Properties of Caputo-Fabrizio fractional operators

In this paper, we have studied some theoretical properties concerning the Caputo-Fabrizio fractional operators. Also we obtained expressions for the fractional integral and derivative of some elementary functions. As auxiliary results, general integration formulas for functions with radicals of any order are presented.


Introduction
The fractional calculus is a name for the theory of integrals and derivatives of arbitrary order, which unify and generalize the notions of integer-order differentiation and n-fold integration. The history of the Fractional Calculus goes back to seventeenth century, when in 1695 the derivative of order α = 1/2 was described by Leibnitz in his letter to L'Hospital [28]. That date is regarded as the exact birthday of the fractional calculus. Since then this branch has been treated by eminent mathematicians, such as Euler, Laplace, Fourier, Liouville, Riemann, Laurent, Weyl and Abel. And therefore many definitions, concerning the fractional operators have been proposed: Grunwald and Letnikov faced the problem of non-integer differentiation [20], generalizing the derivative definition of an integer order, based on the quotient concept incremental, using the following formula In 1917 Weyl [21] defined a fractional integral adequate to periodic functions In 1938 M. Riesz published a number of papers [22] which are centered around the integral R D α t u(t) = In [23], Laurent used a contour given as an open circuit (known as Laurent loop) instead of a closed circuit used by Sonin and Letnikov and thus produced today's definition of the Riemann-Liouville fractional integral The Riemann-Liouville derivative appears to the paper by N. Ya. Sonin in [24][25] where he used Cauchy's integral formula as a starting point to reach differentiation with arbitrary index RL D α at u(t) = Recently Caputo and Fabrizio launched a new fractional derivative and it was followed by some related theoretical and applied results [26]. The interest for this new approach is due to the necessity to describe material heterogeneities and structures with different scales, which cannot be well described by classical local theories [26].
In the last few decades many authors pointed out that derivatives and integrals of non-integer order are very suitable for the description of properties of various real materials. It has been shown that new fractional-order models are more adequate that integer-order models [27]. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected. The advantages of fractional derivatives become apparent in modelling mechanical and electrical properties of real materials, as well as in the description of rheological properties of rocks, and in market behaviour [27,30].
The other large field which requires the use of derivatives of non-integer order is the theory of fractals. The development of the theory of fractals has opened further perspectives for the theory of fractional derivatives, especially in modelling dynamical processes in self-similar and porous structures [27].
In recent years, many researcher have shown that Fractional Calculus is a useful tool in image processing field such as image enhancement, image denoising, image edge detection, image segmentation, image registration, image recognition, image fusion, image encryption, image compression and image restoration [29,37,38]. Particularly, in [34][35][36], authors have shown the advantage of fractional order derivatives to achieve a good trade-off between image denoising and edge preservation. Also in [31][32][33] fractional differential mask has being proposed for contrast image enhancement. And in [37,39,40] it has been demonstrated how using an edge detector based on fractional differentiation can improve the criterion of detection of thin artefacts in the image.
However any application of fractional calculation requires expressions for fractional derivatives and integrals. For example in [27], based on the definition of Riemann-Liouville and Caputo, expressions for fractional derivatives and integrals of some elementary functions are obtained.
Our purpose in this paper is to give some theoretical properties concerning the Caputo-Fabrizio fractional operators and apply these operators to some interesting elementary functions. The paper has been organized as follows, in Section 2, we present basic definitions and formulas. In Section 3, we give theoretical properties of Caputo-Fabrizio fractional derivative. In section 4, we study the composition of Caputo-Fabrizio fractional operators. In section 5, we give examples on using Caputo-Fabrizio fractional integral. In Section 6, we give some examples on using Caputo-Fabrizio fractional derivative. A conclusion is considered in section 7.

Preliminaries and notations
Here, we introduce some definitions concerning the Caputo-Fabrizio fractional derivative and we give some formulas, which would be needed in our proofs later.
We denote by C 0 ([a, b]) the space of all continuous functions on [a, b] with compact support.
Lemma 1. Let n ∈ N and α, a, b ∈ R. Then, for each n, the integral can be written in the form Proof. We will use the principle of mathematical induction. Let P(n) be For our base case, we need to show P(1) is true, meaning that This is trivial, since For the inductive step, assume that for some n, P(n) holds, so We need to show that P(n + 1) holds, meaning that To see this, note that Thus P(n + 1) holds when P(n) is true, so P(n) is true for all natural numbers n.
Lemma 2. Let n ∈ N and a, b, α ∈ R such that α ∈ (0, 1). Then, for each n, the integral can be written in the form Proof. Let P(n) be P(n) ≡ B n (t) We will show that P(n) holds for all n ∈ R by induction. We note that Thus, P(n) is true for n = 1. Asume that P(n) is true for some natural number n > 1, i.e., We need to proof that P(n) is true for n + 1 whenever P(n) is true for n. We have Proof.
Thus, P(n) is true for n+1 whenever P(n) is true for n. Hence, by the principle of mathematical induction, P(n) is true for all natural numbers n.
Proposition 1. For 1 < n ∈ N, one has easily the following:

