Regularization of an Ill-Posed Hyperbolic Problem and the Uniqueness of the Solution by a Wavelet Galerkin Method

Abstract

We consider the problem $K(x)u_{xx} = u_{tt}$, $0 < x < 1$, $t \geq 0$, with the boundary condition $u(0, t) = g(t) \in L^2(R)$ and $u_x(0, t) = 0$, where $K(x)$ is continuous and $0 < \alpha \leq K(x) < +\infty$. This is an ill-posed problem in the sense that, if the solution exists, it does not depend continuously on $g$. Considering the existence of a solution $u(x, \cdot) \in H^2(R)$ and using a wavelet Galerkin method with Meyer multiresolution analysis, we regularize the ill-posedness of the problem. Furthermore, we prove the uniqueness of the solution for this problem.