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Lie and Conditional Symmetries and Their Applications for Solving Nonlinear...
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Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models, II

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

Deadline for manuscript submissions: closed (30 April 2018) | Viewed by 24144

Special Issue Editors

Prof. Dr. Danny Arrigo

E-Mail Website
Guest Editor
Department of Mathematics, University of Central Arkansas, Conway, AR 72035, USA
Interests: symmetry analysis of nonlinear differential equations (both ODEs and PDEs) and techniques for the construction of exact solutions of PDEs; In particular, physically important equations, such as nonlinear heat equations and governing equations modeling of granular materials and nonlinear elasticity
Prof. Dr. Roman M. Cherniha

E-Mail Website
Guest Editor
Institute of Mathematics, National Academy of Sciences of Ukraine, 3, Tereshchenkivs'ka Street, 01004 Kyiv, Ukraine
Interests: non-linear pdes: lie and conditional symmetries, exact solutions and their properties; application of symmetry-based methods for analytical solving nonlinear initial and boundary value problems arising in mathematical physics and mathematical biology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue is a natural continuation of the previous one, “ Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models ”, which was very successful   https://www.mdpi.com/journal/symmetry/special_issues/Lie_Conditional_Symmetries

Nowadays, the most powerful methods for construction of exact solutions of nonlinear partial differential equations (PDEs) are symmetry based methods. These methods originated from the Lie method, which was created by the prominent Norwegian mathematician Sophus Lie in the 19th century. The method was essentially developed using modern mathematical language in the 1960s and 1970s. Although the technique of the Lie method is well-known, the method still attracts the attention of many researchers, and new results are published on a regular basis.

However, it is well-known that the Lie method is not efficient for solving PDEs with a “poor” Lie symmetry (i.e., their maximal algebra of invariance is trivial). Thus, other symmetry-based methods (conditional symmetry, weak symmetry, generalized conditional symmetry etc.) were developed during the last few decades. The best known among them is the method of nonclassical symmetries, proposed by G. Bluman and J. Cole in 1969. Nevertheless, this approach was suggested almost 50 years ago, its successful applications for solving nonlinear equations were accomplished only in the 1990s. Moreover, one may say that progress is still modest in applications of non-Lie methods to systems of PDEs and integro-differential equations, especially those arising in real world applications. Thus, this Special Issue welcomes articles devoted to these topics.

 Articles and reviews devoted to the theoretical foundations of symmetry-based methods and their applications for solving other nonlinear equations (especially reaction-diffusion-convection equations and higher-order PDEs) and nonlinear models (especially for biomedical applications) are also welcome.

Last but not the least, submissions devoted to different aspects of relations of symmetry with integrability and conservation laws  of a given nonlinear PDE are encouraged.

Prof. Dr. Danny Arrigo
Prof. Dr. Roman M. Cherniha
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.

Published Papers (8 papers)

