# On Some Properties of the New Generalized Fractional Derivative with Non-Singular Kernel

Authors: Khalid Hattaf
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper presents some new formulas and properties of the generalized fractional derivative with non-singular kernel that covers various types of fractional derivatives such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. These new properties extend many recent results existing in the literature. Furthermore, the paper proposes some interesting inequalities that estimate the generalized fractional derivatives of some specific functions. These inequalities can be used to construct Lyapunov functions with the aim to study the global asymptotic stability of several fractional-order systems arising from diverse fields of science and engineering.


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## Body

## 1. Introduction

In the literature, there exist manifold types of fractional derivatives which are used to solve several mathematical and engineering problems as well as to model the dynamics of many systems existing in various fields of science such as mechanics, chemistry, finance, ecology, biology, and control theory. For instance, Caputo and Fabrizio [1] proposed a fractional derivative with non-singular kernel to describe the material heterogeneities and the fluctuations of different scales, which cannot be well described by classical local theories or by fractional models with singular kernel. An extension of [1] was proposed by Atangana and Baleanu [2] by using Mittag-Lefler function with one parameter. Due to the importance of weighted fractional derivatives to solve several types of integral equations with elegant ways, Al-Refai [3] introduced the weighted Atangana–Baleanu fractional operators and he studied their properties. A generalized version of all previous fractional derivative operators with non-singular kernel was recently proposed in [4].The objective of this study is to establish some properties and formulas of the new generalized fractional derivative introduced in [4] in order to extend several results presented in recent works. To achieve this goal, the remainder of this article is outlined as follows. The next section deals with preliminaries and fractional derivatives of some specific functions. The estimates of the generalized fractional derivatives of these specific functions can be applied to find Lyapunov candidate functions for demonstrating the global stability of many fractional-order systems. Section 3 is devoted to the new formulas and properties of the generalized fractional derivative with non-singular kernel. The paper ends with a conclusion in Section 4.


