Caputo-type modification of the Hadamard fractional derivatives
© Jarad et al.; licensee Springer 2012
Received: 24 May 2012
Accepted: 9 July 2012
Published: 10 August 2012
Generalization of fractional differential operators was subjected to an intense debate in the last few years in order to contribute to a deep understanding of the behavior of complex systems with memory effect. In this article, a Caputo-type modification of Hadamard fractional derivatives is introduced. The properties of the modified derivatives are studied.
The fractional calculus which is as old as the usual calculus is the generalization of differentiation and integration of integer order to arbitrary ones.
In the last 30 years or so, there has been an important development of the field of fractional calculus as it has been applied to many fields of science. It was pointed out that fractional derivatives and integrals are more convenient for describing real materials, and some physical problems were treated by using derivatives of non-integer orders [1, 2, 5–9, 12].
Maybe the most well-known fractional integral is the one developed by Riemann and Liouville and based on the generalization of the usual Riemann integral .
where , . Here and in the following, represents the Gamma function.
for , .
where α represents the order of the derivative such that . By definition the Caputo fractional derivative of a constant is zero.
where , , , .
The question arises here is whether we can modify the Hadamard fractional derivatives so that the derivatives of a constant is 0 and these derivatives contain physically interpretable initial conditions similar to the ones in Caputo fractional derivatives.
The main goal of this article is to define the Caputo-type modification of the Hadamard fractional derivatives and present properties of such derivatives.
2 Caputo-type Hadamard fractional derivatives
- (i)if , can be represented by(20)and can be represented by(21)
- (ii)if , then(22)
From Lemma 1.2 in , one derives . The second formula in (22) can be proved likewise. □
- (i)if , and can be represented by (20) and (21) respectively and(24)
if , then the formulas in (22) hold.
When , the first formula in (22) holds as a result of Lemma 1.4 in . The second formula can be proved likewise. □
- (ii)If , then and are bounded from to and(29)
Proof (27) and (28) follow from the estimates (25) and (26) keeping in mind that (see formula 1.1.28 in  when ). The estimates in (29) are straightforward. □
and provide operations inverse to and respectively for or . But it is not the case for and .
from which we conclude that , and thus the first identity in (30) holds. The second identity is proved analogously.
Thus we have , and hence (31) holds. (32) can be proved similarly. □
Proof (36) and (37) follow from the identities and respectively. □
The Caputo modifications of the left and right Hadamard fractional derivatives have the same properties 2.7.16 and 2.7.18. in  but they are different from the ones in 2.7.19.
The Mellin transforms of the Caputo modifications of the left and right Hadamard fractional derivatives are the same as Mellin transforms of the left and right Hadamard fractional derivatives 2.7.72 and 2.7.74 in .
- (i)If , then(44)
- (ii)If , then(45)
(45) can be proved similarly. □
DB is on leave of absence from Institute of Space Sciences, P.O. Box MG-23, R 76900, Magurele-Bucharest, Romania.
This work is partially supported by the Scientific and Technical Research Council of Turkey.
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