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Symmetry
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Advanced Analytical and Numerical Methods for Fractional Initial and Boundary...
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Advanced Analytical and Numerical Methods for Fractional Initial and Boundary Value Problems with Symmetry/Asymmetry

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 April 2024) | Viewed by 9605

Special Issue Editors

Prof. Dr. Muhammad I. Syam

E-Mail Website
Guest Editor
Department of Mathematical Sciences, College of Science, UAE University, Al-Ain P.O. Box 15551, United Arab Emirates
Interests: fractional calculus; numerical analysis; ordinary and partial differential equations
Prof. Dr. Abdul-Majid Wazwaz

E-Mail Website
Guest Editor
Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA
Interests: solitons and compactons; N-solitons; ODE and PDE; numerical analysis; mathematical physics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Mohammed Al-Refai

E-Mail Website
Guest Editor
Department of Mathematics, Yarmouk University, Irbid 21163, Jordan
Interests: numerical and analytical methods for ordinary and partial differential equations; maximum principles; fractional calculus; non-local problems

Special Issue Information

Dear Colleagues,

The literature reveals that numerous real-life phenomena are influenced by symmetry and are treated in different branches of science governed by highly nonlinear fractional initial and boundary value problems with unknown analytical solutions. Therefore, such problems have received a great deal of attention from scientists with the aim of finding or approximating their analytical solutions.

The main goal of this Special Issue is to create a multidisciplinary forum of discussions on the most recent results in the field of fractional calculus. More precisely, we will focus on recent symmetric analytical and numerical studies on fractional initial and boundary differential equations related to physics, biology, and engineering. 

In addition, the well-developed analysis of existing symmetric numerical algorithms in terms of efficiency, applicability, convergence, stability and accuracy is important. A discussion of nontrivial analytical numerical examples is especially encouraged.

Potential topics include, but are not limited to:

  • Symmetric methods for solving fractional and ordinary differential equations;
  • Numerical and analytical methods for fractional ordinary differential equations;
  • Numerical and analytical methods for fractional partial differential equations;
  • Numerical and analytical methods for fractional differential equations;
  • Numerical and analytical methods for fractional integro-differential equations with symmetric kernels;
  • Mathematical control theory;
  • Mathematical biology.

Prof. Dr. Muhammad I. Syam
Prof. Dr. Abdul-Majid Wazwaz
Prof. Dr. Mohammed Al-Refai
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • symmetric methods
  • fractional derivatives
  • initial-value problems
  • boundary-value problems
  • difference equations
  • integro-differential equations
  • mathematical biology
  • control theory

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Published Papers (7 papers)

