Fourier Collocation Algorithm for identification of a spacewise dependent source in wave equation from Neumann-type measured data

Abstract

Inverse problem of identifying the unknown spacewise dependent source F(x) in 1D wave equation utt=c2uxx+F(x)G(t)+h(x,t), (x,t)∈(0,1)×(0,T), from the Neumann-type measured output g(t):=ux(0,t) is investigated. Most studies have attempted to reconstruct an unknown spacewise dependent source F(x) from the final observation uT(x):=u(x,T). Since a boundary measured data is most feasible from an engineering viewpoint, the identification problem has wide applications, in particular, in electrical networks governed by harmonically varying source for the linear wave equation utt−uxx=F(x)cos(ωt), where ω>0 is the frequency and F(x) is an unknown source term. In this paper Fourier Collocation Algorithm for reconstructing the spacewise dependent source F(x) is developed. This algorithm is based on Fourier expansion of the direct problem solution applied to the minimization problem for Tikhonov functional, by taking then a partial N-sum of the Fourier expansion. Tikhonov regularization is then applied to the obtained discrete ill-posed problem. To obtain high quality reconstruction in large values of the noise level, a numerical filtering algorithm is used for smoothing the noisy data. As an application, we demonstrate the ability of the algorithm on benchmark problems, in particular, on source identification problem in electrical networks governed by mono-frequency source. Numerical results show that the proposed algorithm allows to reconstruct the spacewise dependent source F(x) with enough high accuracy, in the presence of high noise levels.