Analytical solutions of thermal conductivity boundary problems with free border

Аuthors

Kartashov E. M.^{1, 2*}, Krylov S. S.²

1. MIREA — Russian Technological University (Lomonosov Institute of Fine Chemical Technologies), 78, Vernadsky prospect, Moscow, 119454, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: professor.kartashov@qmail.com

Abstract

The author elaborated analytical approaches to solving the generalized boundary problems of non-stationary thermal conductivity with free boundary, which are often being encountered in various fields of science and engineering. Despite the widespread development of qualitative theory of differential equations in partial derivations, including boundary problems for the parabolic type equations with free boundary, analytical theory of this class of problems, consisting in the search for the exact solutions, has not yet evolved. The presented article develops two self-sufficient analytical approaches to the boundary problems of non- stationary thermal conductivity with free boundary. These are the method of the generalized integral transform in the area with the boundary moving in time and the differential series method. Despite different functional structures of one and the same problem solution the obtained solutions are equivalent, which is particularly effectively clarified, while considering classical Stefan’s problem and problems of the Stefan’s type. The article considered the number of inverse Stefan’s problems and proved the possibility of the incorrect problem reducing to the correct first boundary problem.

Keywords:

boundary value problems, free border, Stefan's problem, analytical solutions

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