*(English)*Zbl 1026.65079

Summary: An inverse problem concerning diffusion equation with source control parameter is considered. Several finite difference schemes are presented for identifying the control parameter which produces at each time a desired temperature at a given point in a spatial domain. These schemes are based on the second-order 3-point forward time centred space (FTCS) explicit formula, and the fourth-order 5-point FTCS explicit scheme, and the second-order 3-point backward time centred space (BTCS) implicit technique and the fourth-order (3, 3) *S. H. Crandall’s* implicit formula [Q. Appl. Math. 13, 318-320 (1955; Zbl 0066.10501)]. The 5-point FTCS scheme has a bounded range of stability, but its fourth-order accuracy is significant. The 3-point BTCS method uses less central processor (CPU) time than the (3, 3) Crandall’s technique, but it is only second-order accurate.

The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the work of *R. F. Warming* and *B. J. Hyett* [J. Comput. Phys. 14,159-179 (1974; Zbl 0291.65023)]. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The results of a numerical experiment are presented, and accuracy and the CPU time needed for the parabolic inverse problem are discussed.

##### MSC:

65M32 | Inverse problems (IVP of PDE, numerical methods) |

65M15 | Error bounds (IVP of PDE) |

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

35R30 | Inverse problems for PDE |

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