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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2019, Volume 12, Issue 1, Pages 82–95 (Mi vyuru473)  

Mathematical Modelling

On some inverse coefficient problems with the pointwise overdetermination for mathematical models of filtration

S. N. Shergina, E. I. Safonova, S. G. Pyatkovba

a Ugra State University, Khanty-Mansyisk, Russian Federation
b South Ural State University, Chelyabinsk, Russian Federation

Abstract: We examine inverse problems of recovering coefficients in a linear pseudoparabolic equation arising in the filtration theory. Boundary conditions of the Neumann type are supplemented with the overtermination conditions which are the values of the solution at some interior points of a domain. We expose existence and uniqueness theorems in the Sobolev spaces. The solution is regular, i. e., it possesses all generalized derivatives occurring in the equation containing in some Lebesgue space. The method of the proof is constructive. The problem is reduced to a nonlinear operator equation with a contraction operator whenever the time interval is sufficiently small. Involving the method of the proof, we construct a numerical algorithm, the corresponding software bundle, and describe the results of numerical experiments in the two-dimensional case in the space variables. The unknowns are a solution to the equation and the piezo-conductivity coefficient of a fissured rock. The main method of numerical solving the problem is the finite element method together with a difference scheme for solving of the corresponding system of ordinary differential equations. Finally, the problem is reduced to a system of nonlinear algebraic equations which solution is found by the iteration procedure. The results show a good convergence of the algorithms.

Keywords: inverse problem, pseudoparabolic equation, filtration, fissured rock, numerical solution.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 02.A03.21.0011
Yugra State University 13-01-20/16
This work was supported by the science foundation of Yugra State University (Grant no. 13-01-20/16) and by the Act 211 of the Government of the Russian Federation, contract no. 02.A03.21.0011.


DOI: https://doi.org/10.14529/mmp190107

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UDC: 517.956
MSC: 35R30, 35Q35, 65M60, 65M32
Received: 07.12.2018
Language:

Citation: S. N. Shergin, E. I. Safonov, S. G. Pyatkov, “On some inverse coefficient problems with the pointwise overdetermination for mathematical models of filtration”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 12:1 (2019), 82–95

Citation in format AMSBIB
\Bibitem{SheSafPya19}
\by S.~N.~Shergin, E.~I.~Safonov, S.~G.~Pyatkov
\paper On some inverse coefficient problems with the pointwise overdetermination for mathematical models of filtration
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2019
\vol 12
\issue 1
\pages 82--95
\mathnet{http://mi.mathnet.ru/vyuru473}
\crossref{https://doi.org/10.14529/mmp190107}
\elib{http://elibrary.ru/item.asp?id=37092205}


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