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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+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<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 (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:
65M32Inverse problems (IVP of PDE, numerical methods)
65M06Finite difference methods (IVP of PDE)
35K15Second order parabolic equations, initial value problems
35R30Inverse problems for PDE