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Fourth-Order Difference Scheme and a Matrix Transform Approach for Solving Fractional PDEs

Author

Listed:
  • Zahrah I. Salman

    (Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan P.O. Box 158-81595, Iran)

  • Majid Tavassoli Kajani

    (Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan P.O. Box 158-81595, Iran)

  • Mohammed Sahib Mechee

    (Information Technology Research and Development Center, University of Kufa, Najaf 540011, Iraq)

  • Masoud Allame

    (Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan P.O. Box 158-81595, Iran)

Abstract

Proposing a matrix transform method to solve a fractional partial differential equation is the main aim of this paper. The main model can be transferred to a partial-integro differential equation (PIDE) with a weakly singular kernel. The spatial direction is approximated by a fourth-order difference scheme. Also, the temporal derivative is discretized via a second-order numerical procedure. First, the spatial derivatives are approximated by a fourth-order operator to compute the second-order derivatives. This process produces a system of differential equations related to the time variable. Then, the Crank–Nicolson idea is utilized to achieve a full-discrete scheme. The kernel of the integral term is discretized by using the Lagrange polynomials to overcome its singularity. Subsequently, we prove the convergence and stability of the new difference scheme by utilizing the Rayleigh–Ritz theorem. Finally, some numerical examples in one-dimensional and two-dimensional cases are presented to verify the theoretical results.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3786-:d:1232165
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    References listed on IDEAS

    as
    1. 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.
    2. Luo, Ziyang & Zhang, Xingdong & Wang, Shuo & Yao, Lin, 2022. "Numerical approximation of time fractional partial integro-differential equation based on compact finite difference scheme," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
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