Analytical Solution of Nonlinear Parabolic Equations with Non-Stationary Exponent of Nonlinearity

We study the evolution of thermal perturbation in a non-linear medium whose thermal conduction coefficient explicitly depends on time and is a power function of temperature, with its exponent also being time-dependent if there is volume absorption of heat in this medium. The presence of lowest terms in the quasi-linear parabolic equation governing these transport processes affects the properties of the heat transport process in the medium under consideration. The analysis of physical properties of the studied process is conducted and its specific characteristics, i. e. the spatial localization mode, and its variations, i. e. stable and metastable localization, are examined. The effect of an instantaneous heat source on propagation of thermal perturbation in an isotropic medium is considered. The study introduces the conditions for existence of a closed-form solution of the Cauchy problem describing the process, as well as the location of the heat wave front that separates the perturbed and unperturbed zones at an arbitrary moment in time. The conditions for the spatial localization of these perturbations are indicated, i. e. the boundaries are predicted beyond which thermal perturbations from this source do not penetrate

The work was supported by the Russian Foundation for Basic Research (project no. 18-07-00269)

References

[1] Zeldovich Ya.B., Kompaneets A.S. [Towards a theory of heat propagation with heat conductivity depending on temperature]. Sb., posvyashchennyy 70-letiyu akad. A.F. Ioffe [Collection, dedica-ted to 70th anniversary of A.F. Ioffe]. Moscow, AN SSSR Publ., 1950, pp. 61–71 (in Russ.).

[2] Martinson L.K., Pavlov K.B. The problem of the three-dimensional localization of thermal perturbations in the theory of non-linear heat conduction. USSR Computational Mathematics and Mathematical Physics, 1972, vol. 12, iss. 4, pp. 261–268. DOI: 10.1016/0041-5553(72)90131-0

[3] Martinson L.K., Malov Yu.I. Differentsialnye uravneniya matematicheskoy fiziki [Differential equations of mathematical physics]. Moscow, Bauman MSTU Publ., 2011. 368 p.

[4] Samarskiy A.A., Galaktionov V.A., Kurdyumov S.P., Mikhaylov A.P. Rezhimy s obostreniem v zadachakh dlya kvazilineynykh parabolicheskikh uravneniy [Blow-up regimes in problems for quasi-linear parabolic equations]. Moscow, Nauka Publ., 1987. 480 p.

[5] Volosevich P.P., Levanov E.I. Avtomodelnye resheniya zadach gazovoy dinamiki i teplopere-nosa [Self-similar solutions of gas dynamics and heat transfer problems]. Moscow, MFTI Publ., 1997. 240 p.

[6] Akhromeeva T.S., Kurdyumov S.P., Malinetskiy G.G., Samarskiy A.A. Struktury i khaos v nelineynykh sredakh [Structure and chaos in non-linear mediums]. Moscow, Fizmatlit Publ., 2007. 488 p.

[7] Kudryashov N.A., Chmykhov N.A. Approximate solutions to one-dimensional nonlinear heat conduction problems with a given flux. Computational Mathematics and Mathematical Physics, 2007, vol. 47, iss. 1, pp. 107–117. DOI: 10.1134/S0965542507010113

[8] Formalev V.F., Rabinskii L.N. Wave heat transfer in anisotropic space with nonlinear characteristics. High Temperature, 2014, vol. 52, iss. 5, pp. 675–680. DOI: 10.1134/S0018151X14050058

[9] Formalev V.F., Kuznetsova E.L., Rabinskiy L.N. Localization of thermal disturbances in nonli-near anisotropic media with absorption. High Temperature, 2015, vol. 53, iss. 4, pp. 548–553. DOI: 10.1134/S0018151X15040100

[10] Granik I.S., Martinson L.K., Pavlov K.B. Temperature waves in moving media. USSR Computational Mathematics and Mathematical Physics, 1974, vol. 14, iss. 5, pp. 240–246. DOI: 10.1016/0041-5553(74)90213-4

[11] Granik I.S. The localization of temperature perturbations in media with volume absorption of heat. USSR Computational Mathematics and Mathematical Physics, vol. 18, iss. 3, 1978, pp. 246–252. DOI: 10.1016/0041-5553(78)90188-X

[12] Granik I.S., Martinson L.K. Heat wave front motion in non-linear medium with absorption. Inzh.-fiz. zhurnal, 1980, vol. 39, no. 4, pp. 728–731 (in Russ.).

[13] Attetkov A.V., Volkov I.K. Self-similar solution of heat transport problems in solid with heat-absorbing coating spherical hot spot. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2016, no. 4, pp. 97−106 (in Russ.). DOI: 10.18698/1812-3368-2016-4-97-106