Analytical solutions for a Helmholtz equation with Dirichlet boundary conditions and arbitrary boundaries

The Helmholtz partial differential equation occurs in many areas of applied mathematics, with solutions required for a wide range of boundary geometries and boundary conditions. In this paper, an analytical series method is presented to solve the Dirichlet boundary value problem, for arbitrary boundary geometries. Previously, analytical series methods for the Helmholtz equation have been restricted to regular boundary geometries. However, recently the classical approach has been extended to cater for arbitrary boundary geometry, using the principle of eigenfunction expansions applied to nonorthogonal basis functions. The solution method is applied to a direct Helmholtz problem, and demonstrated on a two-dimensional example from transport in porous media.