On the solution to a two-dimensional boundary value problem of heat conduction in a degenerating domain

Abstract

The article considers a homogeneous boundary-value problem for the heat equation in the non-cylindrical domain, namely, in an inverted pyramid with a vertex at the origin of coordinates, two faces of which lie in coordinate planes.A solution to the problem is sought in the form of a sum of generalized thermal potentials. There is a need to study the system of two Volterra integral equations of the second kind with singularities of the kernel. It is assumed that densities (heat intensity) depend only on a time variable, i.e. the density in each time section is considered constant. As a result, the system of integral equations is reduced to the homogeneous Volterra integral equation of the second kind. It is shown that this equation is uniquely solvable in the class of continuous functions.