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Symmetry
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Numerical Methods for Differential Problems and Symmetry
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Numerical Methods for Differential Problems and Symmetry

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

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 4310

Special Issue Editors

Dr. Youssri Youssri

E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
Interests: numerical analysis; special functions; fractional differential equations
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Anna Napoli

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, University of Calabria, Via Pietro Bucci, cubo 30/A, 87036 Rende (CS), Italy
Interests: polynomials and their applications in approximation theory; boundary value problems; numerical quadrature; zeros of functions
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Needless to say, many problems in several fields, such as medicine, biology, economics, finance, or engineering, can be described in terms of differential/integral equations or integrodifferential equations, which usually cannot be solved using known analytical methods. For this reason, numerical methods play a tremendous role in treating such problems.

Error analysis of numerical algorithms can judge the method's accuracy in the absence of exact solutions, which greatly helps in different fields of study in pure and applied mathematics.

This Special Issue intends to compile the recent advances in this area. Topics of interest include, but are not limited to: numerical methods for solving ordinary differential equations, partial differential equations, fractional differential equations, integral equations, and integrodifferential equations. In addition, their real-world applications are also especially welcome.

Submitted manuscripts should not have been published previously nor be under consideration for publication elsewhere. All manuscripts will be thoroughly refereed through a fair peer-review process. Researchers need to have their manuscripts checked for plagiarism before submission.

Dr. Youssri Youssri
Prof. Dr. Anna Napoli
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

  • integral and differential equations
  • fractional differential equations
  • partial differential equations
  • time–delay equations
  • deterministic and stochastic dynamics
  • finite difference algorithms
  • finite element algorithms
  • initial and boundary value problems
  • spectral methods
  • special functions and orthogonal polynomials

Published Papers (3 papers)

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Research

20 pages, 915 KiB  
Article
Application of Homotopy Analysis Transform Method for Solving a Fractional Singular One-Dimensional Thermo-Elasticity Coupled System
by Said Mesloub
Symmetry 2023, 15(10), 1952; https://doi.org/10.3390/sym15101952 - 22 Oct 2023
Viewed by 711
Abstract
This article extends the application of fractional-order time derivatives to replace their integer-order counterparts within a system comprising two singular one-dimensional coupled partial differential equations. The resulting model proves invaluable in representing radially symmetric deformation and temperature distribution within a unit disk. The [...] Read more.
This article extends the application of fractional-order time derivatives to replace their integer-order counterparts within a system comprising two singular one-dimensional coupled partial differential equations. The resulting model proves invaluable in representing radially symmetric deformation and temperature distribution within a unit disk. The incorporation of fractional-order derivatives in mathematical models is shown to significantly enhance their capacity for characterizing real-life phenomena in comparison to their integer-order counterparts. To address the studied system numerically, we employ the q-homotopy analysis transform method (q-HATM). We evaluate the efficiency of this method in solving the problem through a series of illustrative examples. The convergence of the derived scheme is assessed visually, and we compare the performance of the q-HATM with that of the Laplace decomposition method (LDM). While both methods excel in resolving the majority of the presented examples, a notable divergence arises in the final example: the numerical solutions obtained using q-HATM converge, whereas those derived from LDM exhibit divergence. This discrepancy underscores the remarkable efficiency of the q-HATM in addressing this specific problem. Full article
(This article belongs to the Special Issue Numerical Methods for Differential Problems and Symmetry)
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18 pages, 580 KiB  
Article
Numerical Investigation of Time-Fractional Phi-Four Equation via Novel Transform
by Nidhish Kumar Mishra, Mashael M. AlBaidani, Adnan Khan and Abdul Hamid Ganie
Symmetry 2023, 15(3), 687; https://doi.org/10.3390/sym15030687 - 09 Mar 2023
Cited by 16 | Viewed by 1254
Abstract
This paper examines two methods for solving the nonlinear fractional Phi-four problem with variable coefficients. One of the distinct states of the Klein–Gordon model yields the Phi-four equation. It is also used to simulate the kink and anti-kink solitary wave connections that have [...] Read more.
This paper examines two methods for solving the nonlinear fractional Phi-four problem with variable coefficients. One of the distinct states of the Klein–Gordon model yields the Phi-four equation. It is also used to simulate the kink and anti-kink solitary wave connections that have recently emerged in biological systems and nuclear particle physics. The approaches that are being suggested consist of the Yang transform, the homotopy perturbation approach, the decomposition approach, and the fractional derivative as stated by Caputo. The advantages of the proposed techniques are their capability of combining two dominant approaches for attaining precise and approximate solutions of nonlinear equations. It is important to keep in mind that the suggested methods can perform better in general as they need less computational effort than the alternative methods, while keeping a high level of numerical precision. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. Additionally, several simulations are used to show the physical behaviors of the found solutions with regard to fractional order. The article’s results possess complimentary properties that relate to the symmetry of partial differential equations. Full article
(This article belongs to the Special Issue Numerical Methods for Differential Problems and Symmetry)
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17 pages, 1895 KiB  
Article
A Tau Approach for Solving Time-Fractional Heat Equation Based on the Shifted Sixth-Kind Chebyshev Polynomials
by Esraa Magdy Abdelghany, Waleed Mohamed Abd-Elhameed, Galal Mahrous Moatimid, Youssri Hassan Youssri and Ahmed Gamal Atta
Symmetry 2023, 15(3), 594; https://doi.org/10.3390/sym15030594 - 25 Feb 2023
Cited by 15 | Viewed by 1365
Abstract
The time-fractional heat equation governed by nonlocal conditions is solved using a novel method developed in this study, which is based on the spectral tau method. There are two sets of basis functions used. The first set is the set of non-symmetric polynomials, [...] Read more.
The time-fractional heat equation governed by nonlocal conditions is solved using a novel method developed in this study, which is based on the spectral tau method. There are two sets of basis functions used. The first set is the set of non-symmetric polynomials, namely, the shifted Chebyshev polynomials of the sixth-kind (CPs6), and the second set is a set of modified shifted CPs6. The approximation of the solution is written as a product of the two chosen basis function sets. For this method, the key concept is to transform the problem governed by the underlying conditions into a set of linear algebraic equations that can be solved by means of an appropriate numerical scheme. The error analysis of the proposed extension is also thoroughly investigated. Finally, a number of examples are shown to illustrate the reliability and accuracy of the suggested tau method. Full article
(This article belongs to the Special Issue Numerical Methods for Differential Problems and Symmetry)
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