7 - Boundary value problems in ellipsoidal geometry

Summary

Expansion of the fundamental solution

Since the ellipsoidal geometry governs natural processes which exhibit directional differentiation, it is obvious that many real-life problems have to be formulated in the framework of the ellipsoidal coordinate system. Furthermore, many problems of scientific and technological interest are postulated as boundary value problems in ellipsoidal domains. Consequently, it is very important to develop systematic techniques to handle these types of problems. Solving boundary value problems in an ellipsoidal environment is much harder than solving problems in a spherical one, and in many instances it is impossible to obtain an analytic solution in closed form. The difficulty with these problems is mainly due to the analytic computational part and not to the understanding of the underlying theory. Today, we do understand the fundamental structure of the theory of ellipsoidal harmonics, and some simple model problems can be solved exactly. These solutions offer a lot of mathematical and physical insight into many problems with anisotropic behavior. Combining these model solutions with the corresponding theory allows us to obtain enough information for the construction of effective hybrid methods, where the computational part can be left to programs of numerical or symbolic computations.

In the present chapter we collect the known basic tools needed to solve boundary value problems in fundamental domains with ellipsoidal boundaries. The fundamental solution of the Laplacian is a core topic in this chapter. One of the standard references for Green's functions is [28] as well as [59, 60, 266, 267].