Constructing the fundamental solution to a problem of heat conduction

Abstract

In this article, we discuss auxiliary initial-boundary value problems which will subsequently be used to solve boundary-value problem of heat conduction with axial symmetry in a degenerating domain. One of the problems is posed with homogeneous boundary conditions in order to construct a fundamental solution that is used to determine thermal potentials. The initial condition contains the Dirac function. The solution to the problems is found explicitly using the Laplace integral transformation. The boundary value problem is also considered in the absence of axial symmetry. It is shown that this problem splits into families of boundary-value problems similar to the problems considered above. In conclusion, we state the boundary value problem of heat conduction with axial symmetry in a degenerating domain, and its fundamental solution, found above, is written out.