Caputo导数的非均匀L1-2公式及其应用

doi:10.12066/j.issn.1007-2861.2658

上海大学学报(自然科学版) ›› 2026, Vol. 32 ›› Issue (1): 166-186.doi: 10.12066/j.issn.1007-2861.2658

• 数学 • 上一篇

Caputo导数的非均匀L1-2公式及其应用

王俊玲¹, 李东霞²

1. 上海大学理学院, 上海 200444;
2. 内蒙古工业大学理学院, 内蒙古呼和浩特 010051

收稿日期:2024-11-11 发布日期:2026-03-16
通讯作者: 李东霞(1996-),女,博士,研究方向为分数阶偏微分方程的数值计算. E-mail:lidongxia96@163.com
基金资助:
国家自然科学基金资助项目(12271339)

The non-uniform L1-2 formula for Caputo derivatives and its applications

WANG Junling¹, LI Dongxia²

1. College of Sciences, Shanghai University, Shanghai 200444, China;
2. College of Sciences, Inner Mongolia University of Technology, Hohhot 010051, Inner Mongolia, China

Received:2024-11-11 Published:2026-03-16

摘要/Abstract

摘要： 构造了α∈(0,1)阶Caputo导数的一种数值逼近公式.考虑到Caputo导数在初始时刻的弱奇异性,在非均匀网格的第一个小区间上使用线性插值,其后的每个小区间上使用二次插值,从而推导出非均匀L1-2公式.证明了其截断误差可以达到(3-α)阶精度,并讨论了相应的系数性质.将得到的公式应用到时间分数阶扩散方程的数值求解中,数值实验验证了该公式的有效性和正确性.

关键词: L1-2公式, 分级网格, 弱正则性, 分数阶扩散方程

Abstract: This paper constructs a numerical approximation formula for the Caputo derivative of order α∈(0，1). Considering the weak regularity of the Caputo derivative at the initial time, linear interpolation is employed over the first subinterval of the non-uniform mesh, while quadratic interpolation is utilized for each subsequent subinterval, leading to the derivation of a non-uniform L1-2 formula. It is proven that the truncation error can achieve (3-fi)-order accuracy, and the corresponding coefficient properties are discussed. The derived formula is applied to the numerical solution of the time-fractional diffusion equation, and numerical experiments have verified the effectiveness and correctness of the formula.

Key words: L1-2 formula, graded mesh, weak regularity, fractional diffusion equation

中图分类号:

O241.82

王俊玲, 李东霞. Caputo导数的非均匀L1-2公式及其应用[J]. 上海大学学报(自然科学版), 2026, 32(1): 166-186.

WANG Junling, LI Dongxia. The non-uniform L1-2 formula for Caputo derivatives and its applications[J]. Journal of Shanghai University（Natural Science Edition）, 2026, 32(1): 166-186.

参考文献

[1] Li C P, Mainardi F. Editorial [J]. The european physical journal special topics, 2011, 193(1): 1-4.
[2] 史蒂芬￠ G. 萨姆科, 阿纳托利￠ A. 克尔巴斯, 奥列格￠ I. 马里切夫. 分数阶积分和导数||理论与应用[M]. 李常品, 李东霞, 译. 北京: 科学出版社, 2025.
[3] Li C P, Cai M. Theory and numerical approximations of fractional integrals and derivatives [M]. Philadelphia PA: SIAM, 2019.
[4] Oldham K B, Spanier J. The fractional calculus [M]. New York: Academic Press, 1974.
[5] Podlubny I. Fractional difierential equations [M]. San Diego: Academic Press, 1999.
[6] Samko S G, Kilbas A A, Marichev O I. Fractional integrals and derivatives: theory and applications [M]. Amsterdam: Gordon and Breach Science Publishers, 1993.
[7] 代巧巧, 李东霞. Caputo型时间分数阶变系数扩散方程的局部间断Galerkin方法[J]. 上海大学学报(自然科学版), 2024, 30(1): 174-190.
[8] Lin Y M, Xu C J. Finite difierence/spectral approximations for the time-fractional difiusion equation[J]. Journal of Computational Physics, 2007, 225(2): 1533-1552.
[9] Sun Z Z. The method of order reduction and its application to the numerical solutions of partial difierential equations [M]. Beijing: Science Press, 2009.
[10] Sun Z Z, Wu X. A fully discrete difierence scheme for a difiusion-wave system [J]. Applied Numerical Mathematics, 2006, 56(2): 193-209.
[11] Gao G H, Sun Z Z, Zhang H W. A new fractional numerical difierentiation formula to approximate the Caputo fractional derivative and its applications [J]. Journal of Computational Physics, 2014, 259: 33-50.
[12] Alikhanov A A. A new difierence scheme for the time fractional difiusion equation [J]. Journal of Computational Physics, 2015, 280: 424-438.
[13] Li C P, Zeng F H. Numerical methods for fractional calculus [M]. Boca Raton: CRC Press, 2015.
[14] Shen J Y, Li C P, Sun Z Z. An H2N2 interpolation for Caputo derivative with order in (1,2), and its application to time-fractional wave equations in more than one space dimension [J]. Journal of Scientific Computing, 2020, 83(2): 38.
[15] Dong Z N, Fan E Y, Shen A, et al. Three kinds of discrete formulae for the Caputo fractional derivative [J]. Communications on Applied Mathematics and Computation, 2023, 5: 1446-1468.
[16] Du R L, Li C P, Sun Z Z. H1-analysis of H3N3-2σ-based difierence method for fractional hyperbolic equations [J]. Computational and Applied Mathematics, 2024, 43(1): 69.
[17] Du R L, Li C P, Su F, et al. H3N3-2σ-based difierence schemes for time multi-term fractional difiusion-wave equation [J]. Computational and Applied Mathematics, 2024, 43(8): 416.
[18] Fan E Y, Li Y X, Zhao Q L. H3N3 approximate formulae for typical fractional derivatives [EB/OL]. (2024-01-15) [2024-03-03]. https://doi.org/10.1007/s42967-024-00395-w.
[19] Stynes M, O’riordan E, Gracia J L. Error analysis of a flnite difierence method on graded meshes for a time-fractional difiusion equation [J]. SIAM Journal on Numerical Analysis, 2017, 55(2): 1057-1079.
[20] Liao H L, Li D F, Zhang J W. Sharp error estimate of the nonuniform L1 formula for linear reaction-subdifiusion equations [J]. SIAM Journal on Numerical Analysis, 2018, 56(2): 1112- 1133.
[21] Chen H, Stynes M. Error analysis of a second-order method on fltted meshes for a timefractional difiusion problem [J]. Journal of Scientific Computing, 2019, 79: 624-647.

Caputo导数的非均匀L1-2公式及其应用

The non-uniform L1-2 formula for Caputo derivatives and its applications

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