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Symmetry
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Symmetries and Applications of ODE's and PDE's in Natural Sciences
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Symmetries and Applications of ODE's and PDE's in Natural Sciences

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 8609

Special Issue Editor

Dr. Jose M. Vega-Guzman

E-Mail Website
Guest Editor
Department of Mathematics, Lamar University, Beaumont, TX 77710, USA
Interests: evolution equations; nonlinear wave propagation; quantum mechanics; nonlinear optics; applications of ODEs and PDEs in natural sciences; mathematical modeling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Earth is full of waves, and equations have been a key element for their understanding. The sovereign role of Ordinary and Partial Equations in exploring wave phenomena around us is essential for the common welfare, and thus is of paramount importance. Both analytical and numerical results have been extremely important in advancing the fundamental knowledge of nonlinear waves within the natural sciences.

Differential equations have been proven to be essential in characterizing and analyzing wave dynamics. They have served as successful tools in understanding the rules that govern the dynamics that arise in physical, chemical, and biological systems. Thus, the study of their propagation is key for the advancement of applied mathematics in the natural sciences.

The aim of this Special Issues is to advance the mathematical knowledge of the dynamics of wave propagation arising in all interdisciplinary fields of the natural sciences such as Physics, Chemistry, Biology, Geosciences, and related branches.

Topics for this Special Issue cover a broad range not limited to the following: Symmetry/Asymmetry of Wave Propagation; Dynamics of Wave Phenomena; Conservation Laws; Qualitative Analysis of Wave Phenomena; Stability and Instability of Nonlinear Waves; Asymptotic Properties; Symmetry Properties of ODEs and PDEs in Natural Sciences.

Dr. Jose M. Vega-Guzman
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

  • nonlinear waves
  • nonlinear dynamics
  • natural sciences
  • stability, instability
  • symmetries
  • invariance
  • differential equations

Published Papers (6 papers)

