On the Neumann function and the method of images in spherical and ellipsoidal geometry

Abstract

The invention of an image system for a boundary value problem adds to a significant understanding of the structure of the problem, both at the mathematical and at the physical level. In this paper, the interior and exterior Neumann functions for the Laplacian in the cases of spherical and ellipsoidal domains are represented in terms of images. Besides isolated images, the presence of the normal derivative in the Neumann condition demands an additional continuous distribution of images, which in the spherical cases, can be restricted to a one-dimensional manifold, whereas for the ellipsoid, both a one-dimensional and a two-dimensional distribution of images is needed.

Image Theory for Neumann Functions in the Prolate Spheroidal Geometry

“…By reasoning as Dassios and Sten did in the case of the ellipsoidal geometry [12], here we consider an image system consisting of a point image with strength at some exterior point r k = ( k , 0, k ) = ( k , k , 0) (since the source is in the -plane, naturally the only point image should be in the -plane as well) as a continuous one-dimensional line image extending from the point r k to infinity along the radial coordinate curve : ( , ) = ( k , 0) with density…”

“…They are intercorrelated through the underdetermined system of (92) and (93), so there could be infinitely many different solutions, each giving us a different image system. Below we consider an approach that is again similar to that proposed by Dassios and Sten in [12] …”

“…In their interesting article [12], Dassios and Sten studied the Neumann function with the constant boundary condition and the corresponding image system in spherical and ellipsoidal geometry. For example, they have found that, in the spherical case, an image system for the exterior spherical Neumann function may consist of (i) a point image at the origin with strength −1, (ii) a point image at the conventional Kelvin image point r k = ( / s ) 2 r s with strength / s , where is the radius of the sphere, and (iii) a uniform line image with strength −1/ on the line segment between the origin and the Kelvin image point r k .…”

“…Although in theory the case to be considered here is a special case of that studied in [12], the authors still believe it is much beneficial to study it separately because the Neumann functions and their image systems in the ellipsoidal geometry have to be constructed using ellipsoidal harmonics, while those in the prolate spheroidal geometry can be constructed using prolate spheroidal harmonics, but the ellipsoidal harmonics are much more complicated to handle than the spheroidal ones. In addition, it seems that [12] contains several mistakes of omission and commission [13]. The present paper is organized as follows.…”

Green’s function and image system for the Laplace operator in the prolate spheroidal geometry

“…They have also constructed image systems for the exterior and interior Neumann functions for the ellipsoidal geometry. 15 In the present work, we shall give image systems for the exterior and interior Green's functions in the prolate spheroidal geometry. Although in theory the case to be considered here is a special case of the result obtained in Refs.…”