Geodesic
On the construction of solutions to a problem with a free boundary for the non-linear heat equation
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 6, pp. 694-707.

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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, existence theorem.
Mots-clés : exact solution 
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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. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 6, pp. 694-707. http://geodesic.mathdoc.fr/item/JSFU_2020_13_6_a3/

[1] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, A.P. Mikhailov, Blow-up in quasilinear parabolic equations, Walter de Gruyte, Berlin–New York, 1995 | MR | Zbl

[2] J.L. Vazquez, The Porous Medium Equation: Mathematical Theory, Clarendon Press, Oxford, 2007 | MR

[3] V.K. Andreev, Yu.A. Gaponenko, O.N. Goncharova, V.V. Pukhnachev, Mathematical models of convection, De Gruyter Studies in Mathematical Physics, Berlin, 2012 | MR | Zbl

[4] M. Yu.Filimonov, L.G. Korzunin, A.F. Sidorov, “Approximate methods for solving nonlinear initial boundary-value problems based on special construction of series”, Rus. J. Numer. Anal. Math. Modelling, 8:2 (1993), 101–125 | DOI | MR | Zbl

[5] A.L. Kazakov, O.A. Nefedova, L.F. Spevak, “Solution of the problem of initiating the heat wave for a nonlinear heat conduction equation using the boundary element method”, Comp. Math. Math. Phys., 59:6 (2019), 1015–1029 | DOI | MR | Zbl

[6] P.K. Banerjee, R. Butterfield, Boundary Element Methods in Engineering Science, McGraw-Hill, London, 1981 | MR | Zbl

[7] M. Tanaka, T. Matsumoto, S. Takakuwa, “Dual reciprocity BEM for time-stepping approach to the transient heat conduction problem in nonlinear materials”, Comput. Methods Appl. Mech. Eng., 195 (2006), 4953–4961 | DOI | MR | Zbl

[8] Z.C. Li, M.G. Lee, H.T. Huang, J.Y. Chiang, “Neumann problems of 2D Laplace's equation by method of fundamental solutions”, Appl. Num. Math., 119 (2017), 126–145 | DOI | MR | Zbl

[9] Z.C. Li, J.Y. Chiang, H.T. Huang, M.G. Lee, “The interior field method for Laplace's equation in circular domains with circular holes”, Eng. Anal. Boundary Elem., 67 (2016), 173–185 | DOI | MR | Zbl

[10] M.G. Lee, Z.C. Li, L. Zhang, H.T. Huang, J.Y. Chiang, “Algorithm singularity of the null-field method for Dirichlet problems of Laplace's equation in annular and circular domains”, Eng. Anal. Boundary Elem., 41 (2014), 160–172 | DOI | MR | Zbl

[11] L.C. Wrobel, C.A. Brebbia, D. Nardini, “The dual reciprocity boundary element formulation for transient heat conduction”, Finite elements in water resources VI, Springer-Verlag, Berlin, 1986, 801–811 | MR

[12] L.V. Ovsiannikov, Group analysis of differential equations, Academic Press, New York–London, 1982 | MR | Zbl

[13] V.K. Andreev, O.V. Kaptsov, V.V. Pukhnachov, V.V. Rodionov, Applications of group-theoretical methods in hydrodynamics, Kluwer Academic Publishers, Dordrecht, 1998 | MR | Zbl

[14] A.D. Polyanin, V.F. Zaitsev, Handbook of nonlinear partial differential equations, Chapman and Hall/CRC, Boca Raton–London–New York, 2011 | MR

[15] V.V. Pukhnachev, “Exact multidimensional solutions of the nonlinear diffusion equation”, J. Appl. Mech. Technical Phys., 36:2 (1995), 169–176 | DOI | MR | Zbl

[16] A.L. Kazakov, “On exact solutions to a heat wave propagation boundary-value problem for a nonlinear heat equation”, Sib. Electronic Math. Reports, 16 (2019), 1057–1068 (in Russian) | DOI | Zbl

[17] A.L. Kazakov, P.A. Kuznetsov, A.A. Lempert, “On a Heat Wave for the Nonlinear Heat Equation: An Existence Theorem and Exact Solution”, Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov's Legacy, Springer, 2020, 223–228 | DOI

[18] A.L. Kazakov, L.F. Spevak, “Numerical and analytical studies of a nonlinear parabolic equation with boundary conditions of a special form”, Appl. Math. Model., 37 (2013), 6918–6928 | DOI | MR | Zbl

[19] A.L. Kazakov, Sv.S. Orlov, S.S. Orlov, “Sonstruction and study of exact solutions to a nonlinear heat equation”, Sib. Math. J., 59:3 (2018), 427–441 | DOI | MR | Zbl

[20] A.L. Kazakov, P.A. Kuznetsov, L.F. Spevak, “Analytical and numerical construction of heat wave type solutions to the nonlinear heat equation with a source”, Journal of Mathematical Sciences, 239:1 (2019), 111–122 | DOI | MR | Zbl