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**An inverse problem of finding a source parameter in a semilinear parabolic equation.**
*(English)*
Zbl 0995.65098

Summary: An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-differences schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centered space (FTCS) explicit formula, and the 5-point FTCS explicit method and the classical backward time centred space (BTCS) implicit scheme, and the Crank-Nicolson implicit method. The classical FTCS explicit formula and the 5-point TFTCS explicit technique are economical to use, are second-order accurate, but have bounded range of stability. The classical BTCS implicit scheme and the Crank-Nicolson implicit method are unconditionally stable, but these schemes use more central processor (CPU) times than the explicit finite difference methods.

The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed by 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 the accuracy and CPU time needed for this inverse problem are discussed.

The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed by 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 the accuracy and CPU time needed for this inverse problem are discussed.

### MSC:

65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |

35K55 | Nonlinear parabolic equations |

35R30 | Inverse problems for PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

### Keywords:

semilinear parabolic equation; stability; finite difference techniques; inverse problem; source parameter; diffusion processes; energy overspecification; control parameter; Crank-Nicolson implicit method; numerical experiment### Citations:

Zbl 0291.65023
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\textit{M. Dehghan}, Appl. Math. Modelling 25, No. 9, 743--754 (2001; Zbl 0995.65098)

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### References:

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