# Numerical recovering of leading coefficient of nonlinear parabolic equation

Keywords:
inverse coeﬃcient problem, parabolic equation, ﬁnite element method, FEniCS

### Abstract

We numerically recover the leading coeﬃcient of one parabolic equation in a multidimensional region. We consider the case when the leading coeﬃcient depends only on solution itself and observations are taken in some interior points of the region as an additional condition. Finite element method implemented by FEniCS library is used for numerical solution of the problem. Several examples of identiﬁcation of the leading coeﬃcient of a two-dimensional parabolic equation are given.

### References

[1] | Alifanov O. M., Artyukhin E. A., and Rumyantsev S. V., Extreme Methods for Solving IllPosed Problems and Their Applications to Inverse Problems of Heat Transfer, Nauka, Moscow, 1988 |

[2] | Samarskii A. A. and Vabishchevich P. N., Numerical Methods for Solving Inverse Problems of Mathematical Physics, Walter de Gruyter, Berlin; New York, 2007 |

[3] | Iskenderov A. D., “On an inverse problem for quasilinear parabolic equations”, Differ. Equ., 10:5 (1974), |

[4] | Muzylev N. V., “Uniqueness theorems for some inverse heat conduction problems”, U.S.S.R. Comput. Math. Math. Phys., 20:2 (1980), |

[5] | Muzylev N. V., “On the uniqueness of the simultaneous determination of the coefficients of thermal conductivity and the volume heat capacity”, U.S.S.R. Comput. Math. Math. Phys., 23:1 (1983), |

[6] | Artyukhin E. A., “The recovering of the temperature dependence of the thermal conductivity coefficient from the solution of the inverse problem”, High Temperature, 19:5 (1981), |

[7] | Logg A., Mardal K. A., Wells G., Automated solution of differential equations by the finite element method: The FEniCS book, Springer Sci. & Business Media, New York, 2012 |

[8] | Farrell P. E., Ham D. A., Funke S. F., Rognes M. E., “Automated derivation of the adjoint of high-level transient finite element programs”, SIAM J. Sci. Comput., 35:4 (2013), |

[9] | Funke S. W., Farrell P. E., “A framework for automated PDE-constrained optimisation”, 2013, arXiv: 1302.3894 |

[10] | Vasiliev F. P., Methods for Solving Extreme Problems, Nauka, Moscow, 1981 |

[11] | Byrd R. H., Lu P., Nocedal J., Zhu C., “A limited memory algorithm for bound constrained optimization”, SIAM J. Sci. Comput V. 16, N 5., 1995, |

Received

26-06-2017

How to Cite

*Mathematical notes of NEFU*, 24(3), pp. 90-99. doi: https://doi.org/10.25587/SVFU.2018.3.10892.

Issue

Section

Mathematical Modeling

This work is licensed under a Creative Commons Attribution 4.0 International License.