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Article

On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term

1
Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, 134 Lermontov St., Irkutsk 664033, Russia
2
Institute of Engineering Science, Ural Branch of the Russian Academy of Sciences, 34 Komsomolskaya St., Ekaterinburg 620049, Russia
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(6), 921; https://doi.org/10.3390/sym12060921
Received: 7 May 2020 / Revised: 20 May 2020 / Accepted: 28 May 2020 / Published: 2 June 2020
The paper deals with two-dimensional boundary-value problems for the degenerate nonlinear parabolic equation with a source term, which describes the process of heat conduction in the case of the power-law temperature dependence of the heat conductivity coefficient. We consider a heat wave propagation problem with a specified zero front in the case of two spatial variables. The solution existence and uniqueness theorem is proved in the class of analytic functions. The solution is constructed as a power series with coefficients to be calculated by a proposed constructive recurrent procedure. An algorithm based on the boundary element method using the dual reciprocity method is developed to solve the problem numerically. The efficiency of the application of the dual reciprocity method for various systems of radial basis functions is analyzed. An approach to constructing invariant solutions of the problem in the case of central symmetry is proposed. The constructed solutions are used to verify the developed numerical algorithm. The test calculations have shown the high efficiency of the algorithm. View Full-Text
Keywords: nonlinear parabolic PDE; heat wave; existence and uniqueness theorem; boundary element method; dual reciprocity method; moving boundary; exact solution nonlinear parabolic PDE; heat wave; existence and uniqueness theorem; boundary element method; dual reciprocity method; moving boundary; exact solution
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MDPI and ACS Style

Kazakov, A.; Spevak, L.; Nefedova, O.; Lempert, A. On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term. Symmetry 2020, 12, 921. https://doi.org/10.3390/sym12060921

AMA Style

Kazakov A, Spevak L, Nefedova O, Lempert A. On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term. Symmetry. 2020; 12(6):921. https://doi.org/10.3390/sym12060921

Chicago/Turabian Style

Kazakov, Alexander, Lev Spevak, Olga Nefedova, and Anna Lempert. 2020. "On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term" Symmetry 12, no. 6: 921. https://doi.org/10.3390/sym12060921

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