Оценки погрешности численного метода для задачи фильтрации с дробными производными Капуто-Фабрицио
DOI:
https://doi.org/10.26577/JMMCS.2022.v114.i2.010Ключевые слова:
метод конечных элементов, дробная производная Капуто-Фабрицио, сходимость, задача фильтрации, трещиновато-пористая средаАннотация
В данной статье изучается модель движения жидкости в трещиноватой пористой среде в предположении равномерного распределения трещин по объему. Данная модель основана на использовании дробно-дифференциального аналога закона Дарси и построена в предположении, что свойства породы и жидкости зависят от давления и его дробной производной. В отличие от предыдущих исследований, в настоящей статье используется дробная производная в смысле Капуто-Фабрицио с несингулярным ядром. В статье предлагается численный метод решения данной начально-краевой задачи и теоретически исследуется порядок его сходимости. Формулировка полностью дискретной схемы основана на применении конечно-разностной аппроксимации для целых и дробных производных по времени и метода Галеркина по пространственной переменной. Для аппроксимации целочисленной производной и дробной производной в смысле Капуто-Фабрицио используется формула второго порядка. Получены априорные оценки как для полудискретной, так и для полностью дискретной схем, из которых следует их сходимость со вторым порядком по временной и пространственной переменным. На примере модельной задачи проведен ряд вычислительных экспериментов для проверки точности схемы. Результаты численных тестов полностью подтверждают результаты теоретического анализа.
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[2] Chechkin A. V., Gorenflo R. and Sokolov I. M.,
"Fractional diffusion in inhomogeneous media", Journal of Physics A: Mathematical And General, 38 (2005): 679–984.
[3] Esen A., Ucar Y., Yagmurlu M. and Tasbozan O., "Solving fractional diffusion and fractional diffusionwave equations by Petrov Galerkin finite element method", Turkish World Mathematical Society Journal of Applied and Engineering Mathematics, 4, no. 2 (2014): 155–168.
[4] Zhu A., Wang Y. and Xu Q., "A weak Galerkin finite element approximation of two-dimensional subdiffusion equation with time fractional derivative", AIMS Mathematics, 5, no. 5 (2020): 4297–4310.
[5] Abiola O. D., Enamul H. M., Kaseem M. and Sidqi A. A., "A modified memory based mathematical model describing fluid flow in porous media", Computers and Mathematics with Applications, 73, no. 6 (2017): 1385–1402.
[6] Gazizov R. K. and Lukashchuk S. Yu., "Drobno-differentsialnyi podkhod k modelirovaniyu protsessov filtratsii v slozhnykh neodnorodnykh poristykh sredakh [Fractional differential approach to modeling filtration processes in complex inhomogeneous porous media]", Vestnik UGATU, 21, no. 4 (2017): 104–112 (in Russian).
[7] Caffarelli L. and Vazquez J. L., "Nonlinear porous medium flow with fractional potential pressure", Archive for Rational Mechanics and Analysis, 202, no. 2 (2011): 537–565.
[8] Gazizov R. K., Kasatkin A. A. and Lukashchuk S. Yu., "Symmetries and exact solutions of fractional filtration equations", AIP Conference Proceedings, 1907, no. 020010 (2017): 1–9.
[9] Meilanov R. R., Akhmedov E. N. and Beybalaev V. D.,
"To the theory of nonlocal nonisothermal filtration in porous medium", IOP Conf. Series: Journal of Physics: Conf. Series, 946, no. 012076 (2018): 1–8.
[10] El Amin M. F., Radwan A. G. and Sun S., "Analytical solution for fractional derivative gasflow equation in porous media", Results in Physics, 7 (2017): 2432–2438.
[11] Ray S. S., "Exact solutions for time fractional diffusion wave equations by decomposition method", Phys. Scr., 75 (2007): 53–61.
[12] Zhang Y. N., Sun Z. Z. and Liao H. L., "Finite difference methods for the time fractional diffusion equation on nonuniform meshes", Journal of Computational Physics, 265 (2014): 195–210.
