GENERAL BOUNDARY-VALUE PROBLEMS FOR THE HEAT CONDUCTION EQUATION WITH PIECEWISE-CONTINUOUS COEFFICIENTS

A constructive scheme for the construction of a solution of a mixed problem for the heat conduction equation with piecewise-continuous coeffi cients coordinate-dependent in the fi nal interval is suggested and validated in the present work. The boundary conditions are assumed to be most general. The scheme is based on: the reduction method, the concept of quasi-derivatives, the currently accepted theory of the systems of linear differential equations, the Fourier method, and the modifi ed method of eigenfunctions. The method based on this scheme should be related to direct exact methods of solving mixed problems that do not employ the procedures of constructing Green′s functions or integral transformations. Here the theorem of eigenfunction expansion is adapted for the case of coefficients that have discontinuity points of the 1st kind. The results obtained can be used, for example, in investigating the process of heat transfer in a multilayer slab under conditions of ideal thermal contact between the layers. A particular case of piecewise-continuous coeffi cients is considered. A numerical example of calculation of a temperature field in a real four-layer building slab under boundary conditions of the 3rd kind (conditions of convective heat transfer) that model the phenomenon of fi re near one of the external surfaces is given.
   
Author:  R. M. Tatsii and O. Yu. Pazen
Keywords:  reduction, quasi-derivative, Cauchy matrix, Fourier method, method of eigenfunctions
Page:  357