Construction of the solutions of boundary-value problems for the laplace equation in domains of revolution with edged boundary

Abstract

A method for the construction of high-precision approximate solutions of boundary-value problems for the Laplace equation in domains with corner points is proposed. We consider boundary-value problems for the three-dimensional Laplace equation in domains in the form of bodies of revolution whose meridional section has corner points. The solutions of the problems are constructed by using variational methods. For the numerical realization of these methods, we construct special solutions of the Laplace equation with singularities (or with singularities of their partial derivatives) on a certain ray originating at a corner point and directed outside the domain. To illustrate the proposed method, we construct the solutions of the Neumann problem and the problem of natural oscillations of ideal liquid in a spherical cavity.