On some inverse coefficient problems with the pointwise overdetermination for mathematical models of filtration
Автор: Shergin S.N., Safonov E.I., Pyatkov S.G.
Рубрика: Математическое моделирование
Статья в выпуске: 1 т.12, 2019 года.
Бесплатный доступ
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.
Inverse problem, pseudoparabolic equation, filtration, fissured rock, numerical solution
Короткий адрес: https://sciup.org/147232932
IDR: 147232932 | УДК: 517.956 | DOI: 10.14529/mmp190107
О некоторых коэффициентных обратных задачах с точечным переопределением для математических моделей фильтрации
Рассматриваются обратные задачи восстановления коэффициентов линейного псевдопараболического уравнения, возникающие в теории фильтрации. Граничные условия типа Неймана дополняются условиями переопределения, которые есть значения решения в некотором наборе внутренних точек области. Мы приводим теоремы существования и единственности решений в пространствах Соболева. Полученное решение является регулярным, то есть обладает всеми обобщенными производными, входящими в уравнение, принадлежащими некоторому пространству Лебега. Метод доказательства является конструктивным. Задача сводится к нелинейному операторному уравнению с сжимающим оператором, если временной промежуток достаточно мал. Используя метод доказательства, мы строим численный алгоритм определения решения, соответствующий программный комплекс и описываем результаты численных экспериментов в двухмерном случае по пространственным переменным. Определению подлежат само решение уравнения и коэффициент пьезопроводимости трещиноватой среды. Основной метод для численного определения решения - метод конечных элементов, который дополняется разностной схемой для решения соответствующей системы обыкновенных дифференциальных уравнений. В конечном счете задача сводится к решению нелинейной алгебраической системы, решение которой находится при помощи итерационной процедуры. Результаты показывают очень хорошую сходимость численного алгоритма.
Список литературы On some inverse coefficient problems with the pointwise overdetermination for mathematical models of filtration
- 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
- Böhm 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
- 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
- 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
- Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, VSP, 2003. DOI: 10.1515/9783110915501
- Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. N.Y., Marcel Dekker, 1999.
- 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
- Kozhanov A.I. Composite Type Equations and Inverse Problems. Utrecht, VSP, 1999.
- DOI: 10.1515/9783110943276
- Asanov A., Atamanov E.R. Nonclassical and Inverse Problems for Pseudoparabolic Equations. Berlin, De Gruyter, 2014.
- Favini A., Lorenzi A. Differential Equations. Inverse an Direct Problems. Abingdon, Tylor and Francis Group, 2006.
- DOI: 10.1201/9781420011135
- Mamayusupov M.Sh. The Problem of Determining Coefficients of a Pseudoparabolic Equation. Studies in Integro-Differential Equations, 1983, vol. 16, pp. 290-297.
- 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
- 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.
- 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.
- Kabanikhin S.I. Inverse and Ill-Posed Problems. Berlin, Boston, De Gruyter, 2012.
- Samarskii A.A., Vabishchevich P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics. Berlin, Boston, De Gruyter, 2007.
- DOI: 10.1515/9783110205794
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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.
- 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
- Triebel H. Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library. Amsterdam, North-Holland Publishing, 1978.
- Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. Linear and Quasi-Linear Equations of Parabolic Type. Providence, American Mathematical Society, 1968.
- DOI: 10.1090/mmono/023
- 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
