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.
The study was funded by RFBR (research project no. 20-07-00407) and by RFBR and MOST (research project no. 20-51-S52003.
Received: 08.06.2020 Received in revised form: 14.07.2020 Accepted: 10.08.2020
Bibliographic databases:
Document Type:
Article
UDC:
517.958:519.633
Language: English
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
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\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.
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\issue 6
\pages 694--707
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\crossref{https://doi.org/10.17516/1997-1397-2020-13-6-694-707}
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Linking options:
https://www.mathnet.ru/eng/jsfu874
https://www.mathnet.ru/eng/jsfu/v13/i6/p694
This publication is cited in the following 2 articles:
Alexander Kazakov, Lev Spevak, “Constructing Exact and Approximate Diffusion Wave Solutions for a Quasilinear Parabolic Equation with Power Nonlinearities”, Mathematics, 10:9 (2022), 1559
Alexander Kazakov, Anna Lempert, “Diffusion-Wave Type Solutions to the Second-Order Evolutionary Equation with Power Nonlinearities”, Mathematics, 10:2 (2022), 232
Citation in format AMSBIB
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