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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.
- Kilbas AA, Srivastava HH, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives - Theory and Applications. Gordon & Breach, Langhorne; 1993.Google Scholar
- Hadamarad J: Essai sur l’etude des fonctions donnes par leur developpment de Taylor. J. Pure Appl. Math. 1892, 4(8):101–186.Google Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Magin RL: Fractional Calculus in Bioengineering. Begell House Publishers, Redding; 2006.Google Scholar
- Heymans N, Podlubny I: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 2006, 45: 765–771. 10.1007/s00397-005-0043-5View ArticleGoogle Scholar
- Jesus IS, Machado JAT: Fractional control of heat diffusion system. Nonlinear Dyn. 2008, 54(3):263–282. 10.1007/s11071-007-9322-2MathSciNetView ArticleGoogle Scholar
- Mainardi F, Luchko Y, Pagnini G: The fundamental solution of the space-time fractional the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 2001, 4(2):153–192.MathSciNetGoogle Scholar
- Scalas E, Gorenflo R, Mainardi F: Uncoupled continuous-time random walks: solution and limiting behavior of the master equation. Phys. Rev. E 2004., 69: Article ID 011107Google Scholar
- Agrawal OP, Baleanu D: Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 2007, 13(9–10):1269–1281. 10.1177/1077546307077467MathSciNetView ArticleGoogle Scholar
- Chen YQ, Vinagre BM, Podlubny I: Continued fraction expansion approaches to discretizing fractional order derivatives-an expository review. Nonlinear Dyn. 2004, 38(1–4):155–170. 10.1007/s11071-004-3752-xMathSciNetView ArticleGoogle Scholar
- Tarasov VE, Zaslavsky GM: Nonholonomic constraints with fractional derivatives. J. Phys. A, Math. Gen. 2006, 39(31):9797–9815. 10.1088/0305-4470/39/31/010MathSciNetView ArticleGoogle Scholar
- Kilbas AA: Hadamard-type fractional calculus. J. Korean Math. Soc. 2001, 38(6):1191–1204.MathSciNetGoogle Scholar
- Kilbas AA, Titjura AA: Hadamard-type fractional integrals and derivatives. Tr. Inst. Mat., Minsk 2002, 11: 79–87.Google Scholar
- Butzer PL, Kilbas AA, Trujillo JJ: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 2002, 269(2):387–400. 10.1016/S0022-247X(02)00049-5MathSciNetView ArticleGoogle Scholar
- Butzer PL, Kilbas AA, Trujillo JJ: Fractional calculus in Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 2002, 269(1):1–27. 10.1016/S0022-247X(02)00001-XMathSciNetView ArticleGoogle Scholar
- Butzer PL, Kilbas AA, Trujillo JJ: Mellin transformation and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 2002, 270(1):1–15. 10.1016/S0022-247X(02)00066-5MathSciNetView ArticleGoogle Scholar
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