Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives
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...
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| Format: | Article |
| Language: | English |
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Al-Farabi Kazakh National University
2022-06-01
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| Series: | Вестник КазНУ. Серия математика, механика, информатика |
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| Online Access: | https://bm.kaznu.kz/index.php/kaznu/article/view/998/665 |
| _version_ | 1797937157636096000 |
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| author | Dossan Baigereyev Nurlana Alimbekova Nikolay Oskorbin |
| author_facet | Dossan Baigereyev Nurlana Alimbekova Nikolay Oskorbin |
| author_sort | Dossan Baigereyev |
| collection | DOAJ |
| description | 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 timederivatives, 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. |
| first_indexed | 2024-04-10T18:40:18Z |
| format | Article |
| id | doaj.art-0c8d41dcdc92410c9a47d415a04e5752 |
| institution | Directory Open Access Journal |
| issn | 1563-0277 2617-4871 |
| language | English |
| last_indexed | 2024-04-10T18:40:18Z |
| publishDate | 2022-06-01 |
| publisher | Al-Farabi Kazakh National University |
| record_format | Article |
| series | Вестник КазНУ. Серия математика, механика, информатика |
| spelling | doaj.art-0c8d41dcdc92410c9a47d415a04e57522023-02-01T14:49:41ZengAl-Farabi Kazakh National UniversityВестник КазНУ. Серия математика, механика, информатика1563-02772617-48712022-06-011142101116https://doi.org/10.26577/JMMCS.2022.v114.i2.010Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivativesDossan Baigereyev0https://orcid.org/0000-0003-4396-9914Nurlana Alimbekova1https://orcid.org/0000-0002-1078-0480Nikolay Oskorbin2https://orcid.org/0000-0003-2902-0964S. Amanzholov East Kazakhstan UniversityS. Amanzholov East Kazakhstan UniversityAltai State UniversityThis 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 timederivatives, 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.https://bm.kaznu.kz/index.php/kaznu/article/view/998/665finite element methodfractional derivative of caputo-fabrizioconvergencefiltration problemfractured porous medium |
| spellingShingle | Dossan Baigereyev Nurlana Alimbekova Nikolay Oskorbin Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives Вестник КазНУ. Серия математика, механика, информатика finite element method fractional derivative of caputo-fabrizio convergence filtration problem fractured porous medium |
| title | Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives |
| title_full | Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives |
| title_fullStr | Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives |
| title_full_unstemmed | Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives |
| title_short | Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives |
| title_sort | error estimates of the numerical method for the filtration problem with caputo fabrizio fractional derivatives |
| topic | finite element method fractional derivative of caputo-fabrizio convergence filtration problem fractured porous medium |
| url | https://bm.kaznu.kz/index.php/kaznu/article/view/998/665 |
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