Nonlinear Reaction–Diffusion Equations with Variable Coefficients: Method for Finding Exact Solutions in an Implicit Form
https://doi.org/10.1134/S2304487X1903009X
Abstract
Various classes of nonlinear reaction–diffusion equations with variable coefficients c(x)ut = [a(x)f(u)ux]x + b(x)g(u), which is based on the representation of the solution in the implicit form ∫ h(u)du = ξ(x)w(t) + η(x), x(u),where the functions h(u), ξ(x), and w(t) are determined during the study of arising functional differential equations. Examples of specific reaction–diffusion equations and their exact solutions are given. The main attention is paid to nonlinear equations of a sufficiently general form, which contain several arbitrary functions that depend on the desired function u and the spatial variable x. Many new generalized traveling-wave solutions and functional separable solutions are described. It is important to note that solutions of such types are usually noninvariant (i. e., they cannot be obtained by classical Lie group analysis of differential equations).
Keywords
nonlinear reaction–diffusion equations,
nonlinear equations with variable coefficients,
exact solutions in an implicit form,
generalized traveling-wave solutions,
functional separable solutions
Supporting agencies: The work was done on the topic of the state tasks (No. state registration AAAA-A17- 117021310385-6) and with partial financial support of the Russian Foundation for Fundamental research (project No. 18-29-10025)
About the Author
A. D. Polyanin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences; National Research Nuclear University MEPhI (Moscow Engineering Physics Institute); Bauman Moscow State Technical University
Russian Federation
Russian Federation
119526
115409
105005
Moscow
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Review
For citations:
Polyanin A.D. Nonlinear Reaction–Diffusion Equations with Variable Coefficients: Method for Finding Exact Solutions in an Implicit Form. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2019;8(4):321-334. (In Russ.) https://doi.org/10.1134/S2304487X1903009X
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