Numerical methods for two-dimensional parabolic inverse problem with energy overspecification

An inverse problem concerning the two-dimensional diffusion equation with s ource control parameter is considered. Four finite-difference schemes are p resented for identifying the control parameter which produces, at any given time, a desired energy distribution in a portion of the spatial domain. Th e fully explicit schemes developed for this purpose, are based on the (1, 5 ) forward time centred space (FTCS) explicit formula, and the (1, 9) FTCS s cheme, are economical to use, are second-order and have bounded range of st ability. The range of stability for the 9-point finite difference scheme is less restrictive than the (1, 5) FTCS formula. The fully implicit finite d ifference schemes employed, are based on the (5, 1) backward time centred s pace (BTCS) formula, and the (5,5) Crank-Nicolson implicit scheme, which ar e unconditionally stable, but use more CPU times than the fully explicit te chniques. The basis of analysis of the finite difference equation considere d here is the modified equivalent partial differential equation approach, d eveloped from the 1974 work of Warming and Hyeet. This allows direct and si mple comparison of the eff ors associated with the equations as well as pro viding a means to develop more accurate finite difference methods. The resu lts of numerical experiments are presented, and central processor (CPU) tim es needed for solving this inverse problem are reported.