Interior solution of azimuthally symmetric case of Laplace equation in orthogonal similar oblate spheroidal coordinates

Abstract

Curvilinear coordinate systems distinct from the rectangular Cartesian coordinate system are particularly valuable in the field calculations as they facilitate the expression of boundary conditions of differential equations in a reasonably simple way when the coordinate surfaces fit the physical boundaries of the problem. The recently finalized orthogonal similar oblate spheroidal (SOS) coordinate system can be particularly useful for a physical processes description inside or in the vicinity of the bodies or particles with the geometry of an oblate spheroid. The solution of the azimuthally symmetric case of the Laplace equation was found for the interior space in the orthogonal SOS coordinates. In the frame of the derivation of the harmonic functions, the Laplace equation was separated by a special separation procedure. A generalized Legendre equation was introduced as the equation for the angular part of the separated Laplace equation. The harmonic functions were determined as relations involving generalized Legendre functions of the first and of the second kind. Several lower-degree functions are reported. Recursion formula facilitating determination of the higher-degree harmonic functions was found. The general solution of the azimuthally symmetric Laplace equation for the interior space in the SOS coordinates is reported.

Analytical solution of orthogonal similar oblate spheroidal coordinate system

“…(13) In the case when V can be separated to R-dependent part and W-dependent part, i.e. V=r(R)F(W), the Laplacian in the SOS coordinates in the azimuthally symmetric case can be rewritten with the help of product and chain rules for derivatives to the form (14) (deduce it from (B20) in the Supplement B of the interior-solution article [11] and from the steps leading to it). The Laplacian ( 14) is a sum of terms which are always a product of the Wdependent expression and the R-dependent expression.…”

“…The angular part of the Laplace equation expressed in s terms in the case of variables separation is (see [11], Eq. ( 68)) .…”

“…(20) In [11], this equation was solved for positive values of the separation constant K d (i.e. for the interior space) with a help of newly defined generalized Legendre functions.…”

“…It was shown that solution of (20) certainly exists when the separation constant K d is a non-negative integer n. Then, the function F(s) is proportional to the generalized Legendre function of the first kind, P n SI (s), or of the second kind, Q n SI (s). In [11], Eq.102, the complete interior solution of the Laplace equation ( 21) is recorded. a n and b n are arbitrary real coefficients, which can be determined for the particular boundary conditions.…”

“…Recently, the Laplace equation solution for the interior space was derived [11] in the SOS coordinates. Nevertheless, the found harmonic functions cannot be used or easily extended to the exterior space (i.e.…”