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Determination of a control parameter in the two-dimensional diffusion equation. (English) Zbl 0982.65103

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 \(\int^1_0 \int^1_0 w(x,y,t) dx dy= E(t)\), \(0< t\leq T\), 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 (N-H) fully implicit technique, and the Peaceman and Rachford (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.

MSC:

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
35R30 Inverse problems for PDEs
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References:

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