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.
关键词
奇异函数/分数阶导数/Hadamard有限部分积分/Chebyshev谱逼近
Key words
singular function/fractional derivative/Hadamard finite part integral/Chebyshev spectral approximation
维普
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