Analytic Solution for the Dirichlet Problem in 2-D

A broad class of steady-state physical problems can be reduced to finding the harmonic functions that satisfy certain boundary conditions. A fundamental equation of applied mathematics is Laplace equation. This equation models important phenomena in engineering and physics, Laplace equation with satisfied boundary values is known as the Dirichlet problem. In this study, an alternative method is presented for the solution of the Dirichlet problem in a cut-ring region and the solution function of the problem is based on the Green function, and therefore on elliptic functions.