Differential and Difference Equations and Symmetries

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

Deadline for manuscript submissions: closed (30 November 2021) | Viewed by 2156

Special Issue Editors

Department of Differential Equations, Belgorod State National Research University, Belgorod, Russia
Interests: singular integrals; pseudo-differential equations; boundary value problems; operator theory; Fourier analysis; computational mathematics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Differential and difference equations play an important role in many branches of mathematics, and they also often appear in other sciences. This fact leads us to more studying such equations and related boundary value problems in more detail, and a theory of solvability and (numerical) solutions for such equations are needed for distinct scientific groups.

Usually, one cannot find an exact solution for such equations, and one then needs to describe its qualitative properties in the appropriate functional spaces as well as to suggest a way of reducing the starting equation to a certain well known studied case, or to suggest some computational algorithm for the numerical solution. These studies are the intermediate points for solving equations.

There are a lot of methods for studying such problems in mathematics, as well as in the theory of differential and difference equations and boundary value problems. We hope this Issue will help mathematicians discover some new mathematical objects, approaches, and methods for their future works.

Symmetry ideas are often invisible in these studies, but they help us to decide on the right way to study them, and to show us the correct direction for future developments.

Prof. Vladimir Vasilyev
Prof. Josef Diblik
Guest Editors

Manuscript Submission Information

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Keywords

  • Ordinary differential equation
  • Partial differential equation
  • Difference equation
  • Symmetry
  • Pseudo-differential operator
  • Solvability
  • Numerical analysis
  • Approximation
  • Fredholm properties
  • Norm inequalities
  • A priori estimates
  • Stability
  • Asymptotic properties

Published Papers (1 paper)

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Research

16 pages, 619 KiB  
Article
Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations
by Yarong Xia, Ruoxia Yao, Xiangpeng Xin and Yan Li
Symmetry 2021, 13(7), 1268; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071268 - 15 Jul 2021
Cited by 1 | Viewed by 1412
Abstract
In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. [...] Read more.
In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions u and v. Full article
(This article belongs to the Special Issue Differential and Difference Equations and Symmetries)
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