ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2018. Vol. 26

On a Three-Dimensional Heat Wave Generated by Boundary Condition Specified on a Time-Dependent Manifold

A. L. Kazakov, P. A. Kuznetsov, L. F. Spevak

The article is devoted to study the nonlinear heat equation (the porous medium equation) in the case of power nonlinearity. Three–dimensional problem of the initiation of a heat wave by boundary condition specified on a time–dependent manifold is considered. The wave has a finite velocity of propagation on the cold (zero) background. A new theorem of existence and uniqueness of the analytical solution (the main theorem) is proved. The solution is constructed in the form of a multiple power series with respect to independent variables. The coefficients of the series are computed recurrently by induction on the total order of differentiation: a system of algebraic equations of increasing dimension is solved at each step. The local convergence of the series is proved by majorant method using Cauchy–Kovalevskaya theorem. Thus, previously obtained results are generalize and reinforced which concern the solution of the problem of heat wave motion on the cold background. Besides, some particular cases are considered when the solution procedure can be reduced to the solution of a second order nonlinear ordinary differential equation unsolved with respect to the highest derivative. As the obtained ordinary differential equation can not be solved in quadratures, qualitative research is performed as well as the numerical experiments with the use of the boundary element method. The obtained results are interpreted with respect to the original problem of the heat wave motion.

About the Authors

Alexandr L. Kazakov, Dr. Sci. (Phys.–Math.), Chief Researcher, Matrosov Institute for System Dynamics and Control Theory of SB RAS, 134, Lermontov st., Irkutsk, 664033, Russian Federation; Institute of Engineering Science of UB RAS, 34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation, e-mail: kazakov@icc.ru

Pavel A. Kuznetsov, Cand. Sci. (Phys.–Math.), Junior Researcher, Matrosov Institute for System Dynamics and Control Theory of SB RAS, 134, Lermontov st., Irkutsk, 664033, Russian Federation; Assoc. Prof., Irkutsk State University,1, K. Marx st., Irkutsk, 664003, Russian Federation, e-mail: pav_ku@mail.ru

Lev F. Spevak, Cand. Sci. (Technics), Head of the Laboratory of Applied Mechanics, Institute of Engineering Science of UB RAS, 34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation, e-mail: lfs@imach.uran.ru

For citation

Kazakov A.L., Kuznetsov P.A., Spevak L.F. On a Three-Dimensional Heat Wave Generated by Boundary Condition Specified on a Time-Dependent Manifold. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 26, pp. 16-34. (In Russian) https://doi.org/10.26516/1997-7670.2018.26.16

nonlinear heat equation, existence theorem, invariant solution, boundary element method, numerical experiment
  1. Barenblatt G.I., Vishik M.I. O konechnoy skorosti rasprostraneniya v zadachakh nestatsyonarnoy fil’tratsii zhidkosti i gaza [On finite velocity of propagation in problems of non-stationary filtration of a liquid or gas]. Prikl. Matematika i Mekhanika, 1956, vol. 20, is. 3, pp. 411-417. (in Russian)
  2. Bautin N.N., Leontovich E.A. Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti [Methods and techniques of the qualitative study of dynamical systems on the plane]. Moscow, Nauka, 1990, 490 p. (in Russian)
  3. Zel’dovich Ya.B., Kompaneets A.S. K teorii rasprostraneniya tepla pri teploprovodnosti, zavisyashchey ot temperatury [Towards a theory of heat conduction with thermal conductivity depending on the temperature]. In Collection of Papers Dedicated to 70th Anniversary of A.F. Ioffe, Moscow, Akad. Nauk SSSR Publ., 1950, pp. 61-71. (in Russian)
  4. Kazakov A.L., Kuznetsov P.A., Spevak L.F. On a Degenerate Boundary Value Problem for the Porous Medium Equation in Spherical Coordinates. Trudy Instituta matematiki i mehaniki UrO RAN, 2014, vol. 20, no. 1, pp. 119-129. (in Russian)
  5. Kazakov A.L., Kuznetsov  P.A. On the Analytic Solutions of a Special Boundary Value Problem for a Nonlinear Heat Equation in Polar Coordinates. Journal of Applied and Industrial Mathematics, 2018, vol. 12, no. 2, pp. 1-11. https://doi.org/10.1134%2FS1990478918020060
  6. Kazakov A.L., Lempert A.A. Analytical and Numerical Investigation of a Nonlinear Filtration Boundary-Value Problem with Degeneration. Vychislitel’nye tehnologii, 2012, vol. 17, no. 1, pp. 57-68. (in Russian)
  7. Kazakov A.L., Lempert A.A. Existence and Uniqueness of the Solution of the Boundary–Value Problem for a Parabolic Equation of Unsteady Filtration. J. Appl. Mech. Tech. Phys., 2013, vol. 54, no. 2, pp. 251–258. https://doi.org/10.1134/S0021894413020107
  8. Kazakov A.L., Orlov Sv.S., Orlov S.S. Construction and Study of Exact Solutions to a Nonlinear Heat Equation. Siberian Mathematical Journal, 2018, vol. 59, no. 3, pp. 427-441. https://doi.org/10.1134%2FS0037446618030060
  9. Kazakov A.L., Spevak L.F. Boundary Elements Method and Power Series Method for One-Dimensional Nonlinear Filtration Problems. The Bulletin of Irkutsk State University. Series Mathematics, 2012, vol. 5, no. 2, pp. 2-17. (in Russian)
  10. Kudryashov N.A., Sinelshchikov D.I. Analytical Solutions for Nonlinear Convection–Diffusion Equations with Nonlinear Sources. Automatic Control and Computer Sciences, 2017, vol. 51, is. 7, pp. 621-626. https://doi.org/10.3103%2FS0146411617070148
  11. Kudryashov N.A., Chmykhov M.A. Approximate Solutions to One–Dimensional Nonlinear Heat Conduction Problems with a Given Flux. Computational Mathematics and Mathematical Physics, 2007, vol. 47, no. 1, pp. 107-117. https://doi.org/10.1134/S0965542507010113
  12. Kuznetsov P.A. On Boundary Value Problem with Degeneration for Nonlinear Porous Medium Equation with Boundary Conditions on the Closed Surface. The Bulletin of Irkutsk State University. Series Mathematics, 2014, vol. 9, pp. 61-74. (in Russian)
  13. Polyanin A.D., Zaitsev V.F., Zhurov A.I. Nelineynye uravneniya matematicheskoy fiziki i mekhaniki. Metody resheniya [Nonlinear equations of mathematical physics and mechanics. Methods of solution]. Moscow, Urait Publishing House, 2017, 256 p. (in Russian)
  14. Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P. Blow-up in Quasilinear Parabolic Equations. NY, Walter de Gruyte Berlin, 1995, 534 p.
  15. Sidorov A.F. Selected Works: Mathematics. Mechanics. Moscow, Fizmatlit Publ., 2001, 576 p. (in Russian)
  16. Banerjee P.К., Butterheld R. Boundary Element Methods in Engineering Science. UK, McGraw-Hill Book Company Limited, 1981, 494 p.
  17. Kazakov A.L., Spevak L.F. An Analytical and Numerical Study of a Nonlinear Parabolic Equation with Degeneration for the Cases of Circular and Spherical Symmetry. Applied Mathematical Modelling, 2016, vol. 40, iss. 2, pp. 1333-1343. https://doi.org/10.1016/j.apm.2015.06.038
  18. Vazquez J.L. The Porous Medium Equation: Mathematical Theory. Oxford, Clarendon Press, 2007, 648 p.

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