Volume 12, no. 1Pages 82 - 95

# On Some Inverse Coefficient Problems with the Pointwise Overdetermination for Mathematical Models of Filtration

S.N. Shergin, E.I. Safonov, S.G. Pyatkov
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.
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Keywords
inverse problem; pseudoparabolic equation; filtration; fissured rock; numerical solution.
References
1. Barenblatt G.I., Zheltov Iu.P., Kochina I.N. Basic Concepts in the Theory of Seepage of Homogeneous Liquids in Fissured Rocks. Journal of Applied Mathematics and Mechanics, 1960, vol. 24, no. 5, pp. 1286-1303. DOI: 10.1016/0021-8928(60)90107-6
2. Bohm M., Showalter R.E. Diffusion in Fissured Media. SIAM Journal on Mathematical Analysis, 1985, vol. 16, no. 3, pp. 500-509. DOI: 10.1137/0516036
3. Lyubanova A.Sh., Tani A. On Inverse Problems for Pseudoparabolic and Parabolic Equations of Filtration. Inverse Problems in Science and Engineering, 2011, vol. 19, no. 7, pp. 1023-1042. DOI: 10.1080/17415977.2011.569712
4. Al'shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-up in Nonlinear Sobolev Type Equations. Berlin, N.Y., De Gruyter, 2011. DOI: 10.1515/9783110255294
5. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, VSP, 2003. DOI: 10.1515/9783110915501
6. Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. N.Y., Marcel Dekker, 1999.
7. Rundell W. Determination of an Unknown Nonhomogeneous Term in a Linear Partial Differential Equation from Overspecified Boundary Data. Journal of Applied Analysis, 1980, vol. 10, no. 3, pp. 231-242. DOI: 10.1080/00036818008839304
8. Kozhanov A.I. Composite Type Equations and Inverse Problems. Utrecht, VSP, 1999. DOI: 10.1515/9783110943276
9. Asanov A., Atamanov E.R. Nonclassical and Inverse Problems for Pseudoparabolic Equations. Berlin, De Gruyter, 2014.
10. Favini A., Lorenzi A. Differential Equations. Inverse an Direct Problems. Abingdon, Tylor and Francis Group, 2006. DOI: 10.1201/9781420011135
11. Mamayusupov M.Sh. The Problem of Determining Coefficients of a Pseudoparabolic Equation. Studies in Integro-Differential Equations, 1983, vol. 16, pp. 290-297.
12. Lyubanova A.Sh. Identification of a Coefficient in the Leading Term of a Pseudoparabolic Equation of Filtration. Siberian Mathematical Journal, 2013, vol. 54, no. 6, pp. 1046-1058. DOI: 10.1134/S0037446613060116
13. Lyubanova A.Sh. The Inverse Problem for the Nonlinear Pseudoparabolic Equation of Filtration Type. Journal of Siberian Federal University. Series: Mathematics and Physics, 2017, vol. 10, no. 1, pp. 4-15.
14. Pyatkov S.G., Shergin S.N. On Some Mathematical Models of Filtration Theory. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 105-116.
15. Kabanikhin S.I. Inverse and Ill-Posed Problems. Berlin, Boston, De Gruyter, 2012.
16. Samarskii A.A., Vabishchevich P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics. Berlin, Boston, De Gruyter, 2007. DOI: 10.1515/9783110205794
17. Ewing R.E. Numerical Solution of Sobolev Partial Differential Equations. SIAM Journal on Numerical Analysis, 1975, vol. 12, no. 3, pp. 345-363. DOI: 10.1137/0712028
18. Cuesta C.M., Pop I.S. Numerical Schemes for a Pseudo-Parabolic Burgers Equation, Discontinuous Data and Long-Time Behaviour. Journal of Computational and Applied Mathematics, 2009, vol. 224, no. 1, pp. 269-283. DOI: 10.1016/j.cam.2008.05.001
19. Beshtokov M.Kh. Differential and Difference Boundary Value Problem for Loaded Third-Order Pseudo-Parabolic Differential Equations and Difference Methods for Their Numerical Solution. Computational Mathematics and Mathematical Physics, 2017, vol. 57, no. 12, pp. 1973-1993. DOI: 10.1134/S0965542517120089
20. Vabishchevich P.N., Grigor'ev A.V. Splitting Schemes for Pseudoparabolic Equations. Differential Equations, 2013, vol. 49, no. 7, pp. 807-814. DOI: 10.1134/S0012266113070033
21. Guezane-Lakoud A., Belakroum D. Time-Discretization Schema for an Integrodifferential Sobolev Type Equation with Integral Conditions. Applied Mathematics and Computation, 2012, vol. 218, no. 9, pp. 4695-4702. DOI: 10.1016/j.amc.2011.11.077
22. Luoa Z.D. Teng F. A Reduced-Order Extrapolated Finite Difference Iterative Scheme Based on POD Method for 2D Sobolev Equation. Applied Mathematics and Computation, 2018, vol. 329, pp. 374-383. DOI: 10.1016/j.amc.2018.02.022
23. Xia H., Luo Z. An Optimized Finite Difference Crank-Nicolson Iterative Scheme for the 2D Sobolev Equation. Advances in Difference Equations, 2017, vol. 196, 12 p. DOI: 10.1186/s13662-017-1253-8.
24. Keller A.V., Zagrebina S.A. Some Generalizations of the Schowalter-Sidorov Problem for Sobolev-Type Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 5-23.
25. Hasan A., Aamo A.M., Foss B. Boundary Control for a Class of Pseudo-Parabolic Differential Equations. Systems and Control Letters, 2013, vol. 62, no. 1, pp. 63-69. DOI: 10.1016/j.sysconle.2012.10.009
26. Triebel H. Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library. Amsterdam, North-Holland Publishing, 1978.
27. Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. tLinear and Quasi-Linear Equations of Parabolic Type. Providence, American Mathematical Society, 1968. DOI: 10.1090/mmono/023
28. Amann H. Operator-Valued Fourier Multipliers, Vector-Valued Besov Spaces and Applications. Mathematische Nachrichten, 1997, vol. 186, no. 1, pp. 5-56. DOI: 10.1002/mana.3211860102