Integral equation methods for the Yukawa-Beltrami equation on the sphere

TL;DR

This paper introduces an integral equation approach for solving the Yukawa-Beltrami equation on spherical surfaces, utilizing a fundamental solution expressed via conical functions, resulting in a well-conditioned linear system.

Contribution

It develops a new integral equation method with a fundamental solution for the Yukawa-Beltrami operator on the sphere, enabling stable numerical solutions.

Findings

The method produces a well-conditioned linear system.

Numerical examples demonstrate the effectiveness of the approach.

The fundamental solution is represented by conical functions.

Abstract

An integral equation method for solving the Yukawa-Beltrami equation on a multiply-connected sub-manifold of the unit sphere is presented. A fundamental solution for the Yukawa-Beltrami operator is constructed. This fundamental solution can be represented by conical functions. Using a suitable representation formula, a Fredholm equation of the second kind with a compact integral operator needs to be solved. The discretization of this integral equation leads to a linear system whose condition number is bounded independent of the size of the system. Several numerical examples exploring the properties of this integral equation are presented.