Determination of a space-dependent force function in the one-dimensional wave equation
For researchers in inverse problems and vibration analysis, this provides a numerical method for a specific ill-posed problem, but it is incremental as it applies existing techniques to a known formulation.
This paper numerically solves an inverse problem to determine a space-dependent force function in the 1D wave equation from Cauchy boundary data, using BEM and Tikhonov regularization. Results show accurate reconstruction for exact data and stability for noisy data.
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