Finding a control parameter in one-dimensional parabolic equations

摘要

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) Crandall’s implicit formula. 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 1974 work of Warming and Hyett. 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.