The Analytical Solutions of the Dirichlet and Neumann Boundary Value Problems with Ellipsoidal Boundary

Abstract

The analytical solutions of the exterior boundary value problems of the disturbing potential with ellipsoidal boundary are derived with an accuracy of O(ε ⁴); for both the Dirichlet and the Neumann boundary value problems, where ε ² is the square of the second eccentricity of the ellipsoid. In addition, an arithmetic example is implemented to verify that the analytical solutions improve the accuracy from O(ε ²) to O(ε ⁴) compared to the spherical approximation. The solutions are given as integrals of closed-form, analytic Green’s functions and are particularly suited to local applications. The singularities of kernel functions for the Dirichlet and the Neumann boundary value problem are equivalent to the Poisson kernel and r ⁻¹ respectively.