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On Finite Part Integrals and Hadamard-Type Fractional Derivatives

[+] Author and Article Information
Li Ma

School of Mathematics,
Hefei University of Technology,
Hefei 230009, China
e-mail: mali20062787@163.com

Changpin Li

Department of Mathematics,
Shanghai University,
Shanghai 200444, China
e-mail: lcp@shu.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 18, 2017; final manuscript received August 29, 2017; published online July 26, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(9), 090905 (Jul 26, 2018) (7 pages) Paper No: CND-17-1317; doi: 10.1115/1.4037930 History: Received July 18, 2017; Revised August 29, 2017

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

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