分数阶导数的不适定性及其相关问题

Ill-posedness of the Fractional Derivatives and Other Related Problems

根据分数阶导数的定义,计算基本初等函数在Riemann-Liouville分数阶导数和Caputo分数阶导数不同定义下的分数阶导数,并对同一基本初等函数不同分数阶导数进行计算,研究分数阶导数的不适定性、相容性和四则运算等问题。研究推断出基本初等函数分数阶导数随着阶数变化而变化的趋势,同时发现,分数阶导数并不具备整数阶导数的乘法和除法法则,而是具有更复杂的分析性质。

To study the ill-posedness,compatibility and basic arithmetic operations of fractional derivatives,the fractional derivatives of the elementary functions in the sense of Riemann-Liouville fractional derivative and the Caputo fractional derivative are calculated respectively,and the different order fractional derivatives for a given elementary function calculated according to the definition of fractional derivatives.Thus,it is inferred that the fractional derivatives of a given elementary function change with the order change of the fractional derivatives,and it is found that the usual multiplication and division rule of integer order derivative does not apply to a fractional derivative,which has more complicated analysis properties.