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Numerical Method for a Filtration Model Involving a Nonlinear Partial Integro-Differential Equation

Author

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  • Dossan Baigereyev

    (Department of Mathematics, High School of Information Technology and Natural Sciences, Amanzholov University, Ust-Kamenogorsk 070002, Kazakhstan)

  • Dinara Omariyeva

    (Department of Engineering Mathematics, Faculty of Basic Engineering Training, East Kazakhstan Technical University, Ust-Kamenogorsk 070003, Kazakhstan)

  • Nurlan Temirbekov

    (National Engineering Academy of the Republic of Kazakhstan, Almaty 050010, Kazakhstan)

  • Yerlan Yergaliyev

    (Department of Mathematics, High School of Information Technology and Natural Sciences, Amanzholov University, Ust-Kamenogorsk 070002, Kazakhstan)

  • Kulzhamila Boranbek

    (Department of Engineering Mathematics, Faculty of Basic Engineering Training, East Kazakhstan Technical University, Ust-Kamenogorsk 070003, Kazakhstan)

Abstract

In this paper, we propose an efficient numerical method for solving an initial boundary value problem for a coupled system of equations consisting of a nonlinear parabolic partial integro-differential equation and an elliptic equation with a nonlinear term. This problem has an important applied significance in petroleum engineering and finds application in modeling two-phase nonequilibrium fluid flows in a porous medium with a generalized nonequilibrium law. The construction of the numerical method is based on employing the finite element method in the spatial direction and the finite difference approximation to the time derivative. Newton’s method and the second-order approximation formula are applied for the treatment of nonlinear terms. The stability and convergence of the discrete scheme as well as the convergence of the iterative process is rigorously proven. Numerical tests are conducted to confirm the theoretical analysis. The constructed method is applied to study the two-phase nonequilibrium flow of an incompressible fluid in a porous medium. In addition, we present two examples of models allowing for prediction of the behavior of a fluid flow in a porous medium that are reduced to solving the nonlinear integro-differential equations studied in the paper.

Suggested Citation

  • Dossan Baigereyev & Dinara Omariyeva & Nurlan Temirbekov & Yerlan Yergaliyev & Kulzhamila Boranbek, 2022. "Numerical Method for a Filtration Model Involving a Nonlinear Partial Integro-Differential Equation," Mathematics, MDPI, vol. 10(8), pages 1-24, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:8:p:1319-:d:794838
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    References listed on IDEAS

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    1. M. H. Daliri Birjandi & J. Saberi-Nadjafi & A. Ghorbani, 2018. "An Efficient Numerical Method for a Class of Nonlinear Volterra Integro-Differential Equations," Journal of Applied Mathematics, Hindawi, vol. 2018, pages 1-7, August.
    2. Dossan Baigereyev & Nurlana Alimbekova & Abdumauvlen Berdyshev & Muratkan Madiyarov, 2021. "Convergence Analysis of a Numerical Method for a Fractional Model of Fluid Flow in Fractured Porous Media," Mathematics, MDPI, vol. 9(18), pages 1-25, September.
    3. 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.
    4. Erfanian, Majid & Mansoori, Amin, 2019. "Solving the nonlinear integro-differential equation in complex plane with rationalized Haar wavelet," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 223-237.
    5. Singh, Ajeet & Shukla, Anurag & Vijayakumar, V. & Udhayakumar, R., 2021. "Asymptotic stability of fractional order (1,2] stochastic delay differential equations in Banach spaces," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    6. Hashan, Mahamudul & Jahan, Labiba Nusrat & Tareq-Uz-Zaman, & Imtiaz, Syed & Hossain, M. Enamul, 2020. "Modelling of fluid flow through porous media using memory approach: A review," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 643-673.
    7. Li, Xiaoli & Rui, Hongxing, 2016. "A two-grid block-centered finite difference method for nonlinear non-Fickian flow model," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 300-313.
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    Cited by:

    1. Zahrah I. Salman & Majid Tavassoli Kajani & Mohammed Sahib Mechee & Masoud Allame, 2023. "Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs," Mathematics, MDPI, vol. 11(17), pages 1-15, September.

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