Stable recovery of the time-dependent source term from one measurement for the wave equation. (25th September 2015) - Rashedi, Kamal; Sini, Mourad
Abstract: In this work, we discuss the inverse problem which consists of the determination of an unknown time-dependent force function from one time-dependent measurement collected in any space point for the one-dimensional wave equation. This problem is motivated by the question of estimating the time-dependent body force which needs to be exerted on a given string to reach a desired shape at the final time. We prove its unique solvability using as data a linear combination of displacement and flux measured at one arbitrary fixed point of the string. We also derive a conditional Hölder stability estimate of this inverse problem. The numerical solution of the problem is investigated by means of the Ritz–Galerkin technique along with the application of the satisfier function to obtain cost-effective and stable results. Some numerical examples are provided to show the performance of the proposed scheme.