On one method for solving transient heat conduction problems

I. V. Kudinov ; E. V. Stefanyuk ; M. P. Skvortsova ; E. V. Kotova ; G. M. Sinyaev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 2, pp. 342-353.

Résumé

Using additional boundary conditions and additional required function in integral method of heat-transfer we obtain approximate analytical solution of transient heat conduction problem for an infinite plate with asymmetric boundary conditions of the first kind. This solution has a simple form of trigonometric polynomial with coefficients exponentially stabilizing in time. With the increase in the count of terms of a polynomial the obtained solution is approaching the exact solution. The introduction of a time-dependent additional required function, setting in the one (point) of the boundary points, allows to reduce solving of differential equation in partial derivatives to integration of ordinary differential equation. The additional boundary conditions are found in the form that the required solution would implement the additional boundary conditions and that implementation would be equivalent to executing the original differential equation in boundary points. In this article it is noted that the execution of the original equation at the boundaries of the area only (via the implementation of the additional boundary conditions) leads to the execution of the original equation also inside that area. The absence of direct integration of the original equation on the spatial variable allows to apply this method to solving the nonlinear boundary value problems with variable initial conditions and variable physical properties of the environment, etc.

Détail

Keywords: transient heat conductivity, infinite speed of heat propagation, integral method of thermal balance, inhomogeneous boundary conditions, approximate analytical solution, additional required function, additional boundary conditions, trigonometric coordinate functions.

Bibliographie

[1] Kudinov V. A., Stefanyuk E. V., “A heat conduction problem by determining the temperature perturbation front”, Izvestiia RAN. Energetika, 2008, no. 5, 141–157 (In Russian)

[2] Lykov A. V., “Methods for solving nonlinear equations of nonstationary heat conduction”, Izvestiia AN SSSR. Energetika i transport, 1970, no. 5, 109–150 (In Russian) | Zbl

[3] Goodman T. R, “Application of integral methods to transient nonlinear heat transfer”, Advances in Heat Transfer, 1 (1964), 51–122 | DOI | Zbl

[4] Biot M. A., Variational Principles in Heat Transfer: A Unified Lagrangian Analysis of Dissipative Phenomena, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1970, x+185 pp. | Zbl

[5] Veinik A. I., Priblizhennyi raschet protsessov teploprovodnosti [An approximate calculation of heat conduction processes], Gosenergoizdat, Moscow, Leningrad, 1959, 184 pp. (In Russian)

[6] Shvets M. E., “The approximate solution of some problems of the hydrodynamics of the boundary layer”, Prikladnaia matematika i mekhanika, 13:3 (1949), 257–266 (In Russian)

[7] Timoshpol'skii V. I., Postol'nik Yu. S., Andrianov D. N., Teoreticheskie osnovy teplofiziki i termomekhaniki v metallurgii [Theoretical foundations of thermal physics and metallurgy thermomechanics], Belorusskaia nauka, Minsk, 2005, 560 pp. (In Russian)

[8] Glazunov Yu. T., Variatsionnye metody [Variation Methods], Reguliarnaia i khaoticheskaia dinamika; Institut komp'iuternykh issledovanii, Moscow, Izhevsk, 2006, 470 pp. (In Russian)

[9] Belyaev N. M., Ryadno A. A., Metody nestatsionarnoi teploprovodnosti [Methods of Transient Heat Conduction], Vyssh. shk., Moscow, 1978, 328 pp. (In Russian)

[10] Fedorov F. M., Granichnyi metod resheniia prikladnykh zadach matematicheskoi fiziki [The boundary method for solving applied problems of mathematical physics], Nauka, Novosibirsk, 2000, 220 pp. (In Russian) | MR | Zbl

[11] Kudryashev L. I., Men'shikh N. L., Priblizhennye resheniia nelineinykh zadach teploprovodnosti [Approximate solution of the nonlinear heat conduction problems], Mashinostroenie, Moscow, 1979, 232 pp. (In Russian)

[12] Kudinov V. A., Stefanyuk E. V., “Analytical solution method for heat conduction problems based on the introduction of the temperature perturbation front and additional boundary conditions”, Journal of Engineering Physics and Thermophysics, 82:3 (2009), 537–555 | DOI | MR

[13] Stefanyuk E. V., Kudinov V. A., “Approximate analytic solution of heat conduction problems with a mismatch between initial and boundary conditions”, Russian Math. (Iz. VUZ), 54:4 (2010), 55–61 | DOI | MR | Zbl

[14] Kudinov V. A., Kudinov I. V., Skvortsova M. P., “Generalized functions and additional boundary conditions in heat conduction problems for multilayered bodies”, Comput. Math. Math. Phys., 55:4 (2015), 666–676 | DOI | DOI | MR | Zbl

[15] Lykov A. V., Teoriia teploprovodnosti [Theory of heat conduction], Vyssh. shk., Moscow, 1967, 600 pp. (In Russian) | Zbl