Volume 13, no. 4Pages 35 - 49

Exact Solutions of the Nonlinear Heat Conduction Model

A.L. Kazakov, P.A. Kuznetsov
This paper continues a large series of our publications devoted to solutions of the nonlinear heat conduction equation. The solutions are heat waves that propagate over a zero background with a finite velocity. We study the problem on constructing exact solutions of the considered type for the nonlinear heat conduction equation with a source (sink) and determining their properties. A feature of such solutions is that the parabolic type of the equation is degenerate at the front of a heat wave, therefore, properties unusual for parabolic equations appear. We consider two types of solutions. The first one is a simple wave that moves at a constant speed and has the form of a solitary wave (soliton). The second one is a heat wave with an exponential law of front motion. In both cases, the construction is reduced to Cauchy problems for second-order ordinary differential equations (ODEs), which inherit the singularity from the original problem. We construct phase portraits of ODEs and establish the properties of trajectories passing through singular points. Also, we obtain the power series expansions of the required solutions and estimate their convergence radii.
Full text
nonlinear heat conduction equation; heat wave; exact solution; phase portrait; series.
1. Zel'dovich Ya.B., Raizer Yu.P. Physics of Shock Waves and High-Temperature Hydrodynamics Phenomena. N.Y., Dover Publications, 2002.
2. Barenblatt G.I., Entov V.M., Ryzhik V.M. Theory of Fluid Flows Through Natural Rocks. Dordrecht, Kluwer Academic Publishers, 1990. DOI: 10.1007/978-94-015-7899-8
3. Murray J. Mathematical Biology: I. An Introduction. N.Y., Springer, 2002. DOI: 10.1007/b98868
4. Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P. Blow-Up in Quasilinear Parabolic Equations. Berlin, Walter de Gruyte, 1995. DOI: 10.1515/9783110889864
5. Ladyzenskaja O.A., Solonnikov V.A., Ural'ceva N.N. Linear and Quasi-Linear Equations of Parabolic Type. Providence, American Mathematical Society, 1988.
6. Vazquez J.L. The Porous Medium Equation: Mathematical Theory. Oxford, Clarendon Press, 2007.
7. Sidorov A.F. Izbrannye trudy: Matematika. Mekhanika [Selected Works: Mathematics. Mechanics]. Moscow, Fizmatlit, 2001.
8. Zel'dovich Ya.B., Kompaneets A.S. [Towards a Theory of Heat Conduction with Thermal Conductivity Depending on the Temperature]. Sbornik, posvyashchennyy 70-letiyu A.F. Ioffe [Collection of Papers Dedicated to 70th Anniversary of A.F. Ioffe], 1950, Moscow, Izd. Akad. nauk SSSR, pp. 61-72. (in Russian)
9. Barenblatt G.I., Vishik M.I. [On Finite Velocity of Propagation in Problems of Non-Stationary Filtration of a Liquid or Gas]. Prikladnaya matematika i mekhanika, 1956, vol. 20, pp. 411-417. (in Russian)
10. Filimonov M. Application of Method of Special Series for Solution of Nonlinear Partial Differential Equations. AIP Conference Proceedings, 2014, vol. 40, pp. 218-223. DOI: 10.1063/1.4902479
11. Antontsev S.N., Shmarev S.I. Evolution PDEs with Nonstandard Growth Conditions. Existence, Uniqueness, Localization, Blow-Up. Paris, Atlantis Press, 2015. DOI: 10.2991/978-94-6239-112-3
12. Polyanin A.D., Zaitsev V.F. Handbook of Nonlinear Partial Differential Equations. Boca Raton, Chapman and Hall/CRC, 2011.
13. Kudryashov N.A., Sinelshchikov D.I. Analytical Solutions for Nonlinear Convection-Diffusion Equations with Nonlinear Sources. Automatic Control and Computer Sciences, 2017, vol. 51, no. 7, pp. 621-626. DOI: 10.3103/S0146411617070148
14. Kosov A.A., Semenov E.I. Exact Solutions of the Nonlinear Diffusion Equation. Siberian Mathematical Journal, 2019, vol. 60, no. 1, pp. 93-107. DOI: 10.1134/S0037446619010117
15. Kazakov A.L. On Exact Solutions to a Heat Wave Propagation Boundary-Value Problem for a Nonlinear Heat Equation. Siberian Electronic Mathematical Reports, 2019, vol. 16, pp. 1057-1068. (in Russian) DOI: 10.33048/semi.2019.16.073
16. Courant R. Methods of Mathematical Physics. Vol. II: Partial Differential Equations. N.Y., Interscience, 2008.
17. Kazakov A.L., Spevak L.F. Boundary Element Method and Power Series Method for One-Dimensional Non-Linear Filtration Problems. The Bulletin of Irkutsk State University. Series Mathematics, 2012, vol. 5, no. 2, pp. 2-17. (in Russian)
18. 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 mekhaniki UrO RAN, 2014, vol. 20, no. 1, pp. 119-129. (in Russian)
19. 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)
20. Kazakov A.L., Orlov S.S. On Some Exact Solutions of the Nonlinear Heat Equation. Trudy Instituta matematiki i mekhaniki UrO RAN, 2016, vol. 22, no. 1, pp. 112-123. (in Russian)
21. Kazakov A.L., Orlov S.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. DOI: 10.1134/S0037446618030060
22. Kazakov A.L. Construction and Investigation of Exact Solutions with Free Boundary to a Nonlinear Heat Equation with Source. Siberian Advances in Mathematics, 2020, vol. 30, no. 2, pp. 91-105. DOI: 10.3103/S1055134420020029
23. Bautin N.N., Leontovich E.A. Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti [Methods and Techniques of the Qualitative Investigation of Dynamical Systems in the Plane]. Moscow, Nauka, 1990. (in Russian)
24. Kazakov A.L., Kuznetsov P.A., Lempert A.A. 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, Cham, Springer, 2020, pp. 223-228. DOI: 10.1007/978-3-030-38870-6_29
25. Sidorov N.A., Sidorov D.N. Small Solutions of Nonlinear Differential Equations Near Branching Points. Russian Mathematics, 2011, vol. 55, no. 5, pp. 43-50. DOI: 10.3103/S1066369X11050070