Separation of Variables Solution of PDE via Sinc Methods

Abstract

In their 1953 text of Morse and Feshbach, list 13 regions of when the three dimensional Laplace and Helmholtz partial differential equation (PDE), can be solved via use of separation of variables, i.e., via use of one–dimensional methods. They describe explicit transformations which make such solutions possible. In this paper we state precise assumptions on the PDE, its piecewise smooth curvilinear spacial boundary and the boundary conditions, i.e., assumptions of analyticity in each variable, which are satisfied, in essence, whenever calculus is used to model the PDE. Under these assumptions we are able to prove that the approximate solution of the PDE has similar analyticity properties. By combining this analyticity assumption with novel Sinc convolution methods, we are able to
solve the PDE to arbitrary uniform accuracy via use of a relatively small sequence of one dimensional matrix operations. The proofs of the above claims are lengthy, and we therefore present such proofs only for PDE in two dimensions. Proofs for the case of three dimensional will be published elsewhere.