Solution of the boundary value problem of heat conduction in a cone

Abstract

In the paper we consider the boundary value problem of heat conduction in a non-cylindrical domain, which is an inverted cone, i.e. in the domain degenerating into a point at the initial moment of time. In this case, the boundary conditions contain a derivative with respect to the time variable; in practice, problems of this kind arise in the presence of the condition of the concentrated heat capacity. We prove a theorem on the solvability of a boundary value problem in weighted spaces of essentially bounded functions. The issues of solvability of the singular Volterra integral equation of the second kind, to which the original problem is reduced, are studied. We use the Carleman-Vekua method of equivalent regularization to solve the obtained singular Volterra integral equation.

Inverse Spectral Problem for Sturm-Liouville Operator with Boundary and Jump Conditions Dependent on the Spectral Parameter

“…Hald has studied the discontinuous Sturm-Liouville problem and shown the direct and inverse spectral theory on the Sturm-Liouville problem with internal discontinuous point conditions [10]. The corresponding direct problems of boundary value problems with transmission conditions and/or eigenparameter-dependent boundary conditions, we refer to [20]- [25] and the references therein.…”