Itogi Nauki i Tekhniki. Seriya "Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory"
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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 20-07-00407
20-51-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

Full text: PDF file (182 kB)
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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}


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