Reconstruction of a time-dependent wave source and the initial condition in a hyperbolic equation

Abstract

This article is concerned with determination of a time-dependent wave source together with the initial condition in a hyperbolic equation. Based on the given data including the flux tensions of a string at end points and a nonlocal integral observation along with the wave displacement at two distinct instants of time, we prove the unique solvability of the inverse problem. Moreover, we estimate the modulus of continuity of the inverse problem under special conditions. The numerical solution of the problem is also presented by means of Bernstein spectral method, which converts the problem to a linear system of algebraic equations and then Tikhonov regularization technique is employed to derive reliable solutions. Regarding the perturbed boundary conditions, we use a regularized numerical technique to obtain stable derivatives. Numerical simulations are provided to show the effectiveness of the proposed scheme.