1$\sim$2 阶 Caputo 分数阶导数的稳定化数值微分

数学物理学报 ›› 2026, Vol. 46 ›› Issue (6): 2277-2290.

1$\sim$2 阶 Caputo 分数阶导数的稳定化数值微分

周洁(), 阮周生^*(), 谢珍珍, 朱俊杰

东华理工大学理学院南昌 330013

收稿日期:2026-07-01 修回日期:2026-09-03 出版日期:2026-12-26 发布日期:2026-08-14
通讯作者: 阮周生 E-mail:2407085901@qq.com;zhshruan@ecut.edu.cn
作者简介:周洁, E-mail: 2407085901@qq.com
基金资助:
国家自然科学基金项目(12061008)

Stable Numerical Differentiation for the Caputo Fractional OrderDerivatives of Order between 1 and 2

Jie Zhou(), Zhousheng Ruan^*(), Zhenzhen Xie, Junjie Zhu

School of Science, East China University of Technology, Nanchang 330013

Received:2026-07-01 Revised:2026-09-03 Online:2026-12-26 Published:2026-08-14
Contact: Zhousheng Ruan E-mail:2407085901@qq.com;zhshruan@ecut.edu.cn
Supported by:
NSFC(12061008)

摘要/Abstract

摘要：

该文首先基于 Lavrentiev 正则化思想, 构造了求解扰动数据 1$\sim$2 阶 Caputo 分数阶导数的一类正则化方法; 然后基于对精确数据的先验假设, 证明了正则化分数阶导数在正则化参数先验与后验选取策略下的收敛率; 最后利用数值积分法则, 设计了正则化分数阶导数的数值计算格式, 并通过数值算例验证了数值计算格式的有效性和稳定性.

关键词: 数值微分, Caputo导数, 正则化, 收敛率

Abstract:

In this paper, a class of regularization methods for solving the 1st to 2nd order Caputo fractional derivatives with perturbation data is proposed based on Lavrentiev regularization idea. Then, under a priori assumptions on the exact data, the convergence rates of the regularized fractional derivatives are proved for both a priori and a posteriori parameter choice strategies.Finally, a numerical calculation scheme for regularized fractional derivatives is designed using the numerical integration rule, and the effectiveness and stability of the numerical calculation scheme are verified through numerical examples.

Key words: numerical differentiation, caputo fractional derivative, regularization, convergence rate.

中图分类号:

O241.4

周洁, 阮周生, 谢珍珍, 朱俊杰. 1$\sim$2 阶 Caputo 分数阶导数的稳定化数值微分[J]. 数学物理学报, 2026, 46(6): 2277-2290.

Jie Zhou, Zhousheng Ruan, Zhenzhen Xie, Junjie Zhu. Stable Numerical Differentiation for the Caputo Fractional OrderDerivatives of Order between 1 and 2[J]. Acta mathematica scientia,Series A, 2026, 46(6): 2277-2290.

图/表 2

参考文献 14

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