## 2. Preliminaries and Fractional Derivatives of Some Specific Functions

In this section, we first recall the definition and special cases of the new generalized fractional derivative with non-singular kernel. After, we present new properties for this fractional derivative that can be used to prove the global dynamics of several fractional-order systems.Definition 1 (see [4]).
Letα∈0,1, β,γ>0, and f∈H1a,b. The new generalized fractional derivative of order α of Caputo sense of the function ft with respect to the weight function wt is defined as follows:(1)Da,t,wα,β,γCft=Nα1−α1wt∫atEβ−μαt−xγddxwfxdx,where w∈C1a,b, w,w′>0 on a, b, Nα is a normalization function obeying N0=N1=1, μα=α/1−α, and Eβt=∑k=0+∞tk/Γβk+1 is the Mittag-Leffler function of parameter β.The importance of the fractional derivative given in the above definition is the generalization of most cases existing in the literature. These particular cases can be summarized in the following remark.Remark 1.
(i)
Ifwt=1 and β=γ=1, then (1) is reduced to the Caputo–Fabrizio fractional derivative [1] given by(2)Da,t,1α,1,1Cft=Nα1−α∫atexp−μαt−xf′xdx.(ii)
Ifwt=1 and β=γ=α, then (1) is reduced to the Atangana–Baleanu fractional derivative [2] given by(3)Da,t,1α,α,αCft=Nα1−α∫atEα−μαt−xαf′xdx.(iii)
Ifβ=γ=α, then (1) is reduced to the weighted Atangana–Baleanu fractional derivative [3] given by(4)Da,t,wα,α,αCft=Nα1−α1wt∫atEα−μαt−xαddxwfxdx.Now, we establish new properties for the generalized fractional derivative. We begin with the following result.Theorem 1.
Letf be a continuous function and u be a continuously differentiable function. For any constant c, the new generalized fractional derivative of the function F defined by(5)Ft=1wt∫cwtutfxdxsatisfies the following property:(6)Da,t,wα,β,γCFt=fwtutDa,t,wα,β,γCut−Nα1−αvawtEβ−μαt−aγ−αγNα1−α21wt∫att−τγ−1Eβ,β+12−μαt−τγvτdτ,where(7)vτ=fwtutwtut−wτuτ+∫wtutwτuτfxdx.
Moreover, we have the following inequalities:(i)
Iff is a increasing function, then(8)Da,t,wα,β,γCFt≤fwtutDa,t,wα,β,γCut.(ii)
Iff is a decreasing function, then(9)Da,t,wα,β,γCFt≥fwtutDa,t,wα,β,γCut.Proof.
By using Definition1, we have(10)Da,t,wα,β,γCFt=Nα1−α1wt∫atEβ−μαt−τγfwτuτwu′τdτ.
Let(11)gt=Da,t,wα,β,γCFt−fwtutDa,t,wα,β,γCut.
Then,(12)gt=Nα1−α1wt∫atEβ−μαt−τγfwτuτ−fwtutwu′τdτ.
Obviously,v′τ=fwτuτ−fwtutwu′τ and vt=0. Integrating by parts the last integral, we find(13)gt=Nα1−α1wtEβ−μαt−τγvτ|τ=aτ=t=αγNα1−α21wt∫att−τγ−1Eβ,β+12−μαt−τγvτdτ.
Sincelimτ⟶tEβ−μαt−τγvτ=0, we have(14)gt=−Nα1−αvawtEβ−μαt−aγ−αγNα1−α21wt∫att−τγ−1Eβ,β+12−μαt−τγvτdτ.
Now, we consider the following function:(15)ψc,fτ=fcc−τ+∫cτfxdx.
It is clear thatψc,f′τ=fτ−fc. If f is a increasing function, then the function ψc,fτ is decreasing on the interval −∞,c and increasing on c,+∞ with ψc,fc=0. Hence, ψc,fτ has the global minimum at τ=c. So,(16)ψc,fτ≥0,for allc,τ∈IR2.
Sincevτ=ψwtut,fwτuτ, we have vτ≥0 for all τ∈IR. This proves (i). In the same way, we can easily prove (ii).Corollary 1.
Letut∈IR be a continuously differentiable function. Then, for any time t≥t0, we have(17)Dt0,t,1α,β,γCu2t≤2utDt0,t,1α,β,γCut.Proof.
Applying (i) of Theorem1 to the function fx=2x, we get(18)Dt0,t,1α,β,γCu2t=Dt0,t,1α,β,γC∫cut2xdx≤2utDt0,t,1α,β,γCut.
This completes the proof.Remark 2.
Corollary1 extends the result presented in Lemma 3.1 of [5] for the new generalized fractional derivative. In fact, it suffices to take γ=β=α in inequality (17).Now, we extend the recent result presented in Lemma 3.2 of [5] by proposing the following corollary.Corollary 2.
Letut∈IR+ be a continuously differentiable function and u∗>0. Then, for any time t≥t0, we have(19)Dt0,t,1α,β,γCut−u∗−u∗lnutu∗≤1−u∗utDt0,t,1α,β,γCut.Proof.
The application of Theorem1 (ii) to the function fx=1/x leads to the following:(20)Dt0,t,1α,β,γCut−u∗−u∗lnutu∗=Dt0,t,1α,β,γCut−u∗Dt0,t,1α,β,γC∫u∗ut1xdx≤1−u∗utDt0,t,1α,β,γCut.
The proof of Corollary2 is completed.


