Analytical Solutions for the Laplace Problem of an Eccentric Annular Domain

Boundary value problems including a source and no source for an eccentric annular domain are both revisited in this paper. Instead of using complex variables, the null-field boundary integral equation in conjunction with the degenerate kernel is employed to analytically solve the problem. Due to the geometry of the eccentric domain, the fundamental solution is expanded to the degenerate kernel under the bipolar coordinates. Once the degenerate kernel is found, the boundary integral equation is nothing more than a linear algebraic system. The solutions derived by using the present method have been compared with those done by using the complex variables. The potential field of the problems are plotted to show the validity of the present method.