Laplace's equation for a point source near a sphere: improved internal solution using spheroidal harmonics

TL;DR

This paper develops an improved internal solution for Laplace's equation near a sphere using spheroidal harmonics, achieving faster convergence and better singularity matching through a Kelvin transformation and irregular harmonics.

Contribution

It extends spheroidal harmonic methods to internal potentials with a Kelvin transformation, providing new relationships and faster converging solutions for point sources inside a sphere.

Findings

Faster convergence of internal solutions using spheroidal harmonics.

New relationships between solid spherical and spheroidal harmonics.

Enhanced matching of singularities in potential solutions.

Abstract

As shown recently [Phys. Rev. E 95, 033307 (2017)], spheroidal harmonics expansions are well suited for the external solution of Laplace's equation for a point source outside a spherical object. Their intrinsic singularity matches the line singularity of the analytic continuation of the solution and the series solution converges much faster than the standard spherical harmonic solution. Here we extend this approach to internal potentials using the Kelvin transformation, i.e. radial inversion, of the spheroidal coordinate system. This transform converts the standard series solution involving regular solid spherical harmonics into a series of irregular spherical harmonics. We then substitute the expansion of irregular spherical harmonics in terms of transformed irregular spheroidal harmonics into the potential. The spheroidal harmonic solution fits the image line singularity of the solution exactly and converges much faster. We also discuss why a solution in terms of regular solid spheroidal harmonics cannot work, even though these functions are finite everywhere in the sphere. We also present the analogous solution for an internal point source, and two new relationships between the solid spherical and spheroidal harmonics.