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Analysis of a fast element-free Galerkin method for the multi-term time-fractional diffusion equation

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  • Hu, Zesen
  • Li, Xiaolin

Abstract

In this paper, a fast element-free Galerkin (EFG) method is proposed for solving the multi-term time-fractional diffusion equation (TFDE). Through the use of the multi-term L2−1σ formula to discrete the multi-term Caputo time-fractional derivative, a fast second-order scheme is presented for the temporal discretization, and then the EFG method is used to perform the spatial discretization. The stability of the temporal discretization scheme is discussed. By verifying the fractional Grönwall inequality, the global error of the proposed EFG method for the multi-term TFDE is analyzed in theory. Numerical results validate the theoretical results and the effectiveness of the method.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:223:y:2024:i:c:p:677-692
    DOI: 10.1016/j.matcom.2024.05.008
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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. 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.
    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. Derakhshan, Mohammad Hossein & Rezaei, Hamid & Marasi, Hamid Reza, 2023. "An efficient numerical method for the distributed order time-fractional diffusion equation with error analysis and stability," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 315-333.
    5. Ju, Yuejuan & Yang, Jiye & Liu, Zhiyong & Xu, Qiuyan, 2023. "Meshfree methods for the variable-order fractional advection–diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 489-514.
    6. 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.
    7. Wei, Leilei & Wang, Huanhuan, 2023. "Local discontinuous Galerkin method for multi-term variable-order time fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 685-698.
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