On the inverse problem for nonlinear strongly damped wave equations with discrete random noise

Abstract

Abstract In this paper, we consider the existence of a solution u(x, t) for the inverse backward problem for the nonlinear strongly damped wave equation with statistics discrete data. The problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. In order to regularize the unstable solution, we use the trigonometric method in non-parametric regression associated with the truncated expansion method. We investigate the convergence rate under some a priori assumptions on an exact solution in both L 2 and H q (q > 0) norms. Moreover, a numerical example is given to illustrate our results.