## 3. New Formulas and Properties

For simplicity, we denoteDa,t,1α,β,βC by Da,wα,β. According to [4], we have the following definition.Definition 2 (see [4]).
The generalized fractional integral operator corresponding toDa,wα,β is defined by(21)ℐa,wα,βft=1−αNαft+αNαℐa,wβRLft,where ℐa,wβRL is the standard weighted Riemann–Liouville fractional integral of order β defined by(22)ℐa,wβRLft=1Γβ1wt∫att−xβ−1wxfxdx.Remark 3.
The Atangana–Baleanu fractional integral operator is a special case of (21), and it suffices to take wt=1 and β=α.Lemma 1.
The fractional derivativeDa,wα,β can be expressed as follows:(23)Da,wα,βft=Nα1−α∑k=0+∞−μαkℐa,wβk+1RLwf′wt.
This series converges locally and uniformly int for any a, α, β, w, and f satisfying the conditions from Definition 1.Proof.
The Mittag-Leffler functionEαt is a sum of an entire series. This series converges locally and uniformly in the whole complex plane. Hence, we can rewrite the fractional derivative Da,wα,β as follows:(24)Da,wα,βft=Nα1−α1wt∑k=0+∞−μαkΓβk+1∫att−xβkwf′xdx=Nα1−α∑k=0+∞−μαk1Γβk+11wt∫att−xβkwf′xdx=Nα1−α∑k=0+∞−μαkℐa,wβk+1RLwf′wt.
This completes the proof.Theorem 2.
Letα,γ∈0,1, β,ρ>0, and f∈H1a,b. Then,(i)
Da,wα,βDa,wγ,ρft=Da,wγ,ρDa,wα,βft.(ii)
ℐa,wα,βℐa,wγ,ρft=ℐa,wγ,ρℐa,wα,βft.
This means that the generalized fractional integrals and derivatives are commutative operators.Proof.
First, we prove (i). By applying Lemma1, we get(25)Da,wα,βDa,wγ,ρft=Nα1−α∑k=0+∞−μαkℐa,wβk+1RLwDa,wγ,ρf′wt=Nα1−α∑k=0+∞−μαkℐa,wβk+1RLNγ1−γ∑n=0+∞−μγnℐa,wρnRLwf′wt=NαNγ1−α1−γ∑k=0+∞∑n=0+∞−μαk−μγnℐa,wβk+ρn+1RLwf′wt.
Since the last expression is symmetric inα and γ and also in β and ρ, we easily deduce (i).
For (ii), we have(26)ℐa,wα,βℐa,wγ,ρft=1−αNαℐa,wγ,ρft+αNαℐa,wβRLℐa,wγ,ρft=1−αNα1−γNγft+γNγℐa,wρRLft+αNαℐa,wβRL1−γNγft+γNγℐa,wρRLft=1−α1−γNαNγft+γ1−αNαNγℐa,wρRLft+α1−γNαNγℐa,wβRLft+αγNαNγℐa,wβ+ρRLft=ℐa,wγ,ρℐa,wα,βft.
This shows (ii).Theorem 3.
Letα∈0,1, β>0, and f∈H1a,b. Then,(i)
ℐa,wα,βDa,wα,βft=ft−wafa/wt.(ii)
Da,wα,βℐa,wα,βft=ft−wafa/wt.Proof.
According Definition2 and Lemma 1, we have(27)ℐa,wα,βDa,wα,βft=1−αNαDa,wα,βft+αNαℐa,wβRLDa,wα,βft=1−αNαNα1−α∑k=0+∞−μαkℐa,wβk+1RLwf′wt+αNαℐa,wβRLNα1−α∑k=0+∞−μαkℐa,wβk+1RLwf′wt=∑k=0+∞−μαkℐa,wβk+1RLwf′wt−∑k=0+∞−μαk+1ℐa,wβk+1+1RLwf′wt.
Hence,(28)ℐa,wα,βDa,wα,βft=∑k=0+∞−μαkℐa,wβk+1RLwf′wt−∑k=1+∞−μαkℐa,wβk+1RLwf′wt=ℐa,w1RLwf′wt=1wt∫atwf′xdx=1wtwtft−wafa.
Now, we demonstrate (ii). From Lemma1, we obtain(29)Da,wα,βℐa,wα,βft=Nα1−α∑k=0+∞−μαkℐa,wβk+1RLwℐa,wα,βf′wt=Nα1−α∑k=0+∞−μαkℐa,wβk+1RL1−αNαwf′w+αNαwℐa,wβRLf′wt=∑k=0+∞−μαkℐa,wβk+1RLwf′wt+μα∑k=0+∞−μαkℐa,wβk+1RLwℐa,wβRLf′wt=∑k=0+∞−μαkℐa,wβkRLft−wfaℐa,wβkRL1wt+μα∑k=0+∞−μαkℐa,wβk+1RLft−wfaℐa,wβk+1RL1wt.
Thus,(30)Da,wα,βℐa,wα,βft=∑k=0+∞−μαkℐa,wβkRLft−wfaℐa,wβkRL1wt−∑k=1+∞−μαkℐa,wβkRLft−wfaℐa,wβkRL1wt=ℐa,w0RLft−wfaℐa,w0RL1wt=ft−wfawt.
This completes the proof.
By applying Theorem3 for wt=1, we get the following important result that extends the Newton–Leibniz formula presented in [6, 7].Corollary 3.
The generalized fractional integral and the generalized fractional derivative satisfy the following Newton–Leibniz formula:(31)ℐa,1α,βDa,1α,βft=Da,1α,βℐa,1α,βft=ft−fa.


## 4. Conclusion

In many fields of science like mathematical biology, the method of constructing Lyapunov functions is often based on quadratic and Volterra-type functions. So, the first aim of this work was to estimate the generalized fractional derivatives of these specific functions. The obtained results extend many elementary lemmas existing in the recent literature for other types of fractional derivatives. Furthermore, the second purpose of this work was to establish a new formula for the generalized fractional derivative in the form of a series of weighted Riemann–Liouville fractional integrals. By means of this formula, various properties of the generalized fractional derivatives and integrals operators have been carefully established. Moreover, the Newton–Leibniz formula has been rigorously extended.The results presented in this study form a fundamental basis for extending the classical Lyapunov theorems [8–10] and LaSalle’s invariance principle [11] to the case of the new generalized fractional derivative. This issue will be the first direction of our future works.



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