On Caputo modification of the Hadamard fractional derivatives
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RESEARCH
Open Access
On Caputo modification of the Hadamard fractional derivatives Yusuf Y Gambo1,2 , Fahd Jarad3* , Dumitru Baleanu2,4,5 and Thabet Abdeljawad2,6 *
Correspondence: [email protected] Department of Logistics Management, Faculty of Management, University of Turkish Aeronautical Association, Etimesgut, Ankara, 06790, Turkey Full list of author information is available at the end of the article 3
Abstract This paper is devoted to the study of Caputo modification of the Hadamard fractional derivatives. From here and after, by Caputo-Hadamard derivative, we refer to this modified fractional derivative (Jarad et al. in Adv. Differ. Equ. 2012:142, 2012, p.7). We present the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo-Hadamard setting. Also, several new related results are presented. Keywords: Caputo-Hadamard fractional derivatives; fundamental theorem of fractional calculus
1 Introduction Fractional calculus started to be considered deeply as a powerful tool to reveal the hidden aspects of the dynamics of the complex or hypercomplex systems [–]. Finding new generalization of the existing fractional derivatives was always the main direction of research within this field. These generalized operators will give us new opportunities to improve the existing results from theoretical and applied viewpoints. Although the works in [–] played important roles in the development of the fractional calculus within the frame of the Hadamard derivative, vast and vital work in this field is still undone. d ) in the definition of Hadamard fracThe presence of the δ-differential operator (δ = x dx tional derivatives could make their study uninteresting and less applicable than RiemannLiouville and Caputo fractional derivatives. More so, this operator appears outside the ind is tegral in the definition of the Hadamard derivatives just like the usual derivative D = dx located outside the integral in the case of Riemann-Liouville, which makes the fractional derivative of a constant of these two types not equal to zero in general. The authors in [] studied and modified the Hadamard derivatives into a more useful type using Caputo definitions. d α ) . This fractional derivative is Hadamard proposed a fractional power of the form (x dx invariant with respect to dilation on the whole axis. The Hadamard approach to fractional integral was based on the generalisation of the nth integral []
n Ja+ f
x
(x) = a
dt t
a
t
dt ··· t
tn–
f (tn ) a
dtn . tn
()
Just like Riemann-Liouville, Hadamard derivative has its own disadvantages as well, one of which is the fact that the derivative of a constant is not equal to in general. The authors in ©2014 Gambo et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Gambo et al. Advance
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