Methods for solving elliptic PDEs in spherical coordinates

摘要

A new method for investigating boundary value problems in two dimensions has recently been introduced by one of the authors. The main achievement of this method is that it yields explicit integral (as oppose to series) representations for a variety of boundary value problems. In addition, this method also provides an alternative, apparently simpler, approach for deriving those solution representations that are traditionally constructed by the method of images and of classical integral transforms. Here, we implement this latter approach to boundary value problems formulated in spherical coordinates. In particular, we do the following: (a) We derive the classical Poisson integral formula for the solutions of the Dirichlet problem for the Poisson equation in the interior of a sphere, the analogous formula for the Neumann problem, and the generalizations of these formulae in n dimensions. (b) We derive the solutions of various boundary value problems for the inhomogeneous Helmholtz equation in the interior of a sphere. (c) We solve the Dirichlet problem for the Laplace equation in the interior of a spherical sector.