Determination of a control parameter in the two-dimensional diffusion equation

This paper considers the problem of finding w = w(x, y, t) and p = p(t) whi ch satisfy w(t) = w(xx) + w(yy) + p(t)w + phi, in R x (0, T], w(x, y, 0) = f(x, y), (x, y) is an element of R = [0, 1] x [0, 1], w is known on the bou ndary of R and also integral (1)(0) integral (1)(0) w(x, y, t) dx dy = E(t) , 0 < t less than or equal to T, where E(t) is known. Three different finit e-difference schemes are presented for identifying the control parameter p( t), which produces, at any given time, a desired energy distribution in a p ortion of the spatial domain. The finite difference schemes developed for t his purpose are based on the (1,5) fully explicit scheme, and the (5, 5) No ye-Hayman (denoted N-H) fully implicit technique, and the Peaceman and Rach ford (denoted P-R) alternating direction implicit (ADI) formula. These sche mes are second-order accurate. The ADI scheme and the 5-point fully explici t method use less central processor (CPU) time than the (5, 5) N-H fully im plicit scheme. The P-R ADI scheme and the (5, 5) N-H fully implicit method have a larger range of stability than the (1,5) fully explicit technique. T he results of numerical experiments are presented, and CPU times needed for this problem are reported. (C) 2001 IMACS. Published by Elsevier Science B .V. All rights reserved.