Special Issue "Symmetry Methods and Applications for Nonlinear Partial Differential Equations"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 1 September 2021.

Special Issue Editors

Prof. Dr. Maria Luz Gandarias
E-Mail Website
Guest Editor
Department of Mathematics, Cádiz University, 11003 Cádiz, Spain
Interests: symmetry methods; applications for differential equations
Special Issues and Collections in MDPI journals
Prof. Dr. Maria Santos Bruzón Gallego
E-Mail Website
Guest Editor
Department of Mathematics, University of Cádiz, 11510 Cádiz, Spain
Interests: group analysis; methods of group transformation: classical symmetries; nonclassical methods; direct methods and conservation laws applied to ordinary differential equations; partial differential equations and systems of partial differential equations
Prof. Dr. Rita Tracinà
E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Catania University, Catania, Italy
Interests: equivalence transformations and their differential invariants; symmetry classifications and exact solutions of PDEs; application of the group methods to diffusion models; conservation laws
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

Many real-world problems which arise in various scientific fields, such as economics, biology, physics, fluid dynamics, and engineering, are modeled by physically and mathematically interesting nonlinear differential partial equations (PDEs). To study the exact properties of such equations, symmetries and conservation laws are powerful tools that can provide explicit solutions, conserved quantities, transformations to simpler equations, tests of numerical schemes, and more.

The aim of this Special Issue is to focus on recent developments in symmetry analysis and conservation law analysis with applications to nonlinear PDEs of physical interest.

Other approaches in finding exact solutions to nonlinear differential equations will also be welcomed. High-quality papers that contain original research results are encouraged.

Prof. Dr. Maria Luz Gandarias
Prof. Dr. Maria Santos Bruzón Gallego
Prof. Dr. Rita Tracinà
Guest Editors

Manuscript Submission Information

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Keywords

  • Symmetry groups
  • Conservation laws
  • Partial differential equations

Published Papers (5 papers)

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Research

Article
Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions
Mathematics 2021, 9(9), 1009; https://doi.org/10.3390/math9091009 - 29 Apr 2021
Viewed by 238
Abstract
In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new [...] Read more.
In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively. Full article
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Article
Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV
Mathematics 2021, 9(9), 926; https://doi.org/10.3390/math9090926 - 22 Apr 2021
Viewed by 248
Abstract
We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal transformations. Lax [...] Read more.
We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal transformations. Lax pairs for RKdV and RmKdV are straightforwardly obtained by means of the aforementioned reciprocal transformations. We have also identified the classical Lie symmetries for the Lax pairs of RKdV and RmKdV. Non-trivial similarity reductions are computed and they yield non-autonomous ordinary differential equations (ODEs), whose Lax pairs are obtained as a consequence of the reductions. Full article
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Article
Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions
Mathematics 2021, 9(4), 345; https://doi.org/10.3390/math9040345 - 09 Feb 2021
Cited by 1 | Viewed by 369
Abstract
This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct [...] Read more.
This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,tτ), where τ>0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”. Full article
Article
Imaging Noise Suppression: Fourth-Order Partial Differential Equations and Travelling Wave Solutions
Mathematics 2020, 8(11), 2019; https://doi.org/10.3390/math8112019 - 12 Nov 2020
Viewed by 354
Abstract
In this paper, we discuss travelling wave solutions for image smoothing based on a fourth-order partial differential equation. One of the recurring issues of digital imaging is the amount of noise. One solution to this is to minimise the total variation norm of [...] Read more.
In this paper, we discuss travelling wave solutions for image smoothing based on a fourth-order partial differential equation. One of the recurring issues of digital imaging is the amount of noise. One solution to this is to minimise the total variation norm of the image, thus giving rise to non-linear equations. We investigate the variational properties of the Lagrange functionals associated with these minimisation problems. Full article
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Article
An Efficient Scheme for Time-Dependent Emden-Fowler Type Equations Based on Two-Dimensional Bernstein Polynomials
Mathematics 2020, 8(9), 1473; https://doi.org/10.3390/math8091473 - 01 Sep 2020
Cited by 1 | Viewed by 445
Abstract
In this study, we introduce an efficient computational method to obtain an approximate solution of the time-dependent Emden-Fowler type equations. The method is based on the 2D-Bernstein polynomials (2D-BPs) and their operational matrices. In the cases of time-dependent Lane–Emden type problems and wave-type [...] Read more.
In this study, we introduce an efficient computational method to obtain an approximate solution of the time-dependent Emden-Fowler type equations. The method is based on the 2D-Bernstein polynomials (2D-BPs) and their operational matrices. In the cases of time-dependent Lane–Emden type problems and wave-type equations which are the special cases of the problem, the method converts the problem to a linear system of algebraic equations. If the problem has a nonlinear part, the final system is nonlinear. We analyzed the error and give a theorem for the convergence. To estimate the error for the numerical solutions and then obtain more accurate approximate solutions, we give the residual correction procedure for the method. To show the effectiveness of the method, we apply the method to some test examples. The method gives more accurate results whenever increasing n,m for linear problems. For the nonlinear problems, the method also works well. For linear and nonlinear cases, the residual correction procedure estimates the error and yields the corrected approximations that give good approximation results. We compare the results with the results of the methods, the homotopy analysis method, homotopy perturbation method, Adomian decomposition method, and variational iteration method, on the nodes. Numerical results reveal that the method using 2D-BPs is more effective and simple for obtaining approximate solutions of the time-dependent Emden-Fowler type equations and the method presents a good accuracy. Full article
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