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Symmetry
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Recent Advances in Symmetries Methods and Other Approaches to Nonlinear...
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Recent Advances in Symmetries Methods and Other Approaches to Nonlinear Differential Equations

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

Deadline for manuscript submissions: 31 October 2024 | Viewed by 5562

Special Issue Editor

Prof. Dr. Chaudry Masood Khalique

E-Mail Website
Guest Editor
International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, Mafikeng Campus, North-West University, Private Bag X2046, Mmabatho 2735, South Africa
Interests: nonlinear differential equations; Lie symmetry method; closed-form solutions; conservation laws; mathematical physics; analytical solution methods

Special Issue Information

Dear colleagues, 

It is well known that most real-world physical phenomena are modeled by nonlinear differential equations (NLDEs). Therefore, to better understand these physical phenomena, it is mandatory to determine closed-form solutions of NLDEs. The symmetry method, developed by Sophus Lie in the latter half of the nineteenth century, is one of the most powerful and efficient methods for finding closed-form solutions of NLDEs. Closed-form solutions may be used as benchmarks for testing numerical methods. On the other hand, over the years, researchers have developed various methods for finding closed-form solutions of NLDEs, for instance, the tanh method, the simplest equation method, the exp-function method, and Kudryashov’s method.

This Special Issue aims to collect a portion of recent developments in symmetry and other methods and their applications to NLDEs in diverse fields, such as the control theory, continuum mechanics, quantum mechanics, economics, numerical analysis, relativity, finance, biology, etc. Physicists, mathematicians, engineers, and other scientists working with NLDEs in their field of research are encouraged to submit their original research articles to this Special Issue.

Prof. Dr. Chaudry Masood Khalique
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 differential equations
  • Lie symmetry method
  • closed-form solutions
  • conservation laws
  • mathematical physics
  • analytical solution methods

Published Papers (5 papers)

