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Symmetry
Special Issues
Differential and Difference Equations and Symmetry
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Differential and Difference Equations and Symmetry

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

Deadline for manuscript submissions: closed (31 March 2020) | Viewed by 13082

Special Issue Editors

Prof. Dr. Vladimir Vasilyev

E-Mail Website
Guest Editor
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
Prof. Josef Diblik

E-Mail Website
Guest Editor
Brno University of Technology, Brno, Czech Republic
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Martin Bohner

E-Mail Website
Co-Guest Editor
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA
Interests: Hamiltonian systems; Sturm–Liouville equations; boundary value problems; difference equations; variational analysis; control theory; optimization; dynamical systems; oscillation; fractional differentiation equations; positivity; matrix analysis; eigenvalue problems; computational mathematics; time scales
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 B. Vasilyev
Prof. Josef Diblik
Prof. Martin Bohner
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

  • 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 (5 papers)

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Research

12 pages, 249 KiB  
Article
Operator Symbols and Operator Indices
by Vladimir Vasilyev
Symmetry 2020, 12(1), 64; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12010064 - 30 Dec 2019
Cited by 2 | Viewed by 2203
Abstract
We suggest a certain variant of symbolic calculus for special classes of linear bounded operators acting in Banach spaces. According to the calculus we formulate an index theorem and give applications to elliptic pseudo-differential operators on smooth manifolds with non-smooth boundaries. Full article
(This article belongs to the Special Issue Differential and Difference Equations and Symmetry)
13 pages, 289 KiB  
Article
Oscillation Criteria for Third Order Neutral Generalized Difference Equations with Distributed Delay
by P. Venkata Mohan Reddy, M. Maria Susai Manuel and Adem Kılıçman
Symmetry 2019, 11(12), 1501; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11121501 - 11 Dec 2019
Viewed by 1876
Abstract
This paper aims to investigate the criteria of behavior of a certain type of third order neutral generalized difference equations with distributed delay. With the technique of generalized Riccati transformation and Philos-type method, we obtain criteria to ensure convergence and oscillatory solutions and [...] Read more.
This paper aims to investigate the criteria of behavior of a certain type of third order neutral generalized difference equations with distributed delay. With the technique of generalized Riccati transformation and Philos-type method, we obtain criteria to ensure convergence and oscillatory solutions and suitable examples are provided to illustrate the main results. Full article
(This article belongs to the Special Issue Differential and Difference Equations and Symmetry)
8 pages, 246 KiB  
Article
Discrete Quantum Harmonic Oscillator
by Alina Dobrogowska and David J. Fernández C.
Symmetry 2019, 11(11), 1362; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11111362 - 03 Nov 2019
Cited by 1 | Viewed by 2855
Abstract
In this paper, we propose a discrete model for the quantum harmonic oscillator. The eigenfunctions and eigenvalues for the corresponding Schrödinger equation are obtained through the factorization method. It is shown that this problem is also connected with the equation for Meixner polynomials. [...] Read more.
In this paper, we propose a discrete model for the quantum harmonic oscillator. The eigenfunctions and eigenvalues for the corresponding Schrödinger equation are obtained through the factorization method. It is shown that this problem is also connected with the equation for Meixner polynomials. Full article
(This article belongs to the Special Issue Differential and Difference Equations and Symmetry)
14 pages, 313 KiB  
Article
A Dynamical System with Random Parameters as a Mathematical Model of Real Phenomena
by Josef Diblík, Irada Dzhalladova and Miroslava Růžičková
Symmetry 2019, 11(11), 1338; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11111338 - 30 Oct 2019
Cited by 1 | Viewed by 2002
Abstract
In many cases, it is difficult to find a solution to a system of difference equations with random structure in a closed form. Thus, a random process, which is the solution to such a system, can be described in another way, for example, [...] Read more.
In many cases, it is difficult to find a solution to a system of difference equations with random structure in a closed form. Thus, a random process, which is the solution to such a system, can be described in another way, for example, by its moments. In this paper, we consider systems of linear difference equations whose coefficients depend on a random Markov or semi-Markov chain with jumps. The moment equations are derived for such a system when the random structure is determined by a Markov chain with jumps. As an example, three processes: Threats to security in cyberspace, radiocarbon dating, and stability of the foreign currency exchange market are modelled by systems of difference equations with random parameters that depend on a semi-Markov or Markov process. The moment equations are used to obtain the conditions under which the processes are stable. Full article
(This article belongs to the Special Issue Differential and Difference Equations and Symmetry)
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18 pages, 1215 KiB  
Article
Numerical Simulation of Partial Differential Equations via Local Meshless Method
by Imtiaz Ahmad, Muhammad Riaz, Muhammad Ayaz, Muhammad Arif, Saeed Islam and Poom Kumam
Symmetry 2019, 11(2), 257; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11020257 - 19 Feb 2019
Cited by 21 | Viewed by 3520
Abstract
In this paper, numerical simulation of one, two and three dimensional partial differential equations (PDEs) are obtained by local meshless method using radial basis functions (RBFs). Both local and global meshless collocation procedures are used for spatial discretization, which convert the given PDEs [...] Read more.
In this paper, numerical simulation of one, two and three dimensional partial differential equations (PDEs) are obtained by local meshless method using radial basis functions (RBFs). Both local and global meshless collocation procedures are used for spatial discretization, which convert the given PDEs into a system of ODEs. Multiquadric, Gaussian and inverse quadratic RBFs are used for spatial discretization. The obtained system of ODEs has been solved by different time integrators. The salient feature of the local meshless method (LMM) is that it does not require mesh in the problem domain and also far less sensitive to the variation of shape parameter as compared to the global meshless method (GMM). Both rectangular and non rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy and ease implementation of the proposed method are shown via test problems. Full article
(This article belongs to the Special Issue Differential and Difference Equations and Symmetry)
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