Construction and Study of Exact Solutions to A Nonlinear Heat Equation
- Authors: Kazakov A.L.1, Orlov S.S.1, Orlov S.S.2
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Affiliations:
- Matrosov Institute for System Dynamics and Control Theory of the Russian Academy of Sciences
- Irkutsk State University
- Issue: Vol 59, No 3 (2018)
- Pages: 427-441
- Section: Article
- URL: https://journals.rcsi.science/0037-4466/article/view/171856
- DOI: https://doi.org/10.1134/S0037446618030060
- ID: 171856
Cite item
Abstract
We construct and study exact solutions to a nonlinear second order parabolic equation which is usually called the “nonlinear heat equation” or “nonlinear filtration equation” in the Russian literature and the “porous medium equation” in other countries. Under examination is the special class of solutions having the form of a heat wave that propagates through cold (zero) background with finite velocity. The equation degenerates on the boundary of a heat wave (called the heat front) and its order decreases. The construction of these solutions by passing to an overdetermined system and analyzing its solvability reduces to integration of nonlinear ordinary differential equations of the second order with an initial condition such that the equations are not solvable with respect to the higher derivative. Some admissible families of heat fronts and the corresponding exact solutions to the problems in question are obtained. A detailed study of the global properties of solutions is carried out by the methods of the qualitative theory of differential equations and power geometry which are adapted for degenerate equations. The results are interpreted from the point of view of the behavior and properties of heat waves with a logarithmic front.
About the authors
A. L. Kazakov
Matrosov Institute for System Dynamics and Control Theory of the Russian Academy of Sciences
Author for correspondence.
Email: kazakov@icc.ru
Russian Federation, Irkutsk
Sv. S. Orlov
Matrosov Institute for System Dynamics and Control Theory of the Russian Academy of Sciences
Email: kazakov@icc.ru
Russian Federation, Irkutsk
S. S. Orlov
Irkutsk State University
Email: kazakov@icc.ru
Russian Federation, Irkutsk
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