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Research

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19 pages, 300 KiB  
Article
Second-Order Conditional Lie–Bäcklund Symmetries and Differential Constraints of Nonlinear Reaction–Diffusion Equations with Gradient-Dependent Diffusivity
by Lina Ji and Wei Feng
Symmetry 2018, 10(7), 267; https://doi.org/10.3390/sym10070267 - 07 Jul 2018
Cited by 2 | Viewed by 2232
Abstract
The radially symmetric nonlinear reaction–diffusion equation with gradient-dependent diffusivity is investigated. We obtain conditions under which the equations admit second-order conditional Lie–Bäcklund symmetries and first-order Hamilton–Jacobi sign-invariants which preserve both signs (≥0 and ≤0) on the solution manifold. The corresponding reductions of the [...] Read more.
The radially symmetric nonlinear reaction–diffusion equation with gradient-dependent diffusivity is investigated. We obtain conditions under which the equations admit second-order conditional Lie–Bäcklund symmetries and first-order Hamilton–Jacobi sign-invariants which preserve both signs (≥0 and ≤0) on the solution manifold. The corresponding reductions of the resulting equations are established due to the compatibility of the invariant surface conditions and the governing equations. Full article
(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models, II)
17 pages, 271 KiB  
Article
Some Approaches to the Calculation of Conservation Laws for a Telegraph System and Their Comparisons
by Eerdun Buhe, G.W. Bluman, Chen Alatancang and Hu Yulan
Symmetry 2018, 10(6), 182; https://doi.org/10.3390/sym10060182 - 24 May 2018
Cited by 3 | Viewed by 2549
Abstract
This paper applies the direct construction method, symmetry/adjoint symmetry pair method (SA method), symmetry action on a known conservation law method, Ibragimov’s conservation theorem (which always yields the same results as the SA method) and a recursion formula to calculate several conservation laws [...] Read more.
This paper applies the direct construction method, symmetry/adjoint symmetry pair method (SA method), symmetry action on a known conservation law method, Ibragimov’s conservation theorem (which always yields the same results as the SA method) and a recursion formula to calculate several conservation laws for nonlinear telegraph systems. In addition, a comparison is made between these methods for conservation laws admitted by nonlinear telegraph systems. Full article
(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models, II)
21 pages, 4034 KiB  
Article
Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model
by Roman Cherniha, Vasyl’ Davydovych and John R. King
Symmetry 2018, 10(5), 171; https://doi.org/10.3390/sym10050171 - 17 May 2018
Cited by 4 | Viewed by 2953
Abstract
A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other is not. A complete [...] Read more.
A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other is not. A complete Lie symmetry classification, including a number of the cases characterised as being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications (tumour growth). Graphical representations of the solutions are provided and a biological interpretation is briefly addressed. The results are generalised on multi-dimensional case under the assumption of the radially symmetrical shape of the tumour. Full article
(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models, II)
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7 pages, 241 KiB  
Article
Nonclassical Symmetries of a Power Law Harry Dym Equation
by Daniel J. Arrigo and Andrea N. Weaver
Symmetry 2018, 10(4), 100; https://doi.org/10.3390/sym10040100 - 06 Apr 2018
Cited by 2 | Viewed by 2760
Abstract
It is generally known that classical point and potential Lie symmetries of differential equations can be different. In a recent paper, we were able to show for a class of nonlinear diffusion equation that the nonclassical potential symmetries possess all nonclassical symmetries of [...] Read more.
It is generally known that classical point and potential Lie symmetries of differential equations can be different. In a recent paper, we were able to show for a class of nonlinear diffusion equation that the nonclassical potential symmetries possess all nonclassical symmetries of the original equation. We question whether this is true for the power law Harry Dym equation. In this paper, we show that the nonclassical symmetries of the power law Harry Dym equation and an equivalent system still possess special separate symmetries. However, we will show that we can extend the nonclassical method so that all nonclassical symmetries of the original power law Harry Dym equation can be obtained through the equivalent system. Full article
(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models, II)
9 pages, 226 KiB  
Article
Analytic Solutions of Nonlinear Partial Differential Equations by the Power Index Method
by Barbara Abraham-Shrauner
Symmetry 2018, 10(3), 76; https://doi.org/10.3390/sym10030076 - 19 Mar 2018
Cited by 3 | Viewed by 3315
Abstract
An updated Power Index Method is presented for nonlinear differential equations (NLPDEs) with the aim of reducing them to solutions by algebraic equations. The Lie symmetry, translation invariance of independent variables, allows for traveling waves. In addition discrete symmetries, reflection, or [...] Read more.
An updated Power Index Method is presented for nonlinear differential equations (NLPDEs) with the aim of reducing them to solutions by algebraic equations. The Lie symmetry, translation invariance of independent variables, allows for traveling waves. In addition discrete symmetries, reflection, or 180 ° rotation symmetry, are possible. The method tests whether certain hyperbolic or Jacobian elliptic functions are analytic solutions. The method consists of eight steps. The first six steps are quickly applied; conditions for algebraic equations are more complicated. A few exceptions to the Power Index Method are discussed. The method realizes an aim of Sophus Lie to find analytic solutions of nonlinear differential equations. Full article
(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models, II)
18 pages, 1258 KiB  
Article
Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion
by Philip Broadbridge, Dimetre Triadis, Dilruk Gallage and Pierluigi Cesana
Symmetry 2018, 10(3), 72; https://doi.org/10.3390/sym10030072 - 17 Mar 2018
Cited by 8 | Viewed by 3329
Abstract
Using the nonclassical symmetry of nonlinear reaction–diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen–Cahn–Hilliard equation. This models a phase field that gives a phenomenological description of a two-phase system near critical temperature. Solutions are given for the changing [...] Read more.
Using the nonclassical symmetry of nonlinear reaction–diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen–Cahn–Hilliard equation. This models a phase field that gives a phenomenological description of a two-phase system near critical temperature. Solutions are given for the changing phase of cylindrical or spherical inclusion, allowing for a “mushy” zone with a mixed state that is controlled by imposing a pure state at the boundary. The diffusion coefficients for transport of one phase through the mixture depend on the phase field value, since the physical structure of the mixture depends on the relative proportions of the two phases. A source term promotes stability of both of the pure phases but this tendency may be controlled or even reversed through the boundary conditions. Full article
(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models, II)
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15 pages, 364 KiB  
Article
Lie Symmetry of the Diffusive Lotka–Volterra System with Time-Dependent Coefficients
by Vasyl’ Davydovych
Symmetry 2018, 10(2), 41; https://doi.org/10.3390/sym10020041 - 03 Feb 2018
Cited by 1 | Viewed by 3174
Abstract
Lie symmetry classification of the diffusive Lotka–Volterra system with time-dependent coefficients in the case of a single space variable is studied. A set of such symmetries in an explicit form is constructed. A nontrivial ansatz reducing the Lotka–Volterra system with correctly-specified coefficients to [...] Read more.
Lie symmetry classification of the diffusive Lotka–Volterra system with time-dependent coefficients in the case of a single space variable is studied. A set of such symmetries in an explicit form is constructed. A nontrivial ansatz reducing the Lotka–Volterra system with correctly-specified coefficients to the system of ordinary differential equations (ODEs) and an example of the exact solution with a biological interpretation are found. Full article
(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models, II)
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Review

