*(English)*Zbl 0982.65103

Summary: This paper considers the problem of finding $w=w(x,y,t)$ and $p=p\left(t\right)$ which satisfy

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 ${\int}_{0}^{1}{\int}_{0}^{1}w(x,y,t)dxdy=E\left(t\right)$, $0<t\le T$, where $E\left(t\right)$ is known.

Three different finite-difference schemes are presented for identifying the control parameter $p\left(t\right)$, 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 | Inverse problems (IVP of PDE, numerical methods) |

65M06 | Finite difference methods (IVP of PDE) |

35K15 | Second order parabolic equations, initial value problems |

35R30 | Inverse problems for PDE |