Open AccessUniqueness, existence and solution of the direct boundary heat exchange problem for a weakly non-linear temperature conductivity coefficient
Author(s) -
В. П. Танана,
B. A. Markov
Publication year - 2021
Publication title -
journal of physics. conference series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 85
eISSN - 1742-6596
pISSN - 1742-6588
DOI - 10.1088/1742-6596/1715/1/012050
Subject(s) - thermal diffusivity , thermal conduction , uniqueness , mathematics , work (physics) , thermal conductivity , heat equation , mathematical analysis , exact solutions in general relativity , discontinuity (linguistics) , thermodynamics , power series , series (stratigraphy) , boundary value problem , piecewise , physics , paleontology , biology
This work is a preliminary result necessary for the formulation and solution of the inverse heat conduction problem for a weakly quasilinear equation. Nevertheless, the article has independent significance. For the direct heat conduction problem a solution is constructed in the form of a series with a small parameter, the existence of the solution, its uniqueness are proved, and a number of estimates are made - in particular, the rate of decrease of the solution at large times. The solution is built for a medium, part of which has a piecewise constant coefficient of thermal conductivity, and part of the medium has slightly non-linear thermal diffusivity. The construction of a classical solution is impossible due to the discontinuity of the thermal diffusivity, therefore, on the sections of the media the matching conditions are set - the conditions for the continuity of the solution and the condition for the continuity of the heat flux. The solution to the weakly quasilinear equation is constructed in the explicit form of a series with a small parameter, for which uniform convergence is proved. The decreasing of the solution at large times is proved as of time raised to the negative third power.