Separation of variables in an asymmetric cyclidic coordinate system

TL;DR

This paper introduces a new asymmetric cyclidic coordinate system for solving Laplace's equation in 3D space, utilizing separation of variables and multiparameter spectral theory to analyze solutions and solve boundary value problems.

Contribution

It presents a novel five-cyclide coordinate system derived from stereographic projection, enabling separation of variables for Laplace's equation in complex geometries.

Findings

Solutions expressed via Fuchsian ODEs with five singularities

Global harmonic solutions constructed using multiparameter spectral theory

Dirichlet problem solved in regions bounded by asymmetric cyclidic surfaces

Abstract

A global analysis is presented of solutions for Laplace's equation on three-dimensional Euclidean space in one of the most general orthogonal asymmetric confocal cyclidic coordinate systems which admit solutions through separation of variables. We refer to this coordinate system as five-cyclide coordinates since the coordinate surfaces are given by two cyclides of genus zero which represent the inversion at the unit sphere of each other, a cyclide of genus one, and two disconnected cyclides of genus zero. This coordinate system is obtained by stereographic projection of sphero-conal coordinates on four-dimensional Euclidean space. The harmonics in this coordinate system are given by products of solutions of second-order Fuchsian ordinary differential equations with five elementary singularities. The Dirichlet problem for the global harmonics in this coordinate system is solved using multiparameter spectral theory in the regions bounded by the asymmetric confocal cyclidic coordinate surfaces.