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

Abstract

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.

Splitting the One-Dimensional Wave Equation. Part I: Solving by Finite-Difference Method and Separation Variables

“…In physics, the wave equation uses vibrations of a spring or membrane, acoustic scattering, etc. (1)(2)(3). The aim of this study: Firstly, to investigate and use finite-difference method (FDM) rather than boundary element method (BEM) in (1,2).…”

“…(1)(2)(3). The aim of this study: Firstly, to investigate and use finite-difference method (FDM) rather than boundary element method (BEM) in (1,2). Secondly, to show numerical results and compare to (3)(4)(5)(6), in addition presented numerical results were not shown in (4).…”

“…Furthermore, numerical solving for inverse problems part by finite-difference method (FDM), and apply the Tikhonov regularization to stable the ill-posed inverse problem with various regularization parameters (2,3). In this research the condition and the theory that insure uniqueness solution is not provided because it is the same in (1)(2)(3)(4)(5). Inverse problem is linear source inverse problem (1)(2)(3).…”

“…In this research the condition and the theory that insure uniqueness solution is not provided because it is the same in (1)(2)(3)(4)(5). Inverse problem is linear source inverse problem (1)(2)(3). The design of the paper is as follows: The mathematical formulation is given in Section 2.…”

Inverse space-dependent force problems for the wave equation

“…Prior to this study, the reconstruction of a space-dependent force in the wave equation from Cauchy data measurements of both displacement and its normal derivative on the boundary has been attempted in [3,10,11]. This inverse formulation is, as expected, improperly posed because the unknown output force f (x) depends on x in the domain Ω, whilst the known input data, say u and ∂ n u, depend on (x, t) on the boundary ∂Ω × (0, T ).…”