Conformal mapping solution of Laplace's equation on a polygon with oblique derivative boundary conditions

摘要

We consider Laplace's equation in a polygonal domain together with the boundary conditions that along each side, the derivative in the direction at a specified oblique angle from the normal should be zero. First we prove that solutions to this problem can always be constructed by taking the real part of an analytic function that maps the domain onto another region with straight sides oriented according to the angles given in the boundary conditions. Then we show that this procedure can be carried out successfully in practice by the numerical calculation of Schwarz-Christoffel transformations. The method is illustrated by application to a Hall effect problem in electronics, and to a reflected Brownian motion problem motivated by queueing theory.