Determination of forcing functions in the wave equation. Part I: the space-dependent case

Abstract

We consider the inverse problem for the wave equation which consists of determining an unknown space-dependent force function acting on a vibrating structure from Cauchy boundary data. Since only boundary data are used as measurements, the study has importance and significance to non-intrusive and non-destructive testing of materials. This inverse force problem is linear, the solution is unique, but the problem is still ill-posed since, in general, the solution does not exist and, even if it exists, it does not depend continuously upon the input data. Numerically, the finite difference method combined with the Tikhonov regularization are employed in order to obtain a stable solution. Several orders of regularization are investigated. 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. An extension to the case of multiple additive forces is also addressed. In a companion paper, in Part II, the time-dependent force identification will be undertaken.

Inverse space-dependent force problems for the wave equation

“…For this, let v 2 be the solution of the adjoint problem (8)- (10) with ζ = G = 0 and ξ = u h (·, T ). From Green's formula (11) applied to the functions u h and v 2 we obtain…”

“…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 ).…”

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).…”

Determination of a time-dependent coefficient in awave equation with unusual boundary condition

“…The work [6] considers the inverse problem for the wave equation which consists in determining an unknown time-dependent force function by applying finite difference method. Same method is used for an unknown space-dependent force function acting on a vibrating structure in the wave equation from Cauchy boundary data in [7]. In [8], inverse problem of finding space-dependent potential or damping coefficients in the wave equation is considered.…”