Two-Dimensional Boundary Value Problem of Heat Conduction in a Cone with Special Boundary Conditions

Abstract

We consider the boundary value problem of heat conduction in a domain that is an inverted cone, while the boundary conditions contain a derivative with respect to the time variable. 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 integral Volterra equation of the second kind, to which the original problem is reduced, are studied. Then we use the Carleman–Vekua regularization method to solve the resulting singular Volterra integral equation.