[13] Qiao H. L., Liu Z. G. and Cheng A. J., "Two unconditionally stable difference schemes for time distributed order differential equation based on Caputo–Fabrizio fractional derivative", Advances in Difference Equations, 2020 (2020): 1–17.
[14] Alikhanov A. A., "A new difference scheme for the time fractional diffusion equation", Journal of Computational Physics, 208 (2015): 424–438.
[15] Du R., Cao W. R. and Sun Z. Z., "A compact difference scheme for the fractional diffusion wave equation", Applied Mathematical Modelling, 34 (2010): 2998–3007.
[16] Xu T., Lu S., Chen W. and Chen H., "Finite difference scheme for multiterm variable order fractional diffusion equation", Advances in Difference Equations, 1 (2018): 1-13.
[17] Huang J., Tang Y., Wang W. and Yang J. "A compact difference scheme for time fractional diffusion equation with Neumann boundary conditions", Communications in Computer and Information Science, 323 (2012): 273–284.
[18] Zhang C., Liu H. and Zhou Z. J., "A priori error analysis for timestepping discontinuous Galerkin finite element approximation of time fractional optimal control problem", Journal of Scientific Computing, 80, no. 2 (2019): 993–1018
[19] Liu J. and Zhou Z., "Finite element approximation of time fractional optimal control problem with integral state constraint", AIMS Mathematics, 6, no. 1 (2020): 979–997.
[20] Liu K., Feckan M., O'Regan D. and Wang J. R., "Hyers–Ulam stability and existence of solutions for differential equations with Caputo–Fabrizio fractional derivative", Mathematics, 333, no. 7 (2019): 1-14.
[21] Coronel Escamilla A., Gomez Aguilar J.F., Torres L. and Escobar Jimenez R.F., "A numerical solution for a variable order reaction diffusion model by usingfractional derivatives with nonlocal and nonsingular kernel", Physica A: Statistical Mechanics and its Applications, 491 (2017): 406–424.
[22] Liu Y., Du Y. and Li H., "A twogrid mixed finite element method for a nonlinear fourthorder reaction diffusion problem with time fractional derivative", Computers and Mathematics with Application, 70 (2015): 2474–2492.
[23] Alimbekova N. B., Baigereyev D. R. and Madiyarov M. N., "Study of a numerical method for solving a boundary value problem for a differential equation with a fractional time derivative", Izvestiya of Altai State University, 114, no. 4 (2020): 64–69.
[24] Alimbekova N. B., Berdyshev A. S. and Baigereyev D. R., "Study of initial boundary value problem for two dimensional differential equation with fractional time derivative in the sense of Caputo", Third International Conference on Material Science, Smart Structures and Applications: (ICMSS 2020) (2021): 1–6.
[25] Baigereyev D., Alimbekova N., Berdyshev A. and Madiyarov M., "Convergence analysis of a numerical method for a fractional model of fluid flow in fractured porous media", Mathematics, 9, no. 2179 (2021): 1-24.
[26] Caputo M., "Models of flux in porous media with memory", Water Resources Research, 36, no. 3 (2000): 693– 705.
[27] Agarwal R., Yadav M. P., Baleanu D. and Purohit S. D., "Existence and uniqueness of miscible flow equation through porous media with a non singular fractional derivative", AIMS Mathematics, 5, no. 2 (2020): 1062–1073.
[28] Akman T., Yildiz B. and Baleanu D., "New discretization of Caputo-Fabrizio derivative", Computational and Applied Mathematics, 37, no. 3 (2017): 3307-3333.
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Опубликован
2022-06-24
Как цитировать
Baigereyev, D., Alimbekova, N., & Oskorbin, N. (2022). Оценки погрешности численного метода для задачи фильтрации с дробными производными Капуто-Фабрицио. Вестник КазНУ. Серия математика, механика, информатика, 114(2). https://doi.org/10.26577/JMMCS.2022.v114.i2.010
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Прикладная математика
