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This paper considers the problem of finding $u=u(x,y,t)$ and
$p=p(t)$ which satisfy $u_t=u_{xx}+u_{yy}+p(t)u+\phi$ in $R\times
(0,T], u(x,y,0)=f(x,y),(x,y)\in R=[0,1]\times [0,1], u$ is known
on the boundary of $R$ and also $u(x_0,y_0,t)=E(t),0\leq t\leq T$,
where $E(t)$ is known and $(x_0,y_0)$ is a given point of $R$.
Three different finite difference schemes are developed for
identifying the control parameter $p(t)$, in this two-dimensional
diffusion equation. These schemes are considered for identifying
the control parameter which produces, at any given time, a desired
temperature distribution at a given point in the above spatial
domain. The numerical methods discussed are based on the 13-point
forward time centred space (FTCS) explicit finite difference
formula, and the (3,9) alternating direction implicit (denoted
ADI) finite difference scheme, and the (9,9) fully implicit finite
difference technique. These schemes have the fourth-order accuracy
with respect to the spatial grid size. The (1,13) FTCS finite
difference scheme has a bounded range of stability, but the (3,9)
ADI formula and the (9,9) fully implicit finite difference method
are unconditionally stable. The results of numerical experiments
are presented, and accuracy and central processor (CPU) times
needed for each of the methods are discussed. The (1,13) FTCS
scheme and the (3,9) ADI technique use less CPU times than the
(9,9) fully implicit finite difference method.
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