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

Abstract

This paper considers the problem of finding w=w(x,y,t) and p=p(t) which satisfy w t=w xx+w yy+p(t)w+φ , in R×(0,T], w(x,y,0)=f(x,y), (x,y)∈R=[0,1]×[0,1] , w is known on the boundary of R and also ∫ 0 1∫ 0 1w(x,y,t) dx dy=E(t) , 0<t⩽T, where E(t) is known. Three different finite-difference schemes are presented for identifying the control parameter p(t), which produces, at any given time, a desired energy distribution in a portion of the spatial domain. The finite difference schemes developed for this purpose are based on the (1,5) fully explicit scheme, and the (5,5) Noye–Hayman (denoted N–H) fully implicit technique, and the Peaceman and Rachford (denoted P–R) alternating direction implicit (ADI) formula. These schemes are second-order accurate. The ADI scheme and the 5-point fully explicit method use less central processor (CPU) time than the (5,5) N–H fully implicit 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. The results of numerical experiments are presented, and CPU times needed for this problem are reported.