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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (4): 533-545.doi: 10.3969/j.issn.1005-3085.2015.04.007

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分数阶微分方程block-by-block算法的最优阶收敛性分析

王自强,   曹俊英   

  1. 贵州民族大学理学院，贵阳 550025
  • 收稿日期:2014-01-27 接受日期:2014-09-11 出版日期:2015-08-15 发布日期:2015-10-15
  • 通讯作者: 曹俊英 E-mail: caojunying1000@126.com
  • 基金资助:
    国家基础研究计划973项目 (2012CB025904); 国家自然科学基金数学天元基金 (11426074)；贵州省科学技术基金 ([2014]2098; [2013]2144)；贵州省教育厅项目 ([2013]405).

Optimal Convergence Order Analysis of a Block-by-block Algorithm for Fractional Differential Equations

WANG Zi-qiang,   CAO Jun-ying   

  1. College of Science, Guizhou Minzu University, Guiyang 550025
  • Received:2014-01-27 Accepted:2014-09-11 Online:2015-08-15 Published:2015-10-15
  • Contact: J. Cao.E-mail address: caojunying1000@126.com
  • Supported by:
    The Special Funds for National Basic Research Program of China (2012CB025904); the Tianyuan Special Funds of the National Natural Science Foundation of China (11426074); the Foundation of Guizhou Science and Technology Department ([2014]2098; [2013]2144); the Foundation of Guizhou Education Department ([2013]405).

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摘要： 经典的block-by-block方法是求解积分方程的一种高效的数值方法．研究者们已经把经典的block-by-block方法成功地用在构造非线性分数阶常微分方程的高阶数值格式上，对该格式的收敛性分析也已经有了初步的结果．但数值实验的结果表明目前的理论分析仍未达到最优阶误差估计．本文将利用Taylor公式和积分中值定理对非线性分数阶常微分方程的block-by-block方法的收敛性进行细致的分析，对其获得了最优阶误差估计，最后通过数值算例验证了理论分析的正确性．

关键词: 分数阶微分方程, block-by-block算法, 收敛性分析, Caputo导数

Abstract:

The classic block-by-block method is a highly efficient numerical method to solve the integral equation. Using the classic block-by-block method, researchers have successfully constructed higher order numerical methods for nonlinear fractional ordinary differential equ-ation, and made preliminary analysis on the convergence of this numerical method. But the results of numerical experiments show that the theoretical analysis does not achieve the optimal error estimate order. Based on the Taylor formula and integral mean value theorem, this article makes a thorough analyses on the block-by-block method of nonlinear fractional ordinary differential equations and obtains the optimal error estimate order. Finally numerical experiments are carried out to support the theoretical claims.

Key words: fractional differential equation, block-by-block algorithm, convergence analysis, Caputo derivative

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