A New Computer-Assisted Analytic Method for the Dirichlet and Neumann Problems

Abstract

The authors introduce a new method for the Dirichlet and Neumann problems (in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$) based on certain theorems of Zaremba and Bergmann. With the help of a symbolic computing language (FORMAC with PL/1), the authors show how it is possible to orthonormalize a complete sequence of harmonic polynomials on the domain $G$ of the solution. The solution is then given by an expansion in these orthonormalized polynomials. Under certain conditions on the boundary values, the expansion coefficients themselves may also be automatically calculated.