Stable recovery of the time-dependent source term from one measurement for the wave equation. (25th September 2015)

• Record Type: Journal Article
• Authors: Rashedi, Kamal; Sini, Mourad
• Is Part Of: Inverse problems. Volume 31:Number 10(2015:Oct.)
• Publication Date: 2015-09-25
• Subjects: Inverse wave problem -- Ritz--Galerkin method -- Landweber iteration; 65N12 -- 65N15 -- 65N20 -- 65N21; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/31/10/105011
• Languages: English
• ISSNs: 0266-5611

Abstract

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.