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Advances in Mathematical Physics
Volume 2018, Article ID 7683929, 13 pages
https://doi.org/10.1155/2018/7683929
Research Article

Image Theory for Neumann Functions in the Prolate Spheroidal Geometry

1School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, Jiangsu 224051, China
2Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA

Correspondence should be addressed to Shaozhong Deng; ude.ccnu@gnedoahs

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.