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Symmetry
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Symmetry in Computational and Mathematical Methods of Fractional Calculus
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Symmetry in Computational and Mathematical Methods of Fractional Calculus

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

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 8418

Special Issue Editor

Prof. Dr. Jorge E. Macías Díaz

E-Mail Website
Guest Editor
1. Department of Mathematics, School of Digital Technologies, Tallinn University, 10120 Tallinn, Estonia
2. Department of Mathematics and Physics, Autonomous University of Aguascalientes, Aguascalientes 20131, Mexico
Interests: fractional calculus; difference equations; differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The theory of fractional-order integro-differential operators (fractional calculus, for short) has been one of those areas in mathematics which has undergone a tremendous development in the last decades. Indeed, various fractional derivatives have been introduced in the literature to extend the traditional calculus and to provide more accurate descriptions of processes in physics, chemistry, and biology. From the analytic point of view, the theory of fractional-order integro-differential operators (along with their symmetry properties) has contributed decisively to the development of other areas in mathematics, including ordinary and partial differential equations, numerical analysis, calculus of variations, optimization theory, and symmetry analysis. From a more pragmatic point of view, some important applications of fractional calculus have been found in physics, biology, chemistry, and engineering, and new applications are frequently proposed to other areas of the sciences and to the development of new technology. Needless to mention that the speed at which this area expands is vertiginous.

The purpose of this Special Issue is to provide a means to communicate recent progresses in the field of fractional-order integro-differential operators and some of its most important symmetry properties. We invite researchers in this area to submit high-quality papers which stress the development of new computational and mathematical methods in fractional calculus. Applications of those methods to the analysis of the existence, uniqueness, symmetry, and regularity of the solutions of systems consisting of fractional integro-differential equations is an important topic considered in this work. The development and rigorous analysis of numerical methods to approximate solutions of systems of fractional-order equations is also a relevant topic in this Special Issue. Among others, optimization problems where the objective or the constraints are described in terms of fractional derivatives, the derivations of exact analytic solutions of systems of partial integro-differential equations, and the determination of new conservation and symmetry laws of these systems are all topics which are considered in this work. Likewise, the application of fractional calculus to modern problems in the sciences and engineering are problems which are covered in this special issue.

The potential topics include, but are not limited to:

  • Fractrional-order integro-differential operator theory;
  • Symmetry and conserved laws and properties;
  • Analysis of the solutions of integro-differential systems;
  • Development of new fractional integrals and derivatives, and their properties;
  • Special solutions of fractional-order systems;
  • Mathematical modeling through fractional integrals and derivatives;
  • Fractional-order variational calculus;
  • Numerical methods for integro-differential equations;
  • Stability and convergence analysis of methods for integro-differential equations;
  • Applications to the sciences and engineering.

Submit your paper and select the Journal “Symmetry” and the Special Issue “Symmetry in Computational and Mathematical Methods of Fractional Calculus” via: MDPI submission system. Our papers will be published on a rolling basis and we will be pleased to receive your submission once you have finished it.

Prof. Dr. Jorge Eduardo Macias-Diaz
Guest Editor

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

  • fractional calculus
  • computational methods
  • mathematical methods
  • mathematical modeling

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Published Papers (5 papers)

