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Paper   IPM / M / 541
School of Mathematics
  Title:   Determination of a control parameter in the two-dimensional diffusion equation
  Author(s):  M. Dehghan
  Status:   Published
  Journal: Appl. Numer. Math.
  Vol.:  37
  Year:  2001
  Pages:   489-502
  Supported by:  IPM
This paper considers the problem of finding w=w(x,y,t) and p=p(t) which satisfy wt=wxx+wyy+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 ∫0101w(x,y,t)dxdy=E(t),0 < tT, 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.

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