Comparison of Definition of Several Fractional Derivatives

DOI

10.2991/icemc-16.2016.114How to use a DOI?

Keywords

Riemann-Liouville fractional derivatives; Caputo's fractional derivative; Grunwald- Letnikov fractional derivatives; Fractional integral.

Abstract

The idea of fractional derivatives was raised first by L'Hospital in 1695. The fractional calculus and its mathematical consequences attracted many mathematicians such as Fourier, Euler, Laplace. Various definitions of non-integer order integral or derivative was given by many mathematicians. In this paper, firstly, we discuss the positive integer higher order derivative of a function, and obtain general formula of high order derivative. Secondly, we introduce the definitions of Grunwald-Letnikov, Riemann-Liouville and Caputo. Finally, we point out the relationship between these definitions. Caputo's integral definition and Grunwald-Letnikov integral definition are consistent with the Riemann-Liouville integral definition. When f has m+1 order continuous derivative and m

Copyright

© 2016, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - CONF
AU  - Yun Ouyang
AU  - Wusheng Wang
PY  - 2016/05
DA  - 2016/05
TI  - Comparison of Definition of Several Fractional Derivatives
BT  - Proceedings of the 2016 International Conference on Education, Management and Computer Science
PB  - Atlantis Press
SP  - 553
EP  - 557
SN  - 1951-6851
UR  - https://doi.org/10.2991/icemc-16.2016.114
DO  - 10.2991/icemc-16.2016.114
ID  - Ouyang2016/05
ER  -