The semi-analytical solution of the Fourier heat equation in beam – 3D inhomogeneous media interaction

The purpose of this paper is to discuss some solutions for the basic heat Fourier equation. It is well known that solutions to the heat equation can only be obtained in simple analytical form only when one is prepared to make a variety of assumptions regarding the spatial and temporal dependence of the heat source and the geometry of the sample. As the description of these boundary conditions becomes more and more rigorous in terms of the actual spatial and temporal dependence of the heat source, and the geometry of the sample, analytical and semi-analytical solutions of the heat equation can no longer be obtained. The main goal of the present paper is to establish how intricate the heat source, the sample description and the interaction between them can be still to get have a semi-analytical solution.