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A transformed L1 method for solving the multi-term time-fractional diffusion problem

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  • She, Mianfu
  • Li, Dongfang
  • Sun, Hai-wei

Abstract

In this paper, we present a novel scheme for solving a time-fractional initial–boundary value problem, where the equation contains a sum of Caputo derivatives with orders between 0 and 1. In order to overcome the difficulty of initial layer, we introduce a change of variable in the temporal direction and investigate the regularity of the solutions of the resulting system. A modified L1 approximation is used to approximate the Caputo derivatives and a standard Galerkin-Spectral method is applied to approximate the spatial derivatives. Unconditional stability and convergence of the fully-discrete scheme are proved by applying a novel discrete fractional Grönwall inequality. Finally, numerical examples are given to confirm our theoretical results.

Suggested Citation

  • She, Mianfu & Li, Dongfang & Sun, Hai-wei, 2022. "A transformed L1 method for solving the multi-term time-fractional diffusion problem," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 584-606.
    • Handle: RePEc:eee:matcom:v:193:y:2022:i:c:p:584-606
    DOI: 10.1016/j.matcom.2021.11.005
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    References listed on IDEAS

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    1. Li, Zhiyuan & Liu, Yikan & Yamamoto, Masahiro, 2015. "Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 381-397.
    2. Kolk, Marek & Pedas, Arvet & Tamme, Enn, 2016. "Smoothing transformation and spline collocation for linear fractional boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 234-250.
    3. 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.
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    Citations

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    Cited by:

    1. Han, Yuxin & Huang, Xin & Gu, Wei & Zheng, Bolong, 2023. "Linearized transformed L1 finite element methods for semi-linear time-fractional parabolic problems," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    2. Tan, Zhijun & Zeng, Yunhua, 2024. "Temporal second-order fully discrete two-grid methods for nonlinear time-fractional variable coefficient diffusion-wave equations," Applied Mathematics and Computation, Elsevier, vol. 466(C).
    3. Feng, Libo & Liu, Fawang & Anh, Vo V., 2023. "Galerkin finite element method for a two-dimensional tempered time–space fractional diffusion equation with application to a Bloch–Torrey equation retaining Larmor precession," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 517-537.
    4. Zhou, Yongtao & Li, Mingzhu, 2024. "Error estimate of a transformed L1 scheme for a multi-term time-fractional diffusion equation by using discrete comparison principle," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 395-404.
    5. Boya Zhou & Xiujun Cheng, 2023. "A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions," Mathematics, MDPI, vol. 11(12), pages 1-10, June.
    6. Li, Yuyu & Wang, Tongke & Gao, Guang-hua, 2023. "The asymptotic solutions of two-term linear fractional differential equations via Laplace transform," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 394-412.
    7. Hu, Zesen & Li, Xiaolin, 2024. "Analysis of a fast element-free Galerkin method for the multi-term time-fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 677-692.

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    More about this item

    Keywords

    Multi-term time-fractional equation; Modified L1 scheme; Chebyshev–Galerkin spectral method; Error estimates;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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