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 Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2021, Volume 196, Pages 36–43 (Mi into847)

On solutions of the traveling wave type for the nonlinear heat equation

A. L. Kazakova, P. A. Kuznetsova, L. F. Spevakb

a Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Irkutsk
b Institute of Engineering Science, Urals Branch, Russian Academy of Sciences, Ekaterinburg

Abstract: In this paper, we consider the problem of finding solutions to a nonlinear heat equation with a power-law nonlinearity, which have the form of a traveling wave and simulate the propagation of disturbances along a cold background with a finite speed. We show that the construction can be reduced to the Cauchy problem for a second-order ordinary differential equation with a singular coefficient of the highest derivative. For this Cauchy problem, the theorem on the existence and uniqueness of a smooth solution is proved. We develop an algorithm for constructing an approximate solution based on the boundary-element method and also present the results of computational experiments with numerical estimates of the parameters of the solution.

Keywords: nonlinear heat equation, exact solution, existence theorem, uniqueness theorem, series, convergence, boundary-element method

 Funding Agency Grant Number Russian Foundation for Basic Research 20-07-0040720-51-S52003 This work was supported by the Russian Foundation for Basic Research and the Ministry of Science and Technology of Taiwan (project Nos. 20-07-00407 and 20-51-S52003).

DOI: https://doi.org/10.36535/0233-6723-2021-196-36-43

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UDC: 517.95, 519.62
MSC: 35K65

Citation: A. L. Kazakov, P. A. Kuznetsov, L. F. Spevak, “On solutions of the traveling wave type for the nonlinear heat equation”, Differential Equations and Optimal Control, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 196, VINITI, Moscow, 2021, 36–43

Citation in format AMSBIB
\Bibitem{KazKuzSpe21} \by A.~L.~Kazakov, P.~A.~Kuznetsov, L.~F.~Spevak \paper On solutions of the traveling wave type for the nonlinear heat equation \inbook Differential Equations and Optimal Control \serial Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. \yr 2021 \vol 196 \pages 36--43 \publ VINITI \publaddr Moscow \mathnet{http://mi.mathnet.ru/into847} \crossref{https://doi.org/10.36535/0233-6723-2021-196-36-43} \elib{https://elibrary.ru/item.asp?id=46664221}