity including one-sided and two-sided confidence intervals are proposed in this paper since the median is a more robust measure of central tendency compared to the mean. These intervals can be used to identify treatment populations that are better than the control or worse than the control in terms of median lifetimes in agriculture, stock market, pharmaceutical industries. Tables of critical values are obtained for practical use. An example of comparing the survival days for four categories of lung cancer in a standard chemotherapeutic agent is given to demonstrate the proposed procedures. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

19 pages, 5153 KiB

Open AccessArticle

On Generalized Fourier’s and Fick’s Laws in Bio-Convection Flow of Magnetized Burgers’ Nanofluid Utilizing Motile Microorganisms

by Ali Saleh Alshomrani

Mathematics 2020, 8(7), 1186; https://doi.org/10.3390/math8071186 - 19 Jul 2020

Cited by 8 | Viewed by 2456

Abstract

This article describes the features of bio-convection and motile microorganisms in magnetized Burgers’ nanoliquid flows by stretchable sheet. Theory of Cattaneo–Christov mass and heat diffusions is also discussed. The Buongiorno phenomenon for nanoliquid motion in a Burgers’ fluid is employed in view of [...] Read more.

This article describes the features of bio-convection and motile microorganisms in magnetized Burgers’ nanoliquid flows by stretchable sheet. Theory of Cattaneo–Christov mass and heat diffusions is also discussed. The Buongiorno phenomenon for nanoliquid motion in a Burgers’ fluid is employed in view of the Cattaneo–Christov relation. The control structure of governing partial differential equations (PDEs) is changed into appropriate ordinary differential equations (ODEs) by suitable transformations. To get numerical results of nonlinear systems, the bvp4c solver provided in the commercial software MATLAB is employed. Numerical and graphical data for velocity, temperature, nanoparticles concentration and microorganism profiles are obtained by considering various estimations of prominent physical parameters. Our computations depict that the temperature field has direct relation with the thermal Biot number and Burgers’ fluid parameter. Here, temperature field is enhanced for growing estimations of thermal Biot number and Burgers’ fluid parameter. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)

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