Integral equation approach for the numerical solution of a Robin problem for the Klein-Gordon equation in a doubly connected domain

TL;DR

This paper develops an integral equation method to numerically solve a Robin boundary value problem for the Klein-Gordon equation in a doubly connected domain, demonstrating unique solvability and exponential convergence through numerical experiments.

Contribution

It introduces a novel integral equation approach with a specific discretization scheme for the Klein-Gordon Robin problem in doubly connected domains, ensuring unique solutions and high convergence.

Findings

Unique solvability of the integral system established.

Exponential order of convergence demonstrated.

Numerical experiments confirm theoretical results.

Abstract

In this paper we consider a Robin problem for the Klein-Gordon equation in a doubly connected domain. The solution domain considered is a bounded smooth doubly connected planar domain bounded by two simple disjoint closed curves. The analysis of the problem is based on the indirect integral equations method. The solution is represented as a sum of two single-layer potentials defined on each of the two boundary curves with unknown densities. To find out the densities the representation is matched with the given Robin data to generate a system of linear integral equations of the second kind with continuous and weakly-singular kernels. It is shown that the operator corresponding to this system is injective and due to its compactness according to Riesz theory there exists a unique solution. To discretize the system we apply Nystrom method with a specifically chosen quadrature rules to obtain an exponential order of convergence of the approximate solution. Numerical experiments are conducted for three testing examples that back up the theoretical reasoning.