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Symmetry
Special Issues
Numerical Analysis and Boundary Value Problems in Symmetry
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Numerical Analysis and Boundary Value Problems in Symmetry

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

Deadline for manuscript submissions: closed (31 January 2024) | Viewed by 10068

Special Issue Editors

Prof. Dr. Mina Abdel Malek

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Guest Editor
Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria, Egypt
Interests: nonlinear partial differential equations; integral equations; water waves; heat conduction; heat convection; sound waves; integral transforms
Prof. Dr. Mariano Torrisi

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Guest Editor
Dipartimento di Matematica e Informatica, University of Catania, Catania, Italy
Interests: group methods for nonlinear differential equations (both ODEs and PDEs); reduction techniques for the search of exact solutions of PDEs; applications of the group methods to reaction diffusion models, such as nonlinear governing equations modeling population dynamics and biomathematical problems; nonlinear diffusion and propagation of heat
Special Issues, Collections and Topics in MDPI journals
Dr. Rita Tracinà

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Guest Editor
Department of Mathematics and Computer Science, Catania University, 95125 Catania, Italy
Interests: symmetry methods; applications for differential equations
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Maria Luz Gandarias

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Guest Editor
Department of Mathematics, Cádiz University, 11003 Cádiz, Spain
Interests: symmetry methods; applications for differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

For nonlinear ordinary and partial differential equations, the general solution usually cannot be given explicitly. It is desirable to have an approach by which it can be determined whether a given nonlinear differential equation is Integrable. One of the powerful methods is the Lie symmetries which was created during the end of the 19th century by the prominent Norwegian mathematician Sophus Lie (1842–1899) who developed the method of their applications, his interesting theory and method have been continuously been in the focus of research of many well-known mathematicians, physicists and engineers. This Special Issue of the journal Symmetry is devoted to recent development of Lie theory for solving boundary value problems as well as it is required to draw the attention to the mathematical methods used in numerical analysis, such as special functions, orthogonal polynomials and their theoretical tools, such as Lie algebra, to study the concepts and properties of some special and advanced methods, which are useful in the description of solutions of linear and nonlinear differential equations.

Prof. Dr. Mina Abdel Malek
Prof. Dr. Mariano Torrisi
Dr. Rita Tracinà
Prof. Dr. Maria Luz Gandarias
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

  • Lie algebra/groups
  • representation of Lie algebra
  • nonlinear boundary value problems
  • symmetry of boundary value problems
  • invariant solutions
  • conditional symmetry
  • orthogonal polynomials

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Published Papers (3 papers)

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Research

17 pages, 297 KiB  
Article
Symmetries and Conservation Laws for a Class of Fourth-Order Reaction–Diffusion–Advection Equations
by Mariano Torrisi and Rita Tracinà
Symmetry 2023, 15(10), 1936; https://doi.org/10.3390/sym15101936 - 19 Oct 2023
Cited by 2 | Viewed by 1348
Abstract
We have studied a class of (1+1)-dimensional equations that models phenomena with heterogeneous diffusion, advection, and reaction. We have analyzed these fourth-order partial differential equations within the framework of group methods. In this class, the diffusion coefficient is [...] Read more.
We have studied a class of (1+1)-dimensional equations that models phenomena with heterogeneous diffusion, advection, and reaction. We have analyzed these fourth-order partial differential equations within the framework of group methods. In this class, the diffusion coefficient is constant, while the coefficients of advection and the reaction term are assumed to depend on the unknown density u(t,x). We have identified the Lie symmetries extending the Principal Algebra along with all the conservation laws corresponding to the different forms of the coefficients, and have derived several brief applications. Full article
(This article belongs to the Special Issue Numerical Analysis and Boundary Value Problems in Symmetry)
22 pages, 500 KiB  
Article
Modelling and Analysis of a Measles Epidemic Model with the Constant Proportional Caputo Operator
by Muhammad Farman, Aamir Shehzad, Ali Akgül, Dumitru Baleanu and Manuel De la Sen
Symmetry 2023, 15(2), 468; https://doi.org/10.3390/sym15020468 - 9 Feb 2023
Cited by 45 | Viewed by 6449
Abstract
Despite the existence of a secure and reliable immunization, measles, also known as rubeola, continues to be a leading cause of fatalities globally, especially in underdeveloped nations. For investigation and observation of the dynamical transmission of the disease with the influence of vaccination, [...] Read more.
Despite the existence of a secure and reliable immunization, measles, also known as rubeola, continues to be a leading cause of fatalities globally, especially in underdeveloped nations. For investigation and observation of the dynamical transmission of the disease with the influence of vaccination, we proposed a novel fractional order measles model with a constant proportional (CP) Caputo operator. We analysed the proposed model’s positivity, boundedness, well-posedness, and biological viability. Reproductive and strength numbers were also verified to examine how the illness dynamically behaves in society. For local and global stability analysis, we introduced the Lyapunov function with first and second derivatives. In order to evaluate the fractional integral operator, we used different techniques to invert the PC and CPC operators. We also used our suggested model’s fractional differential equations to derive the eigenfunctions of the CPC operator. There is a detailed discussion of additional analysis on the CPC and Hilfer generalised proportional operators. Employing the Laplace with the Adomian decomposition technique, we simulated a system of fractional differential equations numerically. Finally, numerical results and simulations were derived with the proposed measles model. The intricate and vital study of systems with symmetry is one of the many applications of contemporary fractional mathematical control. A strong tool that makes it possible to create numerical answers to a given fractional differential equation methodically is symmetry analysis. It is discovered that the proposed fractional order model provides a more realistic way of understanding the dynamics of a measles epidemic. Full article
(This article belongs to the Special Issue Numerical Analysis and Boundary Value Problems in Symmetry)
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19 pages, 1673 KiB  
Article
Bright Soliton Behaviours of Fractal Fractional Nonlinear Good Boussinesq Equation with Nonsingular Kernels
by Gulaly Sadiq, Amir Ali, Shabir Ahmad, Kamsing Nonlaopon and Ali Akgül
Symmetry 2022, 14(10), 2113; https://doi.org/10.3390/sym14102113 - 11 Oct 2022
Cited by 9 | Viewed by 1426
Abstract
In this manuscript, we investigate the nonlinear Boussinesq equation (BEQ) under fractal-fractional derivatives in the sense of the Caputo–Fabrizio and Atangana–Baleanu operators. We use the double modified Laplace transform (LT) method to determine the general series solution of the Boussinesq equation. We study [...] Read more.
In this manuscript, we investigate the nonlinear Boussinesq equation (BEQ) under fractal-fractional derivatives in the sense of the Caputo–Fabrizio and Atangana–Baleanu operators. We use the double modified Laplace transform (LT) method to determine the general series solution of the Boussinesq equation. We study the convergence, existence, uniqueness, boundedness, and stability of the solution of the considered good BEQ under the aforementioned derivatives. The obtained solutions are presented with numerical illustrations considering a particular example by two cases based on both derivatives with suitable initial conditions. The results are illustrated graphically where good agreements are obtained. Our results show that fractal-fractional derivatives are a very effective tool for studying nonlinear systems. Furthermore, when t increases, the solitary waves of the system oscillate. As the fractional order α or fractal dimension β increases, the soliton solutions become coherently close to the exact solution. For compactness, an error analysis is performed. The absolute error reveals an approximate linear evolution in the soliton solutions as time increases and that the system does not blow up nonlinearly. Full article
(This article belongs to the Special Issue Numerical Analysis and Boundary Value Problems in Symmetry)
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