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33 pages, 4833 KiB  
Review
Lie and Q-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions
by Roman Cherniha, Mykola Serov and Oleksii Pliukhin
Symmetry 2018, 10(4), 123; https://doi.org/10.3390/sym10040123 - 20 Apr 2018
Cited by 4 | Viewed by 3129
Abstract
This review is devoted to search for Lie and Q-conditional (nonclassical) symmetries and exact solutions of a class of reaction-diffusion-convection equations with exponential nonlinearities. A complete Lie symmetry classification of the class is derived via two different algorithms in order to show [...] Read more.
This review is devoted to search for Lie and Q-conditional (nonclassical) symmetries and exact solutions of a class of reaction-diffusion-convection equations with exponential nonlinearities. A complete Lie symmetry classification of the class is derived via two different algorithms in order to show that the result depends essentially on the type of equivalence transformations used for the classification. Moreover, a complete description of Q-conditional symmetries for PDEs from the class in question is also presented. It is shown that all the well-known results for reaction-diffusion equations with exponential nonlinearities follow as particular cases from the results derived for this class of reaction-diffusion-convection equations. The symmetries obtained for constructing exact solutions of the relevant equations are successfully applied. The exact solutions are compared with those found by means of different techniques. Finally, an application of the exact solutions for solving boundary-value problems arising in population dynamics is presented. Full article
(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models, II)
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