关于分数阶导数定义的商榷  

作　　者：吴佳[1] 施伟辰[1] 

机构地区：[1]上海海事大学物流工程学院,中国上海201306

出　　处：《科技视界》2015年第11期88-89,共2页Science & Technology Vision

摘　　要：导数的概念最早是由莱布尼茨引入的,记作dny/dxn。当函数导数次数不是整数而是分数时,即为分数阶导数dαy/dxα(0<α<1)。本文利用数学归纳法推导出分数阶导数的两种公式,即所谓的分数阶导数Riemann-liouville(R-L)"定义"、和Caputo"定义"。但他们都是从分数阶导数dαy dxα0<α<1<出发推导得到,并且互为等价,所以只是表达形式不同,含义相同。以此类推,或许存在其他的所谓的"定义"也是从dαy/dxα0<<α<1<出发获得,因而他们也只是分数阶导数不同的计算公式,而非定义。因此,分数阶导数的定义应是dαy/dxα(0<α<1),而所谓的Riemannliouville(R-L)"定义"、和Caputo"定义"只能称作是分数阶导数的两种不同的计算公式。这是本文商榷的问题。The concept of derivative denoted as （d^ny）/（dx^n） was firstly introduced by Leibniz. And the fractional derivative is denoted as （d^αy）/（dx^α）（0〈α〈1） when the derivative order of function is not an integer. In this paper, two fractional derivative formulas which contain so-called Riemann-liouville （R-L） “definition” and Caputo “definition ”are calculated with mathematical induction. But they are all from the fractional derivative expression （d^αy）/（dx^α）（0〈α〈1） and are equivalent with the same meaning. Similarly , perhaps there are other so-called “definitions” which are also from the expression （d^αy）/（dx^α）（0〈α〈1）, thus they are just different fractional derivative formulas rather than real definitions.Therefore, the definition of fractional derivative should be still the expression （d^αy）/（dx^α） （0〈α〈1）, and the so-called R-L “definition” and Caputo“definition ” can just be called as two different formulas to calculate fractional derivative. This is the question to be deliberated in this paper.

关 键 词：分数阶导数 Riemann-Liouville定义 Caputo定义 等价 计算公式 

分 类 号：O37[理学—流体力学]