Determination of a space-dependent force function in the one-dimensional wave equation

TL;DR

This paper presents a numerical method combining boundary element and regularization techniques to accurately determine a space-dependent force in a vibrating string from boundary data, addressing the ill-posed nature of the inverse problem.

Contribution

It introduces a novel numerical approach that effectively stabilizes the solution of the inverse problem using Tikhonov regularization and the L-curve method.

Findings

Accurate solutions with exact data

Stable solutions with noisy data

Effective regularization parameter selection

Abstract

The determination of an unknown spacewice dependent force function acting on a vibrating string from over-specified Cauchy boundary data is investigated numerically using the boundary element method (BEM) combined with a regularized method of separating variables. This linear inverse problem is ill-posed since small errors in the input data cause large errors in the output force solution. Consequently, when the input data is contaminated with noise we use the Tikhonov regularization method in order to obtain a stable solution. The choice of the regularization parameter is based on the L-curve method. Numerical results show that the solution is accurate for exact data and stable for noisy data