Some properties cencerning the Caputo-Fabrizio fractional derivative
Here, we give some theoretical properties concerning the Caputo-Fabrizio fractional derivative.
holds true.
Proof. Let P(n) be We will show, by induction, that P(n) holds for all n ∈ N. Integrating by parts, we note that Thus, P(n) is true for n = 1. Asume that P(n) is true for some natural number n, i.e., We need to proof that P(n) is true for n + 1 whenever P(n) is true for n. We have Theorem 2. Let be u ∈ C n+1 [a, b]. Then the equality holds true for all t ∈ [a, b] Proof. We insert n and n − 1 into the equality (19). This yields and respctively. Combining (25) with (26) we obtain (24). Theorem 2 is thus proved.
. Then the equality holds true.
Proof. We will use the principle of mathematical induction. Let P(n) be For our base case, we need to show P(1) is true, meaning that This is trivial, since For the inductive step, assume that for some n, P(n) holds, so We need to show that P(n + 1) holds, meaning that To see this, note that thus P(n + 1) holds when P(n) is true, so P(n) is true for all natural numbers n.
holds true Proof. Applying (27) for n = 1, we obtain Then integrating (34) with respect to t over (a, b) and considering that D α aa u(a) = 0, we obtain (33).
Proof. We need to proof that D α at u(t), On the one hand, we note that Then we can write And consequently, On the other hand, applying Theorem 3, we obtain

Composition of Fractional Operators
In this section, we give some theoretical properties concerning the composition of Caputo-Fabrizio fractional operators.
Proof. On the one side, from the definition of Caputo-Fabrizio derivative, we deduce that which is equivalent to On the other side, integrating by parts and considering that D β aa u(a) = 0, we obtain Combining (38) with (39), we obtain (35).
Proof. Applying definition of Caputo-Fabrizio derivative, we obtain as required.
Proof. On the one side, using definition of Caputo-Fabrizio integral, we obtain On the other side, applying Theorem 3, we obtain Integrating (43) respect to s over (a,t) to t and considering that D α aa f (a) = 0, we obtain Inserting the right hand side of (44) into (42), we obtain (41) and thus the proof is completed.
Proof. Using the definition of the Caputo-Fabrizio integral and the Riemann-Liouville integral, we obtain where we have used

Fractional Integral of Caputo-Fabrizio for some elementary functions
In this section we give explicit formulas for fractional integral of Caputo-Fabrizio of the following elementary functions x n x 2 + ax + b, where a, b ∈ R, such that both are not zero simultaneously, 1 < n ∈ N and C will represent a generic constant. These formulas will be formulated as propositions. Further it is important to highlight that to obtain these formulas, an intensive auxiliary calculation work was necessary, which will be presented in the form of lemmas.
Then the Caputo-Fabrizio fractional integral of u(x) is given by where p 4 (x) is given as (71).

holds true
Proof. It is easy to see, after integrating by parts, that which is equivalent to Following equalities (18) and (54), we conclude the proof.
Proof. Combining the change of variable with Lemma 4, we deduce that Consequently, we have the assertion (62) with the aid of Lemma 3.
Proof. It is easy to see, after integrating by parts, that which is equivalent to By (62) and (67), we obtain (66).
as required.
We now give the proof of Proposition 2 Proof. Combining definition 1 with the lemma 8, we obtain 52. . Then the Caputo-Fabrizio fractional integral of u(x) is given by where p 2 (x) is given as (99).
To proof Proposition 3, we need the following lemmas Lemma 9. The equality Proof. It is easy to see, after integrating by parts, that which is equivalent to Combining (17) with (77), equality (75) follows.
Proof. By using the change of variable and Lemma 10, we obtain By (75) and (88) as well as (89), we conclude the proof.
Proof. It is easy to see, after integrating by parts, that which is equivalent to From (92), we obtain where β is given as (79). In terms of (93), it then follows that Combining (86) with (94), we get the desired equality (90).
Lemma 13. The following statements are equivalents , where β is given by (79) and ∆ , by Proof.
as required.
We now give the proof of Proposition 3 Proof. Combining definition 1 with the lemma 14, we obtain 74.
Proposition 4. Let be u(x) = n √ x 2 + ax + b. Then the Caputo-Fabrizio fractional integral of u(x) is given by where p 4 (x) y p 5 (x) are given by (71) and (106), respectively.
Proof. It is easy to see that By (70) and (105) as well as (110), it is easy to see (109).
We now give the proof of Proposition 4 Proof. Combining definition 1 with the lemma 16, we obtain (104).
. Then the Caputo-Fabrizio fractional integral of u(x) is given by where p 2 (x) y p 6 (x) are given by (99) and (113), respectively.
Proof. By using the change of variable we obtain By using (98) and (115) as well as (114), we deduce (112).
Proof. It is easy to see that By using (112) and (98) as well as (117), we deduce (116).
We now give the proof of Proposition 5 Proof. Combining definition 1 with the lemma 18, we obtain (111).
Proof. It is easy to see that In terms of (86) and (98) as well as (116), we deduce equality (119).
We now give the proof of Proposition 6.
We now give the proof of Proposition 7.

Fractional derivative of Caputo-Fabrizio for some elementary functions
Here, we consider some examples on Caputo-Fabrizio Fractional Derivative Proof. From definition 2, we have This completes the proof.
For our base case, we need to show P(1) is true, meaning that This is trivial, since For the inductive step, assume that for some n, P(n) holds, so We need to show that P(n + 1) holds, meaning that To see this, note that Thus P(n + 1) holds when P(n) is true, so P(n) is true for all natural numbers n.

Conclusion
In this paper author has studied some theoretical properties concerning the Caputo-Fabrizio fractional derivative. Also composition of fractional operators has been obtained. In the same line it is given explicit formulas for Caputo-Fabrizio fractional operators of some elementary functions. To obtain such formulas, an auxiliary calculation in form of lemmas has been presented. As a future work, author is planning to use the properties presented in this work to analyse the qualitative properties of some Caputo-Fabrizio ordinary fractional differential equations.