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Zbl 0982.65103
Dehghan, Mehdi
Determination of a control parameter in the two-dimensional diffusion equation.
(English)
[J] Appl. Numer. Math. 37, No.4, 489-502 (2001). ISSN 0168-9274

Summary: This paper considers the problem of finding $w= w(x,y,t)$ and $p= p(t)$ which satisfy $$w_t= w_{xx}+ w_{yy}+ p(t)w+ \phi,$$ in $R\times (0,T]$, $w(x,y,0)= f(x,y)$, $(x,y)\in R= [0,1]\times [0,1]$, $w$ is known on the boundary of $R$ and also $\int^1_0 \int^1_0 w(x,y,t) dx dy= E(t)$, $0< t\le T$, where $E(t)$ is known.\par 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 (N-H) fully implicit technique, and the Peaceman and Rachford (P-R) alternating direction implicit (ADI) formula. These schemes are second-order accurate.\par 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.
MSC 2000:
*65M32 Inverse problems
65M06 Finite difference methods (IVP of PDE)
35K15 Second order parabolic equations, initial value problems
35R30 Inverse problems for PDE

Keywords: Noye-Hayman method; Peaceman-Rachford method; inverse problem; diffusion equation; alternating direction implicit method; finite-difference schemes; control parameter; numerical experiments

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