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Research

11 pages, 1315 KiB  
Article
Numerical Solution for the Heat Conduction Model with a Fractional Derivative and Temperature-Dependent Parameters
by Rafał Brociek, Edyta Hetmaniok and Damian Słota
Symmetry 2024, 16(6), 667; https://doi.org/10.3390/sym16060667 - 28 May 2024
Viewed by 758
Abstract
This paper presents the numerical solution of the heat conduction model with a fractional derivative of the Riemann–Liouville type with respect to the spatial variable. The considered mathematical model assumes the dependence on temperature of the material parameters (such as specific heat, density, [...] Read more.
This paper presents the numerical solution of the heat conduction model with a fractional derivative of the Riemann–Liouville type with respect to the spatial variable. The considered mathematical model assumes the dependence on temperature of the material parameters (such as specific heat, density, and thermal conductivity) of the model. In the paper, the boundary conditions of the first and second types are considered. If the heat flux equal to zero is assumed on the left boundary, then the thermal symmetry is obtained, which results in a simplification of the problem and the possibility of considering only half the area. The numerical examples presented in the paper illustrate the effectiveness and convergence of the discussed computational method. Full article
(This article belongs to the Special Issue Advanced Analytical and Numerical Methods for Fractional Initial and Boundary Value Problems with Symmetry/Asymmetry)
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17 pages, 296 KiB  
Article
Solving Boundary Value Problems by Sinc Method and Geometric Sinc Method
by Amer Darweesh, Kamel Al-Khaled and Mohammed Algamara
Symmetry 2024, 16(4), 411; https://doi.org/10.3390/sym16040411 - 1 Apr 2024
Cited by 1 | Viewed by 903
Abstract
This paper introduces an efficient numerical method for approximating solutions to geometric boundary value problems. We propose the multiplicative sinc–Galerkin method, tailored specifically for solving multiplicative differential equations. The method utilizes the geometric Whittaker cardinal function to approximate functions and their geometric derivatives. [...] Read more.
This paper introduces an efficient numerical method for approximating solutions to geometric boundary value problems. We propose the multiplicative sinc–Galerkin method, tailored specifically for solving multiplicative differential equations. The method utilizes the geometric Whittaker cardinal function to approximate functions and their geometric derivatives. By reducing the geometric differential equation to a system of algebraic equations, we achieve computational efficiency. The method not only proves to be computationally efficient but also showcases a valuable symmetric property, aligning with inherent patterns in geometric structures. This symmetry enhances the method’s compatibility with the often-present symmetries in geometric boundary value problems, offering both computational advantages and a deeper understanding of geometric calculus. To demonstrate the reliability and efficiency of the proposed method, we present several examples with both homogeneous and non-homogeneous boundary conditions. These examples serve to validate the method’s performance in practice. Full article
(This article belongs to the Special Issue Advanced Analytical and Numerical Methods for Fractional Initial and Boundary Value Problems with Symmetry/Asymmetry)
21 pages, 4424 KiB  
Article
Analytical Methods for Fractional Differential Equations: Time-Fractional Foam Drainage and Fisher’s Equations
by Abdulrahman B. M. Alzahrani and Ghadah Alhawael
Symmetry 2023, 15(10), 1939; https://doi.org/10.3390/sym15101939 - 19 Oct 2023
Cited by 1 | Viewed by 1295
Abstract
In this research, we employ a dual-approach that combines the Laplace residual power series method and the novel iteration method in conjunction with the Caputo operator. Our primary objective is to address the solution of two distinct, yet intricate partial differential equations: the [...] Read more.
In this research, we employ a dual-approach that combines the Laplace residual power series method and the novel iteration method in conjunction with the Caputo operator. Our primary objective is to address the solution of two distinct, yet intricate partial differential equations: the Foam Drainage Equation and the nonlinear time-fractional Fisher’s equation. These equations, essential for modeling intricate processes, present analytical challenges due to their fractional derivatives and nonlinear characteristics. By amalgamating these distinctive methodologies, we derive precise and efficient solutions substantiated by comprehensive figures and tables showcasing the accuracy and reliability of our approach. Our study not only elucidates solutions to these equations, but also underscores the effectiveness of the Laplace Residual Power Series Method and the New Iteration Method as potent tools for grappling with intricate mathematical and physical models, thereby making significant contributions to advancements in diverse scientific domains. Full article
(This article belongs to the Special Issue Advanced Analytical and Numerical Methods for Fractional Initial and Boundary Value Problems with Symmetry/Asymmetry)
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13 pages, 309 KiB  
Article
Stability of Two Kinds of Discretization Schemes for Nonhomogeneous Fractional Cauchy Problem
by Xiaoping Xu and Lei Xu
Symmetry 2023, 15(7), 1355; https://doi.org/10.3390/sym15071355 - 3 Jul 2023
Viewed by 782
Abstract
The full discrete approximation of solutions of nonhomogeneous fractional equations is considered in this paper. The methods of iteration, finite differences and projection are applied to obtain desired formulas of explicit- and implicit-difference schemes for discretization schemes. The stability of two difference schemes [...] Read more.
The full discrete approximation of solutions of nonhomogeneous fractional equations is considered in this paper. The methods of iteration, finite differences and projection are applied to obtain desired formulas of explicit- and implicit-difference schemes for discretization schemes. The stability of two difference schemes is also discussed using the Trotter–Kato theorem. Full article
