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On high order numerical schemes for fractional differential equations by block-by-block approach

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  • Li, Lili
  • Zhao, Dan
  • She, Mianfu
  • Chen, Xiaoli

Abstract

The exact solutions to nonlinear fractional problems usually have initial singularity. Taking the singularity into account, the change of variable and the block-by-block approach are introduced to propose a novel high-order scheme. It is proved that our proposed scheme can be of order 3+α under the non-smooth solutions. Numerical examples are shown to validate our theoretical results.

Suggested Citation

  • Li, Lili & Zhao, Dan & She, Mianfu & Chen, Xiaoli, 2022. "On high order numerical schemes for fractional differential equations by block-by-block approach," Applied Mathematics and Computation, Elsevier, vol. 425(C).
    • Handle: RePEc:eee:apmaco:v:425:y:2022:i:c:s0096300322001825
    DOI: 10.1016/j.amc.2022.127098
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    References listed on IDEAS

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    1. Li, Lili & Zhou, Boya & Chen, Xiaoli & Wang, Zhiyong, 2018. "Convergence and stability of compact finite difference method for nonlinear time fractional reaction–diffusion equations with delay," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 144-152.
    2. Li, Changpin & Wang, Zhen, 2021. "Non-uniform L1/discontinuous Galerkin approximation for the time-fractional convection equation with weak regular solution," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 838-857.
    3. Ran, Maohua & Zhou, Xiaoyi, 2021. "An implicit difference scheme for the time-fractional Cahn–Hilliard equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 180(C), pages 61-71.
    4. Iomin, Alexander, 2011. "Fractional-time Schrödinger equation: Fractional dynamics on a comb," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 348-352.
    5. Al-Mdallal, Qasem M. & Abu Omer, Ahmed S., 2018. "Fractional-order Legendre-collocation method for solving fractional initial value problems," Applied Mathematics and Computation, Elsevier, vol. 321(C), pages 74-84.
    6. Zhang, Xinguang & Liu, Lishan & Wu, Yonghong & Wiwatanapataphee, B., 2015. "The spectral analysis for a singular fractional differential equation with a signed measure," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 252-263.
    7. Li, Dongfang & Zhang, Chengjian, 2020. "Long time numerical behaviors of fractional pantograph equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 244-257.
    8. Yang, Yin & Wang, Jindi & Zhang, Shangyou & Tohidi, Emran, 2020. "Convergence analysis of space-time Jacobi spectral collocation method for solving time-fractional Schrödinger equations," Applied Mathematics and Computation, Elsevier, vol. 387(C).
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    Cited by:

    1. Yang, Changqing, 2023. "Improved spectral deferred correction methods for fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).

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