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J. Sib. Fed. Univ. Math. Phys., 2020, Volume 13, Issue 6, Pages 694–707 (Mi jsfu874)  

On the construction of solutions to a problem with a free boundary for the non-linear heat equation

Alexander L. Kazakova, Lev F. Spevakb, Lee Ming-Gongc

a Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, Russian Federation
b Institute of Engineering Science, Ural Branch RAS, Ekaterinburg, Russian Federation
c Chung Hua University, Hsinchu City, Taiwan

Abstract: The construction of solutions to the problem with a free boundary for the non-linear heat equation which have the heat wave type is considered in the paper. The feature of such solutions is that the degeneration occurs on the front of the heat wave which separates the domain of positive values of the unknown function and the cold (zero) background. A numerical algorithm based on the boundary element method is proposed. Since it is difficult to prove the convergence of the algorithm due to the non-linearity of the problem and the presence of degeneracy the comparison with exact solutions is used to verify numerical results. The construction of exact solutions is reduced to integrating the Cauchy problem for ODE. A qualitative analysis of the exact solutions is carried out. Several computational experiments were performed to verify the proposed method.

Keywords: non-linear heat equation, heat wave, boundary element method, approximate solution, exact solution, existence theorem.

Funding Agency Grant Number
Russian Foundation for Basic Research 20-07-00407
20-51-S52003
The study was funded by RFBR (research project no. 20-07-00407) and by RFBR and MOST (research project no. 20-51-S52003.


DOI: https://doi.org/10.17516/1997-1397-2020-13-6-694-707

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Bibliographic databases:

UDC: 517.958:519.633
Received: 08.06.2020
Received in revised form: 14.07.2020
Accepted: 10.08.2020
Language:

Citation: 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”, J. Sib. Fed. Univ. Math. Phys., 13:6 (2020), 694–707

Citation in format AMSBIB
\Bibitem{KazSpeLee20}
\by Alexander~L.~Kazakov, Lev~F.~Spevak, Lee~Ming-Gong
\paper On the construction of solutions to a problem with a free boundary for the non-linear heat equation
\jour J. Sib. Fed. Univ. Math. Phys.
\yr 2020
\vol 13
\issue 6
\pages 694--707
\mathnet{http://mi.mathnet.ru/jsfu874}
\crossref{https://doi.org/10.17516/1997-1397-2020-13-6-694-707}
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