Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives
DOI:
https://doi.org/10.26577/JMMCS.2022.v114.i2.010Keywords:
ақырлы элементтер әдісі, Капуто-Фабрицио бөлшек ретті туындысы, жинақтылық, фильтрация есебі, жарықшалы кеуекті ортаAbstract
This paper investigates a model of fluid flow in a fractured porous medium under the assumption of a uniform distribution of fractures throughout the volume. This model is based on the use of a fractional differential analogue of Darcy's law, as well as on the assumption that the properties of rock and fluid depend on pressure and its fractional derivative. Unlike previous studies, this study uses a fractional derivative in the Caputo-Fabrizio sense with a non-singular kernel. In this paper, we propose a numerical method for solving this initial boundary value problem and theoretically investigate the order of its convergence. The formulation of a fully discrete scheme is based on application of the finite difference approximation for integer and fractional time
derivatives, and the Galerkin method in the spatial variable. A second-order formula is used to approximate both integer derivative and the fractional derivative in the sense of Caputo-Fabrizio. A priori estimates are obtained for both semi-discrete and fully discrete schemes, which imply their second-order convergence in time and space variables. A number of computational experiments were carried out on the example of a model problem to validate the accuracy of the scheme. The results of the numerical tests fully confirm the outcome of the theoretical analysis.
References
[1] Caputo M. and Fabrizio M., "A new definition of fractional derivative without singular kernel", Progress in Fractional Differentiation and Applications, 2 (2015): 73–85.
[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.
[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.
Downloads
Published
2022-06-24
Issue
Section
Applied Mathematics