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Research

15 pages, 1818 KiB  
Article
Solitary Solutions for the Stochastic Fokas System Found in Monomode Optical Fibers
by Wael W. Mohammed, Farah M. Al-Askar and Clemente Cesarano
Symmetry 2023, 15(7), 1433; https://doi.org/10.3390/sym15071433 - 17 Jul 2023
Cited by 10 | Viewed by 684
Abstract
The stochastic Fokas system (SFS), driven by multiplicative noise in the Itô sense, was investigated in this study. Novel trigonometric, rational, hyperbolic, and elliptic stochastic solutions are found using a modified mapping method. Because the Fokas system is used to explain nonlinear pulse [...] Read more.
The stochastic Fokas system (SFS), driven by multiplicative noise in the Itô sense, was investigated in this study. Novel trigonometric, rational, hyperbolic, and elliptic stochastic solutions are found using a modified mapping method. Because the Fokas system is used to explain nonlinear pulse propagation in monomode optical fibers, the solutions provided may be utilized to analyze a broad range of critical physical phenomena. In order to explain the impacts of multiplicative noise, the dynamic performances of the different found solutions are illustrated using 3D and 2D curves. We conclude that multiplicative noise eliminates the symmetry of the solutions of the SFS and stabilizes them. Full article
(This article belongs to the Special Issue Recent Advances in Symmetries Methods and Other Approaches to Nonlinear Differential Equations)
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35 pages, 561 KiB  
Article
Two-Fluid Classical and Momentumless Laminar Far Wakes
by Kiara Pillay and David Paul Mason
Symmetry 2023, 15(5), 961; https://doi.org/10.3390/sym15050961 - 23 Apr 2023
Viewed by 918
Abstract
Two-dimensional two-fluid classical and momentumless laminar far wakes are investigated in the boundary layer approximation. The velocity deficit satisfies a linear diffusion equation and the continuity equation in the upper and lower parts of the wakes. By using the multiplier method, conservation laws [...] Read more.
Two-dimensional two-fluid classical and momentumless laminar far wakes are investigated in the boundary layer approximation. The velocity deficit satisfies a linear diffusion equation and the continuity equation in the upper and lower parts of the wakes. By using the multiplier method, conservation laws for the system of partial differential equations (PDEs) in the upper and lower parts of the wake are derived. Lie point symmetries associated with the conserved vectors for the classical and momentumless wakes are obtained. The conserved quantity for the two-fluid classical wake is the total drag on the obstacle, which is rederived. A new conserved quantity for the two-fluid momentumless wake is obtained, which satisfies the condition that the total drag on the obstacle is zero. Using the conserved quantities, it is shown that the equation of the interface is y=kx12, where k is a constant and x and y are Cartesian coordinates with origin at the trailing edge of the obstacle. New invariant solutions for the two-fluid classical and momentumless wakes with k=0 are found. Both solutions depend on the dimensionless parameter χ=(ρ1μ1)/(ρ2μ2) where suffices 1 and 2 refer to the upper and lower parts of the wake. For the special case in which the kinematic viscosity ratio ν2/ν1=1, two further solutions for the two-fluid momentumless wake are derived with k=±6. Full article
(This article belongs to the Special Issue Recent Advances in Symmetries Methods and Other Approaches to Nonlinear Differential Equations)
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15 pages, 4461 KiB  
Article
Exact Solutions for the Generalized Atangana-Baleanu-Riemann Fractional (3 + 1)-Dimensional Kadomtsev–Petviashvili Equation
by Baojian Hong and Jinghan Wang
Symmetry 2023, 15(1), 3; https://doi.org/10.3390/sym15010003 - 20 Dec 2022
Cited by 4 | Viewed by 1156
Abstract
In this article, the generalized Jacobi elliptic function expansion method with four new Jacobi elliptic functions was used to the generalized fractional (3 + 1)-dimensional Kadomtsev–Petviashvili (GFKP) equation with the Atangana-Baleanu-Riemann fractional derivative, and abundant new types of analytical solutions to the GFKP [...] Read more.
In this article, the generalized Jacobi elliptic function expansion method with four new Jacobi elliptic functions was used to the generalized fractional (3 + 1)-dimensional Kadomtsev–Petviashvili (GFKP) equation with the Atangana-Baleanu-Riemann fractional derivative, and abundant new types of analytical solutions to the GFKP were obtained. It is well known that there is a tight connection between symmetry and travelling wave solutions. Most of the existing techniques to handle the PDEs for finding the exact solitary wave solutions are, in essence, a case of symmetry reduction, including nonclassical symmetry and Lie symmetries etc. Some 3D plots, 2D plots, and contour plots of these solutions were simulated to reveal the inner structure of the equation, which showed that the efficient method is sufficient to seek exact solutions of the nonlinear partial differential models arising in mathematical physics. Full article
(This article belongs to the Special Issue Recent Advances in Symmetries Methods and Other Approaches to Nonlinear Differential Equations)
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19 pages, 312 KiB  
Article
Well-Posedness for Nonlinear Parabolic Stochastic Differential Equations with Nonlinear Robin Conditions
by Mogtaba Mohammed
Symmetry 2022, 14(8), 1722; https://doi.org/10.3390/sym14081722 - 18 Aug 2022
Cited by 2 | Viewed by 1053
Abstract
In this paper, we present the existence and uniqueness of strong probabilistic solutions for nonlinear parabolic Stochastic Partial Differential Equations (SPDEs) with nonlinear Robin boundary conditions in a domain with holes. On the boundary of the holes, a nonlinear Robin condition is imposed, [...] Read more.
In this paper, we present the existence and uniqueness of strong probabilistic solutions for nonlinear parabolic Stochastic Partial Differential Equations (SPDEs) with nonlinear Robin boundary conditions in a domain with holes. On the boundary of the holes, a nonlinear Robin condition is imposed, while a homogeneous Dirichlet condition is prescribed on the exterior boundary. The coefficient matrix is assumed to be symmetric, while the nonlinear random forces are assumed to satisfy some types of regularities. We use Galerkin’s approximation method, probabilistic compactness results and some results from stochastic calculus. Full article
(This article belongs to the Special Issue Recent Advances in Symmetries Methods and Other Approaches to Nonlinear Differential Equations)
8 pages, 860 KiB  
Article
Dynamics and Exact Traveling Wave Solutions of the Sharma–Tasso–Olver–Burgers Equation
by Yan Zhou and Jinsen Zhuang
Symmetry 2022, 14(7), 1468; https://doi.org/10.3390/sym14071468 - 18 Jul 2022
Cited by 3 | Viewed by 1170
Abstract
In this paper, to study the Sharma–Tasso–Olver–Burgers equation, we focus on the geometric properties and the exact traveling wave solutions. The corresponding traveling system is a cubic oscillator with damping, and it has time-dependent and time-independent first integral. For all bounded orbits of [...] Read more.
In this paper, to study the Sharma–Tasso–Olver–Burgers equation, we focus on the geometric properties and the exact traveling wave solutions. The corresponding traveling system is a cubic oscillator with damping, and it has time-dependent and time-independent first integral. For all bounded orbits of the traveling system, we give the exact explicit kink wave solutions. Full article
(This article belongs to the Special Issue Recent Advances in Symmetries Methods and Other Approaches to Nonlinear Differential Equations)
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