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Paper IPM / M / 541  


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}^{1}w(x,y,t)dxdy=E(t),0 < t ≤ T, where E(t) is known. Three
different finitedifference 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) NoyeHayman
(denoted NH) fully implicit technique, and the Peaceman and
Rachford (denoted PR) alternating direction implicit (ADI)
formula. These schemes are second order accurate. The ADI scheme
and the 5point fully explicit method use less central processor
(CPU) time than the (5,5) NH fully implicit scheme. The PR ADI
scheme and the (5,5) NH 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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