Determination of a time-dependent potential in the higher-order pseudo-hyperbolic problem

Abstract

The scope of this paper is to determine the time-dependent potential term numerically in the fourth-order pseudo-hyperbolic equation with initial and boundary conditions from an additional measurement condition. From the literature, we already know that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data. For the numerical realization, we apply the Crank–Nicolson finite difference method combined with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.