Numerical recovering of leading coefficient of nonlinear parabolic equation

  • Ivanov Dulus Kh., i.am.djoos@gmail.com M. K. Ammosov North-Eastern Federal University Institute of mathematics and Informatics 48 Kulakovsky Street, Yakutsk 677891, Russia
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  mathscinet
[2]Samarskii A. A. and Vabishchevich P. N., Numerical Methods for Solving Inverse Problems of Mathematical Physics, Walter de Gruyter, Berlin; New York, 2007  mathscinet
[3]Iskenderov A. D., “On an inverse problem for quasilinear parabolic equations”, Differ. Equ., 10:5 (1974), 890–898  mathnet  mathscinet  zmath
[4]Muzylev N. V., “Uniqueness theorems for some inverse heat conduction problems”, U.S.S.R. Comput. Math. Math. Phys., 20:2 (1980), 388–400  mathnet  mathscinet  zmath
[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), 102–108  mathnet  mathscinet
[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), 963–967  mathnet
[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  mathscinet  zmath
[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), 369–393  crossref  mathscinet  scopus
[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  mathscinet
[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, 1190–1208  crossref  mathscinet  zmath
How to Cite
Ivanov, D. ( ) “Numerical recovering of leading coefficient of nonlinear parabolic equation”, Mathematical notes of NEFU, 24(3), pp. 90-99. doi: https://doi.org/10.25587/SVFU.2018.3.10892.
Section
Mathematical Modeling