Advances in Mathematical Physics

Volume 2018 (2018), Article ID 7683929, 13 pages

https://doi.org/10.1155/2018/7683929

## Image Theory for Neumann Functions in the Prolate Spheroidal Geometry

Correspondence should be addressed to Shaozhong Deng

Received 19 September 2017; Accepted 4 February 2018; Published 11 March 2018

Academic Editor: Pavel Kurasov

Copyright © 2018 Changfeng Xue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Interior and exterior Neumann functions for the Laplace operator are derived in terms of prolate spheroidal harmonics with the homogeneous, constant, and nonconstant inhomogeneous boundary conditions. For the interior Neumann functions, an image system is developed to consist of a point image, a line image extending from the point image to infinity along the radial coordinate curve, and a symmetric surface image on the confocal prolate spheroid that passes through the point image. On the other hand, for the exterior Neumann functions, an image system is developed to consist of a point image, a focal line image of uniform density, another line image extending from the point image to the focal line along the radial coordinate curve, and also a symmetric surface image on the confocal prolate spheroid that passes through the point image.

#### 1. Introduction

Let be a smooth closed surface in , with being its interior and its exterior, respectively. An interior Neumann function for the Laplace operator is the solution of the following boundary value problem for the potential :where is the Laplace operator, is a fixed point in the open domain , is the Dirac delta function, is the outward normal derivative on the surface , and is a given function specified on the surface satisfying the constraintThis constraint follows by applying the divergence theorem to (1a). If we further demand the normal derivative to be constant on , then we arrive at probably the most common boundary condition used in developing a Neumann-Green’s function [1]:where stands for the total surface area of .

The solutions of a Neumann problem when they exist, have the following integral representation (see [2], p.286-287):where stands for the* weighted* mean value of the solution on ; namely, , which can be chosen to be zero to simplify the equation. All other solutions to (4a) and (4b) can be obtained by adding an arbitrary constant to this solution.

Likely, an exterior Neumann function for the Laplace operator is the solution of the following boundary value problem for the potential : where now is a given point in the open domain .

Neumann functions are analogous to Green’s functions for Dirichlet problems, so they are often also called Green’s function for the Neumann problem or Green’s function of the second kind. While Dirichlet-Green’s functions are generally used for electrostatic problems where the potential is specified on bounding surfaces, Neumann-Green’s functions are often useful for finding temperature distributions where the bounding surfaces have specified currents. Similar to Dirichlet-Green’s functions, a Neumann-Green function can also be decomposed into a singular and a regular part as [2]where is a harmonic function that secures the satisfaction of the boundary conditions. In what follows, is also called the reflected part of the Neumann function.

An image system for a Neumann function is a system of fictitious sources inside the complement of the fundamental domain that produces a potential equal to the reflected part of the Neumann function. These fictitious sources are commonly called images because they are located not in the real domain of interest for the problem but in its complement. In general, such an image system is not unique. There are all types of images, from isolated point images to continuous distributions of images on lines, curves, and surfaces to combinations of these images. While image theories for Dirichlet-Green’s functions have been studied quite extensively in the literature [3–11], much less work has been done with image theories for Neumann-Green’s functions.

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 , (ii) a point image at the conventional Kelvin image point with strength , where is the radius of the sphere, and (iii) a uniform line image with strength on the line segment between the origin and the Kelvin image point . On the other hand, an image system for the interior spherical Neumann function may consist of (i) a point image at with strength and (ii) a line image that extends from radially to infinity with field-point dependent charge density.

In the present work, we shall study Neumann functions and their image systems for the Laplace operator in the prolate spheroidal geometry. 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. Section 2 provides a brief introduction to the prolate spheroidal coordinates and to the prolate spheroidal harmonics. Then, interior and exterior Neumann functions and their image systems for the Laplace operator in the prolate spheroidal geometry are developed in Sections 3 and 4, respectively. Finally, some concluding remarks are given in Section 5.

#### 2. Elements of Prolate Spheroidal Harmonics

Prolate spheroid is defined bywhere . Here, the focal symmetry axis of the spheroid is aligned with the -axis, and the interfocal distance is with ; see Figure 1.