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| Paper IPM / M / 541 |
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| Abstract: | |
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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 ∫01 ∫01w(x,y,t)dxdy=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.
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