(This article belongs to the Special Issue Advanced Analytical and Numerical Methods for Fractional Initial and Boundary Value Problems with Symmetry/Asymmetry)
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13 pages, 1161 KiB  
Article
Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations
by Abedel-Karrem Alomari, Mohammad Alaroud, Nedal Tahat and Adel Almalki
Symmetry 2023, 15(7), 1296; https://doi.org/10.3390/sym15071296 - 21 Jun 2023
Cited by 7 | Viewed by 1809
Abstract
In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power [...] Read more.
In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ, for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm’s accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of fractional problem. Under Caputo meaning of the symmetry order, the obtained results are illustrated numerically and graphically. Geometrically, the behavior of the obtained solutions declares that the changing of the fractional derivative parameter values in their domain alters the style of these solutions in a symmetric meaning, as well as indicates harmony and symmetry, which leads them to fully coincide at the value of the ordinary derivative. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for integral and integro-differential equations of fractional order. Full article
(This article belongs to the Special Issue Advanced Analytical and Numerical Methods for Fractional Initial and Boundary Value Problems with Symmetry/Asymmetry)
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28 pages, 18941 KiB  
Article
Dynamical Study of Coupled Riemann Wave Equation Involving Conformable, Beta, and M-Truncated Derivatives via Two Efficient Analytical Methods
by Rimsha Ansar, Muhammad Abbas, Pshtiwan Othman Mohammed, Eman Al-Sarairah, Khaled A. Gepreel and Mohamed S. Soliman
Symmetry 2023, 15(7), 1293; https://doi.org/10.3390/sym15071293 - 21 Jun 2023
Cited by 15 | Viewed by 1867
Abstract
In this study, the Jacobi elliptic function method (JEFM) and modified auxiliary equation method (MAEM) are used to investigate the solitary wave solutions of the nonlinear coupled Riemann wave (RW) equation. Nonlinear coupled partial differential equations (NLPDEs) can be transformed into a collection [...] Read more.
In this study, the Jacobi elliptic function method (JEFM) and modified auxiliary equation method (MAEM) are used to investigate the solitary wave solutions of the nonlinear coupled Riemann wave (RW) equation. Nonlinear coupled partial differential equations (NLPDEs) can be transformed into a collection of algebraic equations by utilising a travelling wave transformation. This study’s objective is to learn more about the non-linear coupled RW equation, which accounts for tidal waves, tsunamis, and static uniform media. The variance in the governing model’s travelling wave behavior is investigated using the conformable, beta, and M-truncated derivatives (M-TD). The aforementioned methods can be used to derive solitary wave solutions for trigonometric, hyperbolic, and jacobi functions. We may produce periodic solutions, bell-form soliton, anti-bell-shape soliton, M-shaped, and W-shaped solitons by altering specific parameter values. The mathematical form of each pair of travelling wave solutions is symmetric. Lastly, in order to emphasise the impact of conformable, beta, and M-TD on the behaviour and symmetric solutions for the presented problem, the 2D and 3D representations of the analytical soliton solutions can be produced using Mathematica 10. Full article
(This article belongs to the Special Issue Advanced Analytical and Numerical Methods for Fractional Initial and Boundary Value Problems with Symmetry/Asymmetry)
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22 pages, 2681 KiB  
Article
Analytical and Numerical Methods for Solving Second-Order Two-Dimensional Symmetric Sequential Fractional Integro-Differential Equations
by Sondos M. Syam, Z. Siri, Sami H. Altoum and R. Md. Kasmani
Symmetry 2023, 15(6), 1263; https://doi.org/10.3390/sym15061263 - 15 Jun 2023
Cited by 8 | Viewed by 1433
Abstract
In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate [...] Read more.
In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate the implementation of numerical and analytical approaches. Then, we propose a numerical approach based on the operational matrix method, which involves deriving operational matrices for the differential and integral terms of the equation and combining them to generate a single algebraic system. This method allows for the efficient and accurate approximation of the solution without the need for projection. Our findings demonstrate the effectiveness of the operational matrix method for solving non-homogeneous fractional integro-differential equations. We then provide examples to test our numerical method. The results demonstrate the accuracy and efficiency of the approach, with the graph of exact and approximate solutions showing almost complete overlap, and the approximate solution to the fractional problem converges to the solution of the integer problem as the order of the fractional derivative approaches one. We use various methods to measure the error in the approximation, such as absolute and L2 errors. Additionally, we explore the effect of the derivative order. The results show that the absolute error is on the order of 10−14, while the L2 error is on the order of 10−13. Next, we apply the Laplace transform to find an analytical solution to a class of fractional integro-differential equations and extend the approach to the two-dimensional case. We consider all homogeneous cases. Through our examples, we achieve two purposes. First, we show how the obtained results are implemented, especially the exact solution for some 1D and 2D classes. We then demonstrate that the exact fractional solution converges to the exact solution of the ordinary derivative as the order of the fractional derivative approaches one. Full article
(This article belongs to the Special Issue Advanced Analytical and Numerical Methods for Fractional Initial and Boundary Value Problems with Symmetry/Asymmetry)
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