The solution of the heat conduction problem for a rectangular area with mixed boundary conditions and internal source in general by using fast expansions method

Abstract

The fundamentals of the fast expansions method are outlined, provided that the mixed Dirichlet-Neumann boundary conditions are specified. Then, using the fast expansions method, the solution of the problem in an explicit analytical form with an accuracy of quadrature of Fourier coefficients is obtained. Such a solution allows to explore temperature fields in a rectangle, depending on the given input data of the task, which include the size of the rectangle, the boundary conditions and the internal source. The resulting solution for a rectangle can also be used to construct a solution to a problem in a curvilinear domain, for a two-phase Stefan problem with a curvilinear boundary, and for other applied problems.