东北大学学报（自然科学版）

东北大学学报:自然科学版 ›› 2018, Vol. 39 ›› Issue (4): 604-608.DOI: 10.12068/j.issn.1005-3026.2018.04.030

• 数学 • 上一篇    

分数阶微积分的高精度递推算法

白鹭1,2， 薛定宇1   

  1. (1. 东北大学 信息科学与工程学院， 辽宁 沈阳110819; 2. 沈阳大学 信息工程学院， 辽宁 沈阳110044)
  • 收稿日期:2016-11-06 修回日期:2016-11-06 出版日期:2018-04-15 发布日期:2018-04-10
  • 通讯作者: 白鹭
  • 作者简介:白鹭(1982-)，男，辽宁沈阳人，东北大学博士研究生； 薛定宇(1963-)，男，辽宁沈阳人，东北大学教授，博士生导师.
  • 基金资助:
    国家自然科学基金资助项目(51171041).国家自然科学基金资助项目(61174145，61673094).

High Precision Recursive Algorithm for Computing Fractional-Order Derivative and Integral

BAI Lu1,2， XUE Ding-yu1   

  1. 1. School of Information Science & Engineering， Northeastern University， Shenyang 110819， China; 2. School of Information Engineering， Shenyang University， Shenyang 110044， China.
  • Received:2016-11-06 Revised:2016-11-06 Online:2018-04-15 Published:2018-04-10
  • Contact: XUE Ding-yu
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摘要： 设计了一种计算分数阶微积分的高精度数值算法，提出了一种构造生成函数的简便方法.分析了基于快速Fourier变换的算法，该算法误差较大的原因是应用了不准确的生成函数的系数，而且没有考虑原函数的非零初值条件对计算精度的影响.新算法应用递推公式计算生成函数的系数，并将原函数分解成零初值条件和非零初值条件两部分，分别计算它们的分数阶微分和积分，这样可以减小计算误差.误差分析和计算实例证明新算法具有很高的计算精度.

关键词: 分数阶, 微积分, 生成函数, 高精度, 递推算法

Abstract: A high precision numerical algorithm was designed to compute fractional-order derivative and integral， and a simple method was proposed to construct the generating function. An algorithm based on fast Fourier transform was analyzed. It could be concluded that the reasons of its large computation error were using the inaccurate coefficient of the generating function and no considering the effect of nonzero initial condition of the original function on calculation precision. The recursive formula was used to compute the coefficient of the generating function in the new algorithm， what’s more， the original function was decomposed into two parts， i.e.， zero initial condition and nonzero initial condition， and their fractional-order derivative and integral were computed to decrease the computation error. The error analysis and the illustrative numerical examples showed that the computation accuracy of the new algorithm was very high.

Key words: fractional-order, derivative and integral, generating function, high-precision, recursive algorithm

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