The analytical solution of the Laplace equation with the Robin boundary conditions on a sphere: Applications to some inverse problems

TL;DR

This paper derives a closed-form analytical solution to the Laplace equation with Robin boundary conditions on a sphere using Legendre transform, facilitating inverse problems in physics and engineering.

Contribution

It introduces a novel closed-form solution involving the Appell hypergeometric function for the Laplace equation with Robin boundary conditions on a sphere, addressing computational challenges.

Findings

Solution expressed with Appell hypergeometric function F1

Applicable to inverse problems in various physical domains

Simplifies evaluation over infinite series of Legendre polynomials

Abstract

This paper studies the third boundary problem of the Laplace equation with azimuthal symmetry.Many solutions of the boundary value problems in spherical coordinates are available in the form of infinite series of Legendre polynomials but the evaluation of the summing series shows many computational difficulties. Integral transform is a challenge as it involves an inverse Legendre transform. Here, the closed-form solution of the Laplace equation with the Robin boundary conditions on a sphere is solved by the Legendre transform. This analytical solution is expressed with the Appell hypergeometric function F1. The Robin boundary conditions is a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions. In many experimental approaches, this weight h, the Robin coefficient, is the main unknown parameter for example in transport phenomena where the Robin coefficient is the dimensionless Biot number. The usefulness of this formula is illustrated by some examples of inverse problems in mass and heat transfer, in optics, in corrosion detection and in physical geodesy.