Functional constraints method for constructing exact solutions to delay reaction-diffusion equations and more complex nonlinear equations

Polyanin, Andrei D. and Zhurov, Alexei I. 2014. Functional constraints method for constructing exact solutions to delay reaction-diffusion equations and more complex nonlinear equations. Communications in Nonlinear Science and Numerical Simulation 19 (3) , pp. 417-430. 10.1016/j.cnsns.2013.07.017

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Abstract

We propose a new method for constructing exact solutions to nonlinear delay reaction–diffusion equations of the form ut=kuxx+F(u,w),ut=kuxx+F(u,w), where u=u(x,t),w=u(x,t-τ)u=u(x,t),w=u(x,t-τ), and ττ is the delay time. The method is based on searching for solutions in the form View the MathML sourceu=∑n=1Nξn(x)ηn(t), where the functions ξn(x)ξn(x) and ηn(t)ηn(t) are determined from additional functional constraints (which are difference or functional equations) and the original delay partial differential equation. All of the equations considered contain one or two arbitrary functions of a single argument. We describe a considerable number of new exact generalized separable solutions and a few more complex solutions representing a nonlinear superposition of generalized separable and traveling wave solutions. All solutions involve free parameters (in some cases, infinitely many parameters) and so can be suitable for solving certain problems and testing approximate analytical and numerical methods for nonlinear delay PDEs. The results are extended to a wide class of nonlinear partial differential–difference equations involving arbitrary linear differential operators of any order with respect to the independent variables x and t (in particular, this class includes the nonlinear delay Klein–Gordon equation) as well as to some partial functional differential equations with time-varying delay.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Schools > Medicine
Subjects:	R Medicine > R Medicine (General)
Publisher:	Elsevier
ISSN:	1007-5704
Date of Acceptance:	16 July 2013
Last Modified:	27 Mar 2019 14:33
URI:	https://orca.cardiff.ac.uk/id/eprint/79183

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