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The Bulletin of Irkutsk State University. Series Mathematics, 2015, Volume 11, Pages 39–53 (Mi iigum216)  

This article is cited in 2 scientific papers (total in 2 papers)

On construction of heat wave for nonlinear heat equation in symmetrical case

A. L. Kazakova, P. A. Kuznetsovb, A. A. Lemperta

a Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, 134, Lermontov st., Irkutsk, 664033
b Irkutsk State University, 1, K. Marx st., Irkutsk, 664003

Abstract: The nonlinear second-order parabolic equation with two variables is considered in the article. Under the additional conditions, this equation can be interpreted as the nonlinear heat equation (the porous medium equation) in case of dependence of the unknown function on two variables (time and origin distance). The equation has many applications in continuum mechanics, in particular, it is used for mathematical modeling of filtration of ideal polytropic gas in porous media. The authors research a special class of solutions which are usually called a “heat wave” in literature. The special feature of these solutions is that they are “sewn” together of two continuously butt-joined solutions (trivial and nonnegative). The solution of heat wave's type can has derivative discontinuity on the line of joint which is called as the heat wave's front (the front of filtration), i.e. smoothness of the solution, generally speaking, is broken. The most natural problem which has the solutions of this kind is so-called “the Sakharov problem of the initiation of a heat wave”. New solutions of this problem in kind of multiple power series in physical variables were constructed in the article. The coefficients of the series are determined from tridiagonal systems of linear algebraic equations. Herewith, the elements of matrixes of systems depend on the order of the matrixes and the condition of the diagonal dominance is not executed. The recurrent formulas of the coefficients were obtained.

Keywords: partial differential equations, porous medium equation, heat wave, power series.

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UDC: 517.95

Citation: A. L. Kazakov, P. A. Kuznetsov, A. A. Lempert, “On construction of heat wave for nonlinear heat equation in symmetrical case”, The Bulletin of Irkutsk State University. Series Mathematics, 11 (2015), 39–53

Citation in format AMSBIB
\Bibitem{KazKuzLem15}
\by A.~L.~Kazakov, P.~A.~Kuznetsov, A.~A.~Lempert
\paper On construction of heat wave for nonlinear heat equation in symmetrical case
\jour The Bulletin of Irkutsk State University. Series Mathematics
\yr 2015
\vol 11
\pages 39--53
\mathnet{http://mi.mathnet.ru/iigum216}


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    This publication is cited in the following articles:
    1. A. L. Kazakov, L. F. Spevak, O. A. Nefedova, “Reshenie dvumernoi zadachi o dvizhenii fronta teplovoi volny s ispolzovaniem stepennykh ryadov i metoda granichnykh elementov”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 18 (2016), 21–37  mathnet
    2. A. L. Kazakov, P. A. Kuznetsov, “Ob analiticheskikh resheniyakh zadachi o dvizhenii teplovogo fronta dlya nelineinogo uravneniya teploprovodnosti s istochnikom”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 24 (2018), 37–50  mathnet  crossref
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