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Finding a control parameter in one-dimensional parabolic equations. (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:
65M32Inverse problems (IVP of PDE, numerical methods)
65M15Error bounds (IVP of PDE)
65M06Finite difference methods (IVP of PDE)
35R30Inverse problems for PDE
35K15Second order parabolic equations, initial value problems