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A high-order L2 type difference scheme for the time-fractional diffusion equation

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  • Alikhanov, Anatoly A.
  • Huang, Chengming

Abstract

The present paper is devoted to constructing L2 type difference analog of the Caputo fractional derivative. The fundamental features of this difference operator are studied and it is used to construct difference schemes generating approximations of the second and fourth order in space and the (3−α)th-order in time for the time fractional diffusion equation with variable coefficients. Difference schemes were also constructed for the variable-order diffusion equation and the generalized fractional-order diffusion equation of the Sobolev type. Stability of the schemes under consideration as well as their convergence with the rate equal to the order of the approximation error are proven. The received results are supported by the numerical computations performed for some test problems.

Suggested Citation

  • Alikhanov, Anatoly A. & Huang, Chengming, 2021. "A high-order L2 type difference scheme for the time-fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 411(C).
    • Handle: RePEc:eee:apmaco:v:411:y:2021:i:c:s0096300321006299
    DOI: 10.1016/j.amc.2021.126545
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    References listed on IDEAS

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    1. Alikhanov, Anatoly A., 2015. "Numerical methods of solutions of boundary value problems for the multi-term variable-distributed order diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 12-22.
    2. Wang, Yuan-Ming & Ren, Lei, 2019. "A high-order L2-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 71-93.
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    Cited by:

    1. Kedia, Nikki & Alikhanov, Anatoly A. & Singh, Vineet Kumar, 2024. "Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 219(C), pages 337-354.
    2. Zhou, Ziyi & Zhang, Haixiang & Yang, Xuehua, 2024. "CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics," Applied Mathematics and Computation, Elsevier, vol. 474(C).
    3. Xuhao Li & Patricia J. Y. Wong, 2022. "gL 1 Scheme for Solving a Class of Generalized Time-Fractional Diffusion Equations," Mathematics, MDPI, vol. 10(8), pages 1-14, April.
    4. Srivastava, Nikhil & Singh, Vineet Kumar, 2023. "L3 approximation of Caputo derivative and its application to time-fractional wave equation-(I)," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 532-557.

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