On Caputo modification of the Hadamard fractional derivatives
© Gambo et al.; licensee Springer. 2014
Received: 19 September 2013
Accepted: 5 December 2013
Published: 7 January 2014
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.
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 [8–10] 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.
The presence of the δ-differential operator () in the definition of Hadamard fractional derivatives could make their study uninteresting and less applicable than Riemann-Liouville and Caputo fractional derivatives. More so, this operator appears outside the integral in the definition of the Hadamard derivatives just like the usual derivative is 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.
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 0 in general. The authors in  resolved these problems by modifying the derivative into a more suitable one having physically interpretable initial conditions similar to the ones in the Caputo settings.
In [12–14], the authors recovered the concepts of fractional integrals and fractional derivatives in different forms and introduced a new version of FTFC in Caputo settings, which is regarded as a generalization of the classical fundamental theorem of calculus. This ignites our curiosity in the possibility of generalizing FTFC in the sense our new definitions given in  as Hadamard and Riemann-Liouville (for example) cannot be used for this generalization (see Section 3). Using the generalization or otherwise, we then formulate new results and theorems.
We study much of this modified derivative thereby formulating some important theorems and results. The Caputo-Hadamard fractional derivatives are used to develop the FTFC, and then the new results are applied in the formulation of some other theorems. As we shall see later, some interesting properties of the modified derivatives are necessary in order to formulate some important outcomes. Section 2 gives some definitions and known results which have been used in this paper, whilst both Sections 3 and 4 are devoted to the original results. Section 5 concludes the paper.
2 Auxiliary results
Below, we begin with some basic definitions and results.
Property 1 [, p.112]
Lemma 1 [, pp.114-116]
- (a)If and , then for ,(10)
Equations (10) and (11) are called semigroup properties of Hadamard fractional integrals and derivatives.
Theorem 1 [, p.4]
- (a)if ,(17)(18)
- (b)if ,(19)
Lemma 2 [, p.5]
Let , and .
Lemma 3 [, p.6]
3 FTFC in the Caputo-Hadamard setting
replaces tedious computations of the limit of sums of rectangular areas with a more easier way of finding an anti-derivative.
are the left-sided Riemann-Liouville fractional integral and the fractional derivative, respectively. While the Hadamard fractional integral and the fractional derivative, and respectively, are given by (2) to (5).
Thus, (30) cannot be considered as the fractional generalization of FTC in the form of (24). Similarly, using Lemma 2.5 of , we can see that Riemann-Liouville fractional integrals and derivatives cannot be used to generalize FTFC in the form of (24) as well.
In most cases, we would only be using the left-sided definitions of fractional derivatives or integrals where the definitions are quite similar to the right-sided ones. Therefore (33) can be considered as a fractional generalization of FTC in the form of (24).
In the next theorem, we give the FTFC in the Caputo-Hadamard setting.
Theorem 2 (Fundamental theorem of fractional calculus)
- (a)If or , then(34)
- (b)Using (17), we have(37)
In this case we can apply the semigroup proper (10), unlike in the cases of Hadamard and Riemann-Liouville fractional derivatives where and , respectively, are located outside the integrals.
Hence gives (35). The right-sided case can be proven in a similar way. □
This is where we make the first use of Theorem 2.
Rearranging (43) gives (41). This completes the proof. □
Note that the right-sided case can also be proven in a similar way.
Applying the mean value theorem for integral and simplifying as before, we obtain (44). □
Thus we have (47). Then if , implies and from (47), we have (48). We can get an immediate consequence of Lemma 6. □
The proof is straightforward.
is the remainder of the terms in the expansion.
However, we may relax the conditions on φ in Corollary 1 as in the next result.
Lemma 7 Let and such that and . More so, suppose that is continuous on for some . Then is continuous and .
Thus, is continuous and by (51). This completes the proof of the lemma. □
4 Semigroup properties of Caputo-Hadamard operators
Theorem 3 (Semigroup property for Caputo-Hadamard derivatives)
This ends the proof. □
In the next lemma, we give the generalization of Theorem 3.
where , and .
Proof The proof follows immediately from Theorem 3 and using mathematical induction. □
Observe that Theorem 4 is the generalization of Lemma 2.4(i) of  where .
Both Theorem 3 and Lemma 8 deal with the reduction of higher fractional order differential systems to lower order systems for Caputo-Hadamard fractional derivatives. However, in some instances it may also be useful to involve the Caputo-Hadamard and the Hadamard differential operator.
- 1.If , then by (19) and from (2.7.13) of , (59) becomes
- 2.Otherwise, since , then by definitions
Since the Caputo-Hadamard fractional derivatives were introduced in , not much about the modified derivatives were studied despite the fact that the derivatives have many advantages over the usual Hadamard fractional derivative. We proved that the Hadamard fractional derivatives cannot be used to generalize the FTFC whereas the Caputo-Hadamard derivative works perfectly. The FTFC is then used in formulating other results whose applications to fractional vector calculus in the study of Green’s theorem, Stoke’s theorem and so forth, as well as in the study of anomalous diffusion is a further work. Many new results such as the semigroup properties for the modified derivatives are studied in detail.
The first author wishes to give special thanks to His Excellency, the Executive Governor of Kano State of Nigeria, Engineer (Dr.) Rabi’u Musa Kwankwaso, for his endless, patriotic and tireless support and constant encouragement.
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