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Mathematics 2015, 3(2), 171-189; doi:10.3390/math3020171

Asymptotic Expansions of Fractional Derivatives andTheir Applications

1
Graduate School of Information Sciences, Tohoku University, Sendai 980-8577, Japan
2
College of Engineering, Nihon University, Koriyama 963-8642, Japan
*
Author to whom correspondence should be addressed.
Academic Editor: Hari M. Srivastava
Received: 3 March 2015 / Accepted: 7 April 2015 / Published: 15 April 2015
(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)
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Abstract

We compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two fIs. When this happens, this fact occurs also for the fractional derivative (fD).This method is applied to the derivation of the asymptotic expansion of the confluenthypergeometric function, which is a solution of Kummer’s differential equation. In thepresent paper, the solutions of the equation in the forms of the Riemann–Liouville fI orfD and the Liouville fI or fD are obtained by using the method, which Nishimoto used insolving the hypergeometric differential equation in terms of the Liouville fD. View Full-Text
Keywords: fractional derivative; asymptotic expansion; Kummer’s differential equation;confluent hypergeometric function fractional derivative; asymptotic expansion; Kummer’s differential equation;confluent hypergeometric function
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Morita, T.; Sato, K.-I. Asymptotic Expansions of Fractional Derivatives andTheir Applications. Mathematics 2015, 3, 171-189.

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