Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media
This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the initial-boundary value problem was constructed, and...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-10-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/8/11/640 |
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| author | Nurlana Alimbekova Aibek Bakishev Abdumauvlen Berdyshev |
| author_facet | Nurlana Alimbekova Aibek Bakishev Abdumauvlen Berdyshev |
| author_sort | Nurlana Alimbekova |
| collection | DOAJ |
| description | This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the initial-boundary value problem was constructed, and a theoretical study of the stability and convergence of the method was carried out using the method of a priori estimates. The results were confirmed through a comparative analysis of the empirical and theoretical orders of convergence based on computational experiments. Furthermore, we analyzed the effect of variable-order functions of fractional derivatives on the process of fluid flow in a heterogeneous medium, presenting new practical results in the field of modeling the fluid flow in complex media. This work is an important contribution to the numerical modeling of filtration in porous media with variable orders of fractional derivatives and may be useful for specialists in the field of hydrogeology, the oil and gas industry, and other related fields. |
| format | Article |
| id | doaj-art-41299de777bb45e5bab531b2b6a0ccb2 |
| institution | OA Journals |
| issn | 2504-3110 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-41299de777bb45e5bab531b2b6a0ccb22025-08-20T02:04:57ZengMDPI AGFractal and Fractional2504-31102024-10-0181164010.3390/fractalfract8110640Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous MediaNurlana Alimbekova0Aibek Bakishev1Abdumauvlen Berdyshev2Department of Mathematics, Higher School of IT and Natural Sciences, Sarsen Amanzholov East Kazakhstan University, 148 Shakarim Ave., Ust-Kamenogorsk 070002, KazakhstanDepartment of Mathematics, Higher School of IT and Natural Sciences, Sarsen Amanzholov East Kazakhstan University, 148 Shakarim Ave., Ust-Kamenogorsk 070002, KazakhstanInstitute of Information and Computational Technologies, 28 Shevchenko Str., Almaty 050010, KazakhstanThis paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the initial-boundary value problem was constructed, and a theoretical study of the stability and convergence of the method was carried out using the method of a priori estimates. The results were confirmed through a comparative analysis of the empirical and theoretical orders of convergence based on computational experiments. Furthermore, we analyzed the effect of variable-order functions of fractional derivatives on the process of fluid flow in a heterogeneous medium, presenting new practical results in the field of modeling the fluid flow in complex media. This work is an important contribution to the numerical modeling of filtration in porous media with variable orders of fractional derivatives and may be useful for specialists in the field of hydrogeology, the oil and gas industry, and other related fields.https://www.mdpi.com/2504-3110/8/11/640fractional differential equationsvariable order of fractional derivativeconvergencestabilityfiltrationheterogeneous porous medium |
| spellingShingle | Nurlana Alimbekova Aibek Bakishev Abdumauvlen Berdyshev Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media Fractal and Fractional fractional differential equations variable order of fractional derivative convergence stability filtration heterogeneous porous medium |
| title | Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media |
| title_full | Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media |
| title_fullStr | Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media |
| title_full_unstemmed | Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media |
| title_short | Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media |
| title_sort | numerical method for the variable order fractional filtration equation in heterogeneous media |
| topic | fractional differential equations variable order of fractional derivative convergence stability filtration heterogeneous porous medium |
| url | https://www.mdpi.com/2504-3110/8/11/640 |
| work_keys_str_mv | AT nurlanaalimbekova numericalmethodforthevariableorderfractionalfiltrationequationinheterogeneousmedia AT aibekbakishev numericalmethodforthevariableorderfractionalfiltrationequationinheterogeneousmedia AT abdumauvlenberdyshev numericalmethodforthevariableorderfractionalfiltrationequationinheterogeneousmedia |