Solution of Dirichlet problem for a rectangular region in terms of elliptic functions

Abstract

In this study, the Green function of the (interior) Dirichlet problem for the Laplace (also Poisson) differential equation in a rectangular domain is expressed in terms of elliptic functions and the solution of the problem is based on the Green function and therefore on the elliptic functions. The method of solution for the Dirichlet problem by the Green function is presented; the Green function and transformation required for the solution of the Dirichlet problem in the rectangular region is found and the problem is solved in the rectangular region. An example for the problem in the rectangular region is given in order to present an application of the solution of Dirichlet problem. The equation is solved first by the known method of separation of variables and then in terms of elliptic functions; the results of both methods are compared. The results are found to be consistent but the advantage of this method is that the solution is obtained in terms of elementary functions.