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The Bulletin of Irkutsk State University. Series Mathematics, 2016, Volume 18, Pages 21–37 (Mi iigum276)  

Solution of a two-dimensionel problem on the motion of a heat wave front with the use of power series and the boundary element method

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

a Matrosov Institute for System Dynamics and Control Theory SB RAS, 134, Lermontov st., Irkutsk, 664033
b Institute of Engineering Science UB RAS, 34, Komsomolskaya st., Ekaterinburg, 620049

Abstract: The paper discusses a nonlinear parabolic equation describing the process of heat conduction for the case of the power dependence of the heat conductivity factor on temperature. Besides heat distribution in space, it describes filtration of a polytropic gas in a porous medium, whereupon, in the English-language literature, this equation is generally referred to as the porous medium equation. A distinctive feature of this equation is the degeneration of its parabolic type when the required function becomes zero, whereupon the equation acquires some properties typical of first-order equations. Particularly, in some cases, it proves possible to substantiate theorems of the existence and uniqueness of heat-wave (filtration-wave) type solutions for it. This paper proves a theorem of the existence and uniqueness of the solution to the problem of the motion of a heat wave with a specified front in the instance of two independent variables. At that, since the front has the form of a closed plane curve, a transition to the polar coordinate system is performed. The solution is constructed in the form of a series, a constructible recurrent procedure for calculating its coefficients being proposed. The series convergence is proved by the majorant method. A boundary-element-based computation algorithm in the form of a computer program has been developed and implemented to solve the problem under study. Test examples are considered, the calculations made by a program designed by the authors being compared with the truncated series. A good agreement of the obtained results has been established.

Keywords: nonlinear equations with partial derivatives, heat wave, power series, existence and uniqueness theorem, boundary element method, computational experiment.

Funding Agency Grant Number
Ural Branch of the Russian Academy of Sciences 15-7-1-17
Russian Foundation for Basic Research 16-01-00608_


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Document Type: Article
UDC: 517.958:519.633
MSC: 35K65

Citation: A. L. Kazakov, L. F. Spevak, O. A. Nefedova, “Solution of a two-dimensionel problem on the motion of a heat wave front with the use of power series and the boundary element method”, The Bulletin of Irkutsk State University. Series Mathematics, 18 (2016), 21–37

Citation in format AMSBIB
\Bibitem{KazSpeNef16}
\by A.~L.~Kazakov, L.~F.~Spevak, O.~A.~Nefedova
\paper Solution of a two-dimensionel problem on the motion of a heat wave front with the use of power series and the boundary element method
\jour The Bulletin of Irkutsk State University. Series Mathematics
\yr 2016
\vol 18
\pages 21--37
\mathnet{http://mi.mathnet.ru/iigum276}


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