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

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 ellipsoida...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2012-03, Vol.35 (4), p.482-496
Main Authors: Dassios, George, Sten, Johan C.-E.
Format: Article
Language:English
Subjects:
ellipsoidal geometry
ellipsoidal harmonics
image system
Neumann boundary condition
Neumann function
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Summary: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. Copyright © 2012 John Wiley & Sons, Ltd.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.1595