论函数的可导与分数阶可导  

On the Differentiability and Fractional Order Differentiability of Functions

作　　者：刘书霞 詹华税[1] LIU Shuxia;ZHAN Huashui(School of Mathematics and Statistics,Xiamen University of Technology,Xiamen 361024,China)

机构地区：[1]厦门理工学院数学与统计学院,福建厦门361024

出　　处：《厦门理工学院学报》2022年第3期75-80,共6页Journal of Xiamen University of Technology

摘　　要：从是否存在一点可导的相关函数和求导法则间相互关系的视角讨论函数的可导性问题,在分析一元分段函数在分界点处的导数问题的基础上,引进Riemann-Liouville分数阶导数定义和Caputo分数阶导数的定义,探讨分数阶导数与整数阶导数的相容性问题,研究分数阶可导问题。结果表明:仅在一点可导的函数及其他相关函数是存在的;导数的加法运算在四则运算中最为重要,复合函数的求导法在求导方法中最重要;Riemann-Liouville分数阶导数与经典整数阶导数具有相容性,Caputo分数阶导数与经典整数阶导数的相容性略差。This paper discusses the differentiability of function from the perspective of whether there is a differentiable correlation function and the relationship between derivation rules.Based on the analysis of the derivative of univariate piecewise function at the boundary point,the definitions of Riemann-Liouville fractional derivative and Caputo fractional derivative are introduced to discuss the compatibility between fractional derivative and integer derivative,and the fractional derivative problem is studied.The results show that functions and other related functions that can be derived only at one point exist;the addition operation of derivative is the most important in the four operations,and the derivation method of composite function is the most important in the derivation method;Riemann-Liouville fractional derivative is compatible with classical integer derivative,and Caputo fractional derivative is slightly incompatible with classical integer derivative.

关 键 词：函数 整数阶导数 分数阶导数 可导性 相容性 Riemann-Liouville分数阶导数 CAPUTO分数阶导数 

分 类 号：O172.1[理学—数学]