Regularization of initial inverse problem for strongly damped wave equation

Abstract

In this paper, we consider the problem of finding the function , from the final data and where is a linear, unbounded, self-adjoint and positive definite operator. This problem is known as the inverse initial problem for non-linear strongly damped wave and is ill-posed in the sense of Hadamard. In order to obtain a stable numerical solution, we propose new quasi-boundary value method to solve the non-linear problem, i.e. for replacing by with the operator will be defined later and satisfies (1.8). Moreover, we show that the regularized solutions converge to the exact solution strongly with respect to under a priori assumption on the exact solution in Gevrey space.