An inverse problem of finding a source parameter in a semilinear parabolicequation

An inverse problem concerning diffusion equation with source control parame ter is considered. Several finite-difference schemes are presented for iden tifying the control parameter. These schemes are based on the classical for ward time centred space (FTCS) explicit formula, and the 5-point FTCS expli cit method and the classical backward time centred space (BTCS) implicit sc heme, and the Crank-Nicolson implicit method. The classical FTCS explicit f ormula and the 5-point FTCS explicit technique are economical to use, are s econd-order accurate, but have bounded range of stability. The classical BT CS implicit scheme and the Crank-Nicolson implicit method are unconditional ly stable, but these schemes use more central processor (CPU) times than th e explicit finite difference mehods. The basis of analysis of the finite di fference equations considered here is the modified equivalent partial diffe rential equation approach, developed from the 1974 work of Warming and Hyet t. This allows direct and simple comparison of the errors associated with t he equations as well as providing a means to develop more accurate finite d ifference schemes. The results of a numerical experiment are presented, and the accuracy and CPU time needed for this inverse problem are discussed. ( C) 2001 Elsevier Science Inc. All rights reserved.