Numerical solution of the linear inverse wave equation

In this paper, a numerical method is proposed for the numerical solution of a linear wave equation with initial and boundary conditions by using the cubic B-spline method to determine the unknown boundary condition. We apply the cubic B-spline for the spatial variable and the derivatives, which generate an ill-posed linear system of equations. In this regard, to overcome, this drawback, we employ the Tikhonov regularization (TR) method for solving the resulting linear system. It is proved that the proposed method has the order of convergence O((Δt)2+h2). Also, the conditional stability by using the Von-Neumann method is established under suitable assumptions. Finally, some numerical experiments are reported to show the efficiency and capability of the proposed method for solving inverse problems.