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.
Highlights
Let S be a smooth closed surface in R3, with Ω being its interior and Ωc its exterior, respectively
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
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 a symmetric surface image on the confocal prolate spheroid that passes through the point image
Summary
Let S be a smooth closed surface in R3, with Ω being its interior and Ωc its exterior, respectively. An interior Neumann function for the Laplace operator is the solution of the following boundary value problem for the potential Ni(r, rs): ΔNi (r, rs) = δ (r − rs) , r ∈ Ω,. An exterior Neumann function for the Laplace operator is the solution of the following boundary value problem for the potential Ne(r, rs): ΔNe (r, rs) = δ (r − rs) , r ∈ Ωc,. 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.
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