Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations

We consider a finite difference approximation to an inverse problem of determining an unknown source parameter p(t) which is a coefficient of the solution u in a linear parabolic equation subject to the specification of the solution u at an internal point along with the usual initial boundary conditions. The backward Euler scheme is studied and its convergence is proved via an application of the discrete maximum principle for a transformed problem. Error estimates For u and p involve numerical differentiation of the approximation to the transformed problem. Some experimental numerical results using the newly proposed numerical procedure are discussed.