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Research

16 pages, 541 KiB  
Article
Fractional Numerical Simulation of Coupled Approximate Long Wave and Modified Boussinesq Equations Involving Mittag-Leffler Kernel
by Aisha Abdullah Alderremy
Symmetry 2022, 14(8), 1632; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081632 - 08 Aug 2022
Viewed by 872
Abstract
This study examines approximate long wave and the modified Boussinesq equations, as well as their complexities with the Atangana–Baleanu fractional derivative operator in the Caputo sense. The analytical solution of the aforementioned model is discussed using the Elzaki transform and the Adomian decomposition [...] Read more.
This study examines approximate long wave and the modified Boussinesq equations, as well as their complexities with the Atangana–Baleanu fractional derivative operator in the Caputo sense. The analytical solution of the aforementioned model is discussed using the Elzaki transform and the Adomian decomposition method. These problems are indispensable for defining the characteristics of surface water waves by applying a particular relationship of dispersion. We used Elzaki transformation on time-fractional approximate long wave and modified Boussinesq equations, followed by inverse Elzaki transformation, to achieve the results of the equations. To validate the methodology, we concentrated on two systems and compared them to the actual solutions. The numerical and graphical results demonstrate that the proposed method is computationally precise and straightforward for investigating and resolving fractionally coupled nonlinear phenomena that occur in scientific and technological. Full article
(This article belongs to the Special Issue Symmetries and Applications of ODE's and PDE's in Natural Sciences)
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10 pages, 273 KiB  
Article
Analysis of the Fractional-Order Local Poisson Equation in Fractal Porous Media
by Manal Alqhtani, Khaled M. Saad, Rasool Shah, Wajaree Weera and Waleed M. Hamanah
Symmetry 2022, 14(7), 1323; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071323 - 27 Jun 2022
Cited by 15 | Viewed by 1505
Abstract
This paper investigates the fractional local Poisson equation using the homotopy perturbation transformation method. The Poisson equation discusses the potential area due to a provided charge with the possibility of area identified, and one can then determine the electrostatic or gravitational area in [...] Read more.
This paper investigates the fractional local Poisson equation using the homotopy perturbation transformation method. The Poisson equation discusses the potential area due to a provided charge with the possibility of area identified, and one can then determine the electrostatic or gravitational area in the fractal domain. Elliptic partial differential equations are frequently used in the modeling of electromagnetic mechanisms. The Poisson equation is investigated in this work in the context of a fractional local derivative. To deal with the fractional local Poisson equation, some illustrative problems are discussed. The solution shows the well-organized and straightforward nature of the homotopy perturbation transformation method to handle partial differential equations having fractional derivatives in the presence of a fractional local derivative. The solutions obtained by the defined methods reveal that the proposed system is simple to apply, and the computational cost is very reliable. The result of the fractional local Poisson equation yields attractive outcomes, and the Poisson equation with a fractional local derivative yields improved physical consequences. Full article
(This article belongs to the Special Issue Symmetries and Applications of ODE's and PDE's in Natural Sciences)
13 pages, 729 KiB  
Article
Analysis of Fractional-Order System of One-Dimensional Keller–Segel Equations: A Modified Analytical Method
by Humaira Yasmin and Naveed Iqbal
Symmetry 2022, 14(7), 1321; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071321 - 26 Jun 2022
Cited by 4 | Viewed by 1195
Abstract
In this paper, an analytical method is implemented to solve fractional-order Keller–Segel equations. The Yang transformation along with the Adomian decomposition method is implemented to obtain the solution of the given problems. The present method has an edge over other techniques as it [...] Read more.
In this paper, an analytical method is implemented to solve fractional-order Keller–Segel equations. The Yang transformation along with the Adomian decomposition method is implemented to obtain the solution of the given problems. The present method has an edge over other techniques as it does not need extra calculations and materials. The validity of the suggested technique is verified by considering some numerical problems. The results obtained confirm the better accuracy of the current technique. The suggested technique has a lesser number of calculations and is straightforward to apply and therefore can be applied to other fractional-order partial differential equations. Full article
(This article belongs to the Special Issue Symmetries and Applications of ODE's and PDE's in Natural Sciences)
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14 pages, 2991 KiB  
Article
Transient Axisymmetric Flows of Casson Fluids with Generalized Cattaneo’s Law over a Vertical Cylinder
by Husna Izzati Osman, Dumitru Vieru and Zulkhibri Ismail
Symmetry 2022, 14(7), 1319; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071319 - 26 Jun 2022
Cited by 5 | Viewed by 947
Abstract
Unsteady axial symmetric flows of an incompressible and electrically conducting Casson fluid over a vertical cylinder with time-variable temperature under the influence of an external transversely magnetic field are studied. The thermal transport is described by a generalized mathematical model based on the [...] Read more.
Unsteady axial symmetric flows of an incompressible and electrically conducting Casson fluid over a vertical cylinder with time-variable temperature under the influence of an external transversely magnetic field are studied. The thermal transport is described by a generalized mathematical model based on the time-fractional differential equation of Cattaneo’s law with the Caputo derivative. In this way, our model is able to highlight the effect of the temperature gradient history on heat transport and fluid motion. The generalized mathematical model of thermal transport can be particularized to obtain the classical Cattaneo’s law and the classical Fourier’s law. The comparison of the three models could offer the optimal model of heat transport. The problem solution has been determined in the general case when cylinder surface temperature is described by a function f(t); therefore, the obtained solutions can be used to study different convective flows over a cylinder. In the particular case of surface temperature varying exponentially in time, it is found that fractional models lead to a small temperature rise according to the Cattaneo model. Full article
(This article belongs to the Special Issue Symmetries and Applications of ODE's and PDE's in Natural Sciences)
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11 pages, 890 KiB  
Article
Analytical and Data-Driven Wave Approximations of an Extended Schrödinger Equation
by Rachel Klauss, Aaron Phillips and José M. Vega-Guzmán
Symmetry 2022, 14(3), 465; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14030465 - 25 Feb 2022
Cited by 1 | Viewed by 1443
Abstract
Using both analytical and numerical techniques, we discuss wave solutions within the framework of an extended nonlinear Schrödinger equation with constant coefficients equipped with spatiotemporal dispersion, self-steepening effects, and a Raman scattering term. We present the exact traveling wave solution of the system [...] Read more.
Using both analytical and numerical techniques, we discuss wave solutions within the framework of an extended nonlinear Schrödinger equation with constant coefficients equipped with spatiotemporal dispersion, self-steepening effects, and a Raman scattering term. We present the exact traveling wave solution of the system in terms of Jacobi elliptic functions and mention some symmetry results as they relate to the resulting ordinary differential equation. A constructed bright soliton solution serves as the base to compare a numerical solution of the system using spectral Fourier methods with a precise statistical low-rank approximation using a data-driven approach aided by the Koopman operator theory. We found that the spatiotemporal feature added to the model serves as a regularizing tool that enables a precise reconstruction of the original solution. Full article
(This article belongs to the Special Issue Symmetries and Applications of ODE's and PDE's in Natural Sciences)
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17 pages, 6526 KiB  
Article
Coupled Fractional Traveling Wave Solutions of the Extended Boussinesq–Whitham–Broer–Kaup-Type Equations with Variable Coefficients and Fractional Order
by Jin Hyuk Choi and Hyunsoo Kim
Symmetry 2021, 13(8), 1396; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13081396 - 01 Aug 2021
Cited by 4 | Viewed by 1429
Abstract
In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider the fractional BWBK equations, the fractional Whitham–Broer–Kaup (WBK) equations and the fractional Boussinesq equations with variable coefficients by setting proper smooth functions that are derived [...] Read more.
In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider the fractional BWBK equations, the fractional Whitham–Broer–Kaup (WBK) equations and the fractional Boussinesq equations with variable coefficients by setting proper smooth functions that are derived from the proposed equation. We obtain uniformly coupled fractional traveling wave solutions of the considered equations by employing the improved system method, and subsequently their asymmetric behaviors are visualized graphically. The result shows that the improved system method is effective and powerful to find explicit traveling wave solutions of the fractional nonlinear evolution equations. Full article
(This article belongs to the Special Issue Symmetries and Applications of ODE's and PDE's in Natural Sciences)
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