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Numerical Method for Fractional-Order Generalization of the Stochastic Stokes–Darcy Model

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  • Abdumauvlen Berdyshev

    (Institute of Information and Computational Technologies, 28 Shevchenko Str., Almaty 050010, Kazakhstan
    Department of Mathematics and Mathematical Modeling, Abai Kazakh National Pedagogical University, 86 Tole bi Str., Almaty 050012, Kazakhstan)

  • Dossan Baigereyev

    (Institute of Information and Computational Technologies, 28 Shevchenko Str., Almaty 050010, Kazakhstan
    Department of Mathematics, Higher School of IT and Natural Sciences, Sarsen Amanzholov East Kazakhstan University, 148 Shakarim Ave., Ust-Kamenogorsk 070002, Kazakhstan)

  • Kulzhamila Boranbek

    (Department of Mathematics, Higher School of IT and Natural Sciences, Sarsen Amanzholov East Kazakhstan University, 148 Shakarim Ave., Ust-Kamenogorsk 070002, Kazakhstan)

Abstract

This paper is aimed at efficient numerical implementation of the fractional-order generalization of the stochastic Stokes–Darcy model, which has important scientific, applied, and economic significance in hydrology, the oil industry, and biomedicine. The essence of this generalization of the stochastic model is the introduction of fractional time derivatives in the sense of Caputo’s definition to take into account long-term changes in the properties of media. An efficient numerical method for the implementation of the fractional-order Stokes–Darcy model is proposed, which is based on the use of a higher-order approximation formula for the fractional derivative, higher-order finite difference relations, and a finite element approximation of the problem in the spatial direction. In the paper, a rigorous theoretical analysis of the stability and convergence of the proposed numerical method is carried out, which is confirmed by numerous computational experiments. Further, the proposed method is applied to the implementation of the fractional-order stochastic Stokes–Darcy model using an ensemble technique, in which the approximation is carried out in such a way that the resulting systems of linear equations have the same coefficient matrix for all realizations. Furthermore, evaluation of the discrete fractional derivatives is carried out with the use of parallel threads. The efficiency of applying both approaches has been demonstrated in numerical tests.

Suggested Citation

  • Abdumauvlen Berdyshev & Dossan Baigereyev & Kulzhamila Boranbek, 2023. "Numerical Method for Fractional-Order Generalization of the Stochastic Stokes–Darcy Model," Mathematics, MDPI, vol. 11(17), pages 1-27, September.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3763-:d:1231350
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    References listed on IDEAS

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    1. Wenwen Zhong & Changpin Li & Jisheng Kou, 2013. "Numerical Fractional-Calculus Model for Two-Phase Flow in Fractured Media," Advances in Mathematical Physics, Hindawi, vol. 2013, pages 1-7, August.
    2. Dragan Poljak & Silvestar Šesnić & Mario Cvetković & Anna Šušnjara & Hrvoje Dodig & Sébastien Lalléchère & Khalil El Khamlichi Drissi, 2018. "Stochastic Collocation Applications in Computational Electromagnetics," Mathematical Problems in Engineering, Hindawi, vol. 2018, pages 1-13, May.
    3. Yang, Xuehua & Wu, Lijiao & Zhang, Haixiang, 2023. "A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity," Applied Mathematics and Computation, Elsevier, vol. 457(C).
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    Cited by:

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    2. Nurlana Alimbekova & Abdumauvlen Berdyshev & Muratkan Madiyarov & Yerlan Yergaliyev, 2024. "Finite Element Method for a Fractional-Order Filtration Equation with a Transient Filtration Law," Mathematics, MDPI, vol. 12(16), pages 1-20, August.

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