奇异函数分数阶导数的Hadamard有限部分积分表示形式

Hadamard finite part integral representations for fractional derivatives of singular functions

针对包含奇点的函数,研究其Riemann-Liouville和Caputo分数阶导数,给出它们的Hadamard有限部分积分表示形式.利用该形式求得分数阶导数在初始点的Psi级数展开式.另外,该形式可以方便地使用Hadamard有限部分积分算法进行高精度计算.最后设计了一种奇点分离的Chebyshev谱逼近方法,通过数值算例验证了分数阶导数的Hadamard积分表示形式及其数值算法的正确性和有效性.

The Riemann-Liouville and Caputo fractional derivatives are studied for functions involving a singular point,and their Hadamard finite part integral representations are given.Then the representations are used to derive the Psi series expan-sions for the fractional derivatives about the origin,which accurately describe the singular behavior of the fractional derivatives.In addition,the representations can conveniently be used to calculate the fractional derivatives by using the high-accuracy al-gorithm evaluating the Hadamard finite part integrals.Finally,a Chebyshev spectral approximation method with singularity sep-aration is designed,and the correctness and effectiveness of the Hadamard representation of fractional derivative and its nu-merical algorithm are verified by numerical examples.