The fully-implicit finite difference method for solving nonlinear inverse parabolic problems with unknown source term

A numerical procedure based on a fully implicit finite difference method for an inverse problem of identification of an unknown source in a heat equation is presented. The approach of the proposed method is to approximate unknown function from the solution of the minimisation problem based on the overspecified data. This problem is ill-posed, in the sense that the solution (if it exist) does not depend continuously on the data. To regularise this ill-conditioned, we apply the Tikhonov regularisation 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. A stability analysis shows that this numerical scheme approximation is unconditionally stable. Numerical results for two inverse source identification problems show that the proposed numerical algorithm is simple, accurate, stable and computationally efficient.