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Research

14 pages, 1842 KiB  
Article
Solving Fractional Gas Dynamics Equation Using Müntz–Legendre Polynomials
by Haifa Bin Jebreen and Carlo Cattani
Symmetry 2023, 15(11), 2076; https://doi.org/10.3390/sym15112076 - 16 Nov 2023
Viewed by 1007
Abstract
To solve the fractional gas dynamic equation, this paper presents an effective algorithm using the collocation method and Müntz-Legendre (M-L) polynomials. The approach chooses a solution of a finite-dimensional space that satisfies the desired equation at a set of collocation points. The collocation [...] Read more.
To solve the fractional gas dynamic equation, this paper presents an effective algorithm using the collocation method and Müntz-Legendre (M-L) polynomials. The approach chooses a solution of a finite-dimensional space that satisfies the desired equation at a set of collocation points. The collocation points in this study are selected to be uniformly spaced meshes or the roots of shifted Legendre and Chebyshev polynomials. Müntz-Legendre polynomials have the interesting property that their fractional derivative is also a Müntz-Legendre polynomial. This property ensures that these bases do not face the problems associated with using the classical orthogonal polynomials when solving fractional equations using the collocation method. The numerical simulations illustrate the method’s effectiveness and accuracy. Full article
(This article belongs to the Special Issue Symmetry in Computational and Mathematical Methods of Fractional Calculus)
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15 pages, 9274 KiB  
Article
The Implicit Numerical Method for the Radial Anomalous Subdiffusion Equation
by Marek Błasik
Symmetry 2023, 15(9), 1642; https://doi.org/10.3390/sym15091642 - 25 Aug 2023
Cited by 2 | Viewed by 1043
Abstract
This paper presents a numerical method for solving a two-dimensional subdiffusion equation with a Caputo fractional derivative. The problem considered assumes symmetry in both the equation’s solution domain and the boundary conditions, allowing for a reduction of the two-dimensional equation to a one-dimensional [...] Read more.
This paper presents a numerical method for solving a two-dimensional subdiffusion equation with a Caputo fractional derivative. The problem considered assumes symmetry in both the equation’s solution domain and the boundary conditions, allowing for a reduction of the two-dimensional equation to a one-dimensional one. The proposed method is an extension of the fractional Crank–Nicolson method, based on the discretization of the equivalent integral-differential equation. To validate the method, the obtained results were compared with a solution obtained through the Laplace transform. The analytical solution in the image of the Laplace transform was inverted using the Gaver–Wynn–Rho algorithm implemented in the specialized mathematical computing environment, Wolfram Mathematica. The results clearly show the mutual convergence of the solutions obtained via the two methods. Full article
(This article belongs to the Special Issue Symmetry in Computational and Mathematical Methods of Fractional Calculus)
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12 pages, 4614 KiB  
Article
A Symmetry Chaotic Model with Fractional Derivative Order via Two Different Methods
by Mohamed Elbadri, Mohamed A. Abdoon, Mohammed Berir and Dalal Khalid Almutairi
Symmetry 2023, 15(6), 1151; https://doi.org/10.3390/sym15061151 - 25 May 2023
Cited by 16 | Viewed by 1520
Abstract
In this article, we have investigated solutions to a symmetry chaotic system with fractional derivative order using two different methods—the numerical scheme for the ABC fractional derivative, and the Laplace decomposition method, with help from the MATLAB and Mathematica platforms. We have explored [...] Read more.
In this article, we have investigated solutions to a symmetry chaotic system with fractional derivative order using two different methods—the numerical scheme for the ABC fractional derivative, and the Laplace decomposition method, with help from the MATLAB and Mathematica platforms. We have explored progressive and efficient solutions to the chaotic model through the successful implementation of two mathematical methods. For the phase portrait of the model, the profiles of chaos are plotted by assigning values to the attached parameters. Hence, the offered techniques are relevant for advanced studies on other models. We believe that the unique techniques that have been proposed in this study will be applied in the future to build and simulate a wide range of fractional models, which can be used to address more challenging physics and engineering problems. Full article
(This article belongs to the Special Issue Symmetry in Computational and Mathematical Methods of Fractional Calculus)
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22 pages, 433 KiB  
Article
Existence Results for Nonlinear Coupled Hilfer Fractional Differential Equations with Nonlocal Riemann–Liouville and Hadamard-Type Iterated Integral Boundary Conditions
by Sunisa Theswan, Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon
Symmetry 2022, 14(9), 1948; https://doi.org/10.3390/sym14091948 - 19 Sep 2022
Cited by 6 | Viewed by 1769
Abstract
We introduce and study a new class of nonlinear coupled Hilfer differential equations with nonlocal boundary conditions involving Riemann–Liouville and Hadamard-type iterated fractional integral operators. By applying the Leray–Schauder alternative and Krasnosel’skiĭ’s fixed point theorem, two results presenting different criteria for the existence [...] Read more.
We introduce and study a new class of nonlinear coupled Hilfer differential equations with nonlocal boundary conditions involving Riemann–Liouville and Hadamard-type iterated fractional integral operators. By applying the Leray–Schauder alternative and Krasnosel’skiĭ’s fixed point theorem, two results presenting different criteria for the existence of solutions to the given problem are proven. The third result provides a sufficient criterion for the existence of a unique solution to the problem at hand. Numerical examples are constructed to demonstrate the application of the results obtained. Two graphs show asymmetric solutions when a Hilfer parameter is varied. The work presented in this paper is novel and significantly enriches the literature on the topic. Full article
(This article belongs to the Special Issue Symmetry in Computational and Mathematical Methods of Fractional Calculus)
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18 pages, 729 KiB  
Article
Some Fuzzy Inequalities for Harmonically s-Convex Fuzzy Number Valued Functions in the Second Sense Integral
by Jorge E. Macías-Díaz, Muhammad Bilal Khan, Hleil Alrweili and Mohamed S. Soliman
Symmetry 2022, 14(8), 1639; https://doi.org/10.3390/sym14081639 - 9 Aug 2022
Cited by 15 | Viewed by 1692
Abstract
Many fields of mathematics rely on convexity and nonconvexity, especially when studying optimization issues, where it stands out for a variety of practical aspects. Owing to the behavior of its definition, the idea of convexity also contributes significantly to the discussion of inequalities. [...] Read more.
Many fields of mathematics rely on convexity and nonconvexity, especially when studying optimization issues, where it stands out for a variety of practical aspects. Owing to the behavior of its definition, the idea of convexity also contributes significantly to the discussion of inequalities. The concepts of symmetry and convexity are related and we can apply this because of the close link that has grown between the two in recent years. In this study, harmonic convexity, also known as harmonic s-convexity for fuzzy number valued functions (F-NV-Fs), is defined in a more thorough manner. In this paper, we extend harmonically convex F-NV-Fs and demonstrate Hermite–Hadamard (H.H) and Hermite–Hadamard Fejér (H.H. Fejér) inequalities. The findings presented here are summaries of a variety of previously published studies. Full article
(This article belongs to the Special Issue Symmetry in Computational and Mathematical Methods of Fractional Calculus)
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