Fourth-order techniques for identifying a control parameter in the parabolic equations

This paper considers the problem of finding u = u (x, y, t) and p = p(t) which satisfy u(t) = u(xx) + u(yy) +p(t)u + phi in R x (0,T], u(x,y,0) =f(x,y), (x,y) epsilon R = [0,1] x [0, 1], u is known on the boundary of R and also u (x(0), y(0), t) = E(t) 0 less than or equal to t less than or equal to T, where E(t) is known and (x(0), y(0)) is a given point of R. Three different finite difference schemes are developed for identifying the control parameter p(t), in this two-dimensional diffusion equation. These schemes are considered for identifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the above spatial domain. The numerical methods discussed are based on the 13-point forward time centred space (FTCS) explicit finite difference formula, and the (3,9) alternating direction implicit (denoted ADI) finite difference scheme, and the (9, 9) fully implicit finite difference technique. These schemes have the fourth-order accuracy with respect to the spatial grid size. The (1, 13) FTCS finite difference scheme has a bounded range of stability, but the (3 9) ADI formula and the (9,9) fully implicit finite difference method are unconditionally stable. The results of numerical experiments are presented, and accuracy and central processor (CPU) times needed for each of the methods are discussed. The (1, 13) FTCS scheme and the (3,9) ADI technique use less CPU times than the (9, 9) fully implicit finite difference method. (C) 2002 Elsevier Science Ltd. All rights reserved.