Special Issue "Conservation Laws and Symmetries of Differential Equations"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: 31 January 2020.

Special Issue Editors

Prof. Maria Luz Gandarias
E-Mail Website
Guest Editor
Department of Mathematics, Universidad de Cádiz, 11510 Puerto Real (Cádiz), Spain
Tel. 606176324
Interests: partial differential equations
Prof. Maria Santos Bruzón Gallego
E-Mail Website
Co-Guest Editor
University of Cádiz, Cádiz, Spain
Interests: group analysis; methods of group transformations: classical symmetries, nonclassical methods, direct methods and conservation laws applied to ordinary differential equations and partial differential equations
Prof. Rita Tracinà
E-Mail
Co-Guest Editor
Dipartimento di Matematica e Informatica, University of 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
Prof. Mariano Torrisi
E-Mail Website
Co-Guest Editor
Dipartimento di Matematica e Informatica, University of Catania, Italy
Interests: group methods for nonlinear differential equations (both ODEs and PDEs); reduction techniques for the search of exact solutions of PDEs; applications of the group methods to reaction diffusion models, such as nonlinear governing equations modeling population dynamics and biomathematical problems; nonlinear diffusion and propagation of heat
Special Issues and Collections in MDPI journals
Prof. Chaudry Masood Khalique
E-Mail Website
Co-Guest Editor
1. Affiliation: North-West University, Mafikeng Campus, South Africa
2. Affiliation: International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

Special Issue Information

Dear Colleagues,

Conservation laws play a vital role in the reduction and solution process of the differential equations.  It is well known that the integrability of the differential equations is strongly related to the existence of conservation laws. Conservation laws are used for existence, uniqueness and stability analysis and for the development of numerical methods. Recently, they have been applied to find exact solutions of certain partial differential equations.

Symmetry analysis for differential equations was developed by Sophus Lie in the latter half of the nineteenth century.  It systematically unifies and extends the well-known ad hoc methods to construct closed form solutions for differential equations, in particular for nonlinear differential equations. These methods are highly algorithmic and hence responsive to symbolic computation.

Conservation laws and symmetry analysis have applications to genuine physical systems of differential equations that are found in diverse fields as continuum mechanics, classical mechanics, quantum mechanics, relativity, numerical analysis, tumour growth, finance, and economics and so on.

The main aim of this Special Issue is to focus on some recent developments in methods and applications of conservation laws and symmetries of differential equations. Mathematicians, engineers, physicists and other scientists for whom differential equations are treasured research apparatuses are encouraged to submit their research to this special issue.

Prof. Maria Luz Gandarias
Prof. Chaudry Masood Khalique
Prof. Mariano Torrisi
Assist. Prof. Rita Tracinà
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 papers will be 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 1400 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

  • Differential equations
  • Conservation laws
  • Exact solutions
  • Symmetry

Published Papers (4 papers)

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Research

Open AccessArticle
First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach
Symmetry 2019, 11(10), 1244; https://doi.org/10.3390/sym11101244 - 04 Oct 2019
Abstract
The aim of the present work is to classify the Noether-like operators of two-dimensional physical systems whose dynamics is governed by a pair of Lane-Emden equations. Considering first-order Lagrangians for these systems, we construct corresponding first integrals. It is seen that for a [...] Read more.
The aim of the present work is to classify the Noether-like operators of two-dimensional physical systems whose dynamics is governed by a pair of Lane-Emden equations. Considering first-order Lagrangians for these systems, we construct corresponding first integrals. It is seen that for a number of forms of arbitrary functions appearing in the set of equations, the Noether-like operators also fulfill the classical Noether symmetry condition for the pairs of real Lagrangians and the generated first integrals are reminiscent of those we obtain from the complex Lagrangian approach. We also investigate the cases in which the underlying systems are reducible via quadrature. We derive some interesting results about the nonlinear systems under consideration and also find that the algebra of Noether-like operators is Abelian in a few cases. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
Open AccessArticle
Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method
Symmetry 2019, 11(3), 394; https://doi.org/10.3390/sym11030394 - 18 Mar 2019
Abstract
In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial [...] Read more.
In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial discretization, which convert the mentioned partial differential equations into a system of ordinary differential equations. The obtained system has been solved by the forward Euler difference formula. An upwind technique is utilized in the case of the convection-dominated coupled Burgers’ model equation. Having no need for the mesh in the problem domain and being less sensitive to the variation of the shape parameter as compared to global meshless methods are the salient features of the local meshless method. Both rectangular and non-rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy, and the ease of implementation of the proposed method are shown via test problems. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
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Open AccessArticle
Nonlocal Symmetries for Time-Dependent Order Differential Equations
Symmetry 2018, 10(12), 771; https://doi.org/10.3390/sym10120771 - 19 Dec 2018
Abstract
A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order [...] Read more.
A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
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Open AccessArticle
New Stability Criterion for the Dissipative Linear System and Analysis of Bresse System
Symmetry 2018, 10(11), 542; https://doi.org/10.3390/sym10110542 - 25 Oct 2018
Cited by 2
Abstract
In this article, we introduce a new approach to obtain the property of the dissipative structure for a system of differential equations. If the system has a viscosity or relaxation term which possesses symmetric property, Shizuta and Kawashima in 1985 introduced the suitable [...] Read more.
In this article, we introduce a new approach to obtain the property of the dissipative structure for a system of differential equations. If the system has a viscosity or relaxation term which possesses symmetric property, Shizuta and Kawashima in 1985 introduced the suitable stability condition called in this article Classical Stability Condition for the corresponding eigenvalue problem of the system, and derived the detailed relation between the coefficient matrices of the system and the eigenvalues. However, there are some complicated physical models which possess a non-symmetric viscosity or relaxation term and we cannot apply Classical Stability Condition to these models. Under this situation, our purpose in this article is to extend Classical Stability Condition for complicated models and to make the relation between the coefficient matrices and the corresponding eigenvalues clear. Furthermore, we shall explain the new dissipative structure through the several concrete examples. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
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