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Zbl 1026.65079
Dehghan, Mehdi
Finding a control parameter in one-dimensional parabolic equations. (English)
[J] Appl. Math. Comput. 135, No.2-3, 491-503 (2003). ISSN 0096-3003

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) {\it 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.\par The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the work of {\it R. F. Warming} and {\it 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 2000:: *65M32 Inverse problems
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

Keywords: error bounds; finite difference methods; control parameter; temperature overspecification; inverse problem; diffusion equation; numerical experiment

Citations: Zbl 0066.10501; Zbl 0291.65023

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