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 Zh. Vychisl. Mat. Mat. Fiz., 2019, Volume 59, Number 6, Pages 1047–1062 (Mi zvmmf10914)

Solution of the problem of initiating the heat wave for a nonlinear heat conduction equation using the boundary element method

A. L. Kazakova, O. A. Nefedovab, L. F. Spevakb

a Matrosov Institute for System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Irkutsk, 664033 Russia
b Institute of Engineering Science, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 620049 Russia

Abstract: The paper is devoted to constructing approximate heat wave solutions propagating along the cold front at a finite speed for a nonlinear (quasi-linear) heat conduction equation with a power nonlinearity. The coefficient of the higher derivatives vanishes on the front of the heat wave, i.e., the equation degenerates. One- and two-dimensional problems about the initiation of a heat wave by the boundary mode specified on a given fixed manifold are studied. Algorithms for solving this problem based on the boundary element method and a special change of variables as a result of which the unknown function and the independent spatial variable exchange their roles are proposed. The solution of the transformed problem in the form of a converging power series is constructed. These algorithms are implemented in computer programs, and test computations are performed. Their results are compared with truncated power series mentioned above and with the known exact solutions; the results are in good agreement.

Key words: nonlinear heat conduction equation, heat wave, boundary element method, special series, numerical solution.

DOI: https://doi.org/10.1134/S0044466919060085

English version:
Computational Mathematics and Mathematical Physics, 2019, 59:6, 1015–1029

Bibliographic databases:

UDC: 751.958:519.633
Revised: 08.02.2019
Accepted:08.02.2019

Citation: A. L. Kazakov, O. A. Nefedova, L. F. Spevak, “Solution of the problem of initiating the heat wave for a nonlinear heat conduction equation using the boundary element method”, Zh. Vychisl. Mat. Mat. Fiz., 59:6 (2019), 1047–1062; Comput. Math. Math. Phys., 59:6 (2019), 1015–1029

Citation in format AMSBIB
\Bibitem{KazNefSpe19} \by A.~L.~Kazakov, O.~A.~Nefedova, L.~F.~Spevak \paper Solution of the problem of initiating the heat wave for a nonlinear heat conduction equation using the boundary element method \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2019 \vol 59 \issue 6 \pages 1047--1062 \mathnet{http://mi.mathnet.ru/zvmmf10914} \crossref{https://doi.org/10.1134/S0044466919060085} \elib{https://elibrary.ru/item.asp?id=37462921} \transl \jour Comput. Math. Math. Phys. \yr 2019 \vol 59 \issue 6 \pages 1015--1029 \crossref{https://doi.org/10.1134/S0965542519060083} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000473489900014} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85068540683} 

• http://mi.mathnet.ru/eng/zvmmf10914
• http://mi.mathnet.ru/eng/zvmmf/v59/i6/p1047

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Alexander L. Kazakov, Lev F. Spevak, Lee Ming-Gong, “On the construction of solutions to a problem with a free boundary for the non-linear heat equation”, Zhurn. SFU. Ser. Matem. i fiz., 13:6 (2020), 694–707
2. A. L. Kazakov, P. A. Kuznetsov, L. F. Spevak, “O resheniyakh tipa beguschei volny dlya nelineinogo uravneniya teploprovodnosti”, Differentsialnye uravneniya i optimalnoe upravlenie, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 196, VINITI RAN, M., 2021, 36–43