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 Trudy Inst. Mat. i Mekh. UrO RAN, 2016, Volume 22, Number 1, Pages 112–123 (Mi timm1265)

On some exact solutions of the nonlinear heat equation

A. L. Kazakov, S. S. Orlov

Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Irkutsk

Abstract: The paper is devoted to finding invariant solutions of the nonlinear heat (filter) equation without sources or sinks in the case of one spatial variable and a power dependence of the thermal conduction coefficient on the temperature. The construction procedure is reduced to Cauchy problems for ordinary differential equations with a singularity at the highest derivative. An existence and uniqueness theorem is proved for solutions of such problems in the class of analytic functions (in the form of a converging series). An estimate is obtained for the convergence domain of this series in one particular case.

Keywords: partial differential equations, nonlinear heat (filter) equation, invariant solution, Cauchy problem.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00608 À16-31-00291 ìîë_à

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Document Type: Article
UDC: 517.95

Citation: A. L. Kazakov, S. S. Orlov, “On some exact solutions of the nonlinear heat equation”, Trudy Inst. Mat. i Mekh. UrO RAN, 22, no. 1, 2016, 112–123

Citation in format AMSBIB
\Bibitem{KazOrl16} \by A.~L.~Kazakov, S.~S.~Orlov \paper On some exact solutions of the nonlinear heat equation \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2016 \vol 22 \issue 1 \pages 112--123 \mathnet{http://mi.mathnet.ru/timm1265} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3497189} \elib{http://elibrary.ru/item.asp?id=25655602} 

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This publication is cited in the following articles:
1. A. L. Kazakov, Sv. S. Orlov, S. S. Orlov, “Construction and study of exact solutions to a nonlinear heat equation”, Siberian Math. J., 59:3 (2018), 427–441
2. A. A. Kosov, È. I. Semenov, “Exact solutions of the nonlinear diffusion equation”, Siberian Math. J., 60:1 (2019), 93–107
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