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Volume 18, Issue 2
L1 Schemes for Time-Fractional Differential Equations: A Brief Survey and New Development

Jingjing Xiao, Yanping Chen, Fanhai Zeng & Zhongqiang Zhang

DOI: 10.4208/nmtma.OA-2024-0085

Numer. Math. Theor. Meth. Appl., 18 (2025), pp. 544-574.

Published online: 2025-05

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  • Abstract

We summarize the L1 method and its variants for the discretization of the Caputo derivative and develop a weighted L1 (WL1) method. The WL1 method includes the classical L1 method as a special case but improves the L1 method in accuracy while accommodating the intrinsic weak singularity of solutions to fractional differential equations. The WL1 method inherits several properties of the L1 method and the analysis of the new method is presented in a simple way. Compared to several variants of the L1 method, the new method accommodates the intrinsic weak singularity of the time-fractional derivatives in a more flexible way, especially for variable-order fractional differential equation. We also develop the fast WL1 method and apply it to initial and boundary value problems. Numerical simulations and comparisons verify our theoretical analysis and demonstrate the flexibility and efficiency of our method.

  • Keywords

The weighted interpolation, fractional initial value problem, fractional subdiffusion equation, fast convolution method, variable-order fractional operator.

  • AMS Subject Headings

26A33, 65M06, 65M12, 65M15, 35R11

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-18-544, author = {Xiao , JingjingChen , YanpingZeng , Fanhai and Zhang , Zhongqiang}, title = {L1 Schemes for Time-Fractional Differential Equations: A Brief Survey and New Development}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2025}, volume = {18}, number = {2}, pages = {544--574}, abstract = {

We summarize the L1 method and its variants for the discretization of the Caputo derivative and develop a weighted L1 (WL1) method. The WL1 method includes the classical L1 method as a special case but improves the L1 method in accuracy while accommodating the intrinsic weak singularity of solutions to fractional differential equations. The WL1 method inherits several properties of the L1 method and the analysis of the new method is presented in a simple way. Compared to several variants of the L1 method, the new method accommodates the intrinsic weak singularity of the time-fractional derivatives in a more flexible way, especially for variable-order fractional differential equation. We also develop the fast WL1 method and apply it to initial and boundary value problems. Numerical simulations and comparisons verify our theoretical analysis and demonstrate the flexibility and efficiency of our method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2024-0085}, url = {http://global-sci.org/intro/article_detail/nmtma/24075.html} }
TY - JOUR T1 - L1 Schemes for Time-Fractional Differential Equations: A Brief Survey and New Development AU - Xiao , Jingjing AU - Chen , Yanping AU - Zeng , Fanhai AU - Zhang , Zhongqiang JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 544 EP - 574 PY - 2025 DA - 2025/05 SN - 18 DO - http://doi.org/10.4208/nmtma.OA-2024-0085 UR - https://global-sci.org/intro/article_detail/nmtma/24075.html KW - The weighted interpolation, fractional initial value problem, fractional subdiffusion equation, fast convolution method, variable-order fractional operator. AB -

We summarize the L1 method and its variants for the discretization of the Caputo derivative and develop a weighted L1 (WL1) method. The WL1 method includes the classical L1 method as a special case but improves the L1 method in accuracy while accommodating the intrinsic weak singularity of solutions to fractional differential equations. The WL1 method inherits several properties of the L1 method and the analysis of the new method is presented in a simple way. Compared to several variants of the L1 method, the new method accommodates the intrinsic weak singularity of the time-fractional derivatives in a more flexible way, especially for variable-order fractional differential equation. We also develop the fast WL1 method and apply it to initial and boundary value problems. Numerical simulations and comparisons verify our theoretical analysis and demonstrate the flexibility and efficiency of our method.

Xiao , JingjingChen , YanpingZeng , Fanhai and Zhang , Zhongqiang. (2025). L1 Schemes for Time-Fractional Differential Equations: A Brief Survey and New Development. Numerical Mathematics: Theory, Methods and Applications. 18 (2). 544-574. doi:10.4208/nmtma.OA-2024-0085
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