# Improvement on Conformable Fractional Derivative and Its Applications in Fractional Differential Equations

Authors: Feng Gao; Chunmei Chi
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we made improvement on the conformable fractional derivative. Compared to the original one, the improved conformable fractional derivative can be a better replacement of the classical Riemann-Liouville and Caputo fractional derivative in terms of physical meaning. We also gave the definition of the corresponding fractional integral and illustrated the applications of the improved conformable derivative to fractional differential equations by some examples.


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## Body

## 1. Introduction

The fractional order derivative has always been an interesting research topic in the theory of functional space for many years [1–11]. Various types of fractional derivatives were introduced, among which the following Riemann-Liouville and Caputo are the most widely used ones.
(1)
Riemann-Liouville Definition. For α∈n−1,n, the α derivative of f is(1)DαaRLft=1Γn−αdndtn∫atfxt−xα−n+1dx.(2)
Caputo Definition. For α∈n−1,n, the α derivative of f is(2)DαaCft=1Γn−α∫atfnxt−xα−n+1dx.Both Riemann-Liouville definition and Caputo definition are defined via fractional integrals. Therefore, these two fractional derivatives inherit some nonlocal behaviors including historical memory and future dependence. All definitions including (1) and (2) above satisfy the property that the fractional derivative is linear. This is the only property inherited from the 1st derivative. However, the existing fractional derivatives do not satisfy the following properties which the integral derivatives have.(1)
Most of the fractional derivatives except Caputo-type derivatives do not satisfyDα1=0, if α is not a natural number(2)
All fractional derivatives do not obey the familiar product rule for two functions:(3)Dαfg=fDαg+gDαf(3)
All fractional derivatives do not obey the familiar quotient rule for two functions:(4)Dαfg=gDαf−fDαgg2(4)
All fractional derivatives do not obey the chain rule:(5)Dαfgt=fαgtDαg(5)
Fractional derivatives do not have corresponding Rolle’s theorem(6)
Fractional derivatives do not have a corresponding mean value theorem(7)
In general, all fractional derivatives do not obey(6)DαDβf=Dα+βf(8)
The Caputo definition assumes that the functionf is differentiableTo overcome some of these difficulties, Khalil et al. [12] proposed a new interesting factional derivative definition called conformable derivative that extends the familiar limit definition of the derivative of a function given by the following.Definition 1.
Given a functionf:0,+∞⟶R, then the conformable fractional derivative of f of order α is defined by
(7)Tαft=limε→0ft+εt1−α−ftε,for t>0 and α∈0,1. If f is α differentiable in some 0,a, a>0, and limt→0+Tαft exists, then define Tαf0=limt→0+Tαft. It is easy to see that if f is differentiable, then Tαft=t1−αf′t. One can find functions which are α-differentiable at a point but not differentiable at this point.As a result of the above definition, the authors in [12] showed that the conformable derivative obeys the product rule and quotient rule and has results similar to Rolle’s theorem and the mean value theorem in classical calculus.The conformable fractional derivative has two advantages over the classical fractional derivatives. First, the conformable fractional derivative definition is natural and it satisfies most of the properties which the classical integral derivative has such as linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, Rolle’s theorem, and mean value theorem. Second, the conformable derivative bring us a lot of convenience when it is applied for modelling many physical problems, because the differential equations with conformable fractional derivative are easier to solve numerically than those associated with the Riemann-Liouville or Caputo fractional derivative. In fact, many researchers have already applied conformable fractional derivative to many fields and a lot of corresponding techniques were developed [13–20].However, there are still shortcomings or disadvantages for the conformable derivative. If we look at the Riemann-Liouville and Caputo fractional derivative definition, we have(8)limα→0aRLDαft=limα→01Γ1−αddt∫atfxt−xαdx=ft,limα→0aCDαft=limα→01Γ1−α∫atf′xt−xαdx=ft−fa,for α∈0,1, and
(9)limα→n−1aRLDαft=limα→n−11Γn−αdndtn∫atfxt−xα−n+1dx=fn−1t,limα→n−1aCDαft=limα→n−11Γn−α∫atfnxt−xα−n+1dx=fn−1t−fn−1a,for α∈n−1,n,n=2,3,⋯, whereas
(10)limα→n−1Tαft=limα→n−1tn−αfnt=tfnt,for α∈n−1,n,n=1,2,⋯.That means the physical meaning of the classical Riemann-Liouville and Caputo fractional derivative is quite different from that of the conformable derivative, especially whenα∈n−1,n and close to n−1,n=1,2,⋯. We graph the conformable derivatives of y=sinx and y=ex in Figures 1 and 2 and make comparison with the Riemann-Liouville and Caputo fractional derivatives. We can see if the Riemann-Liouville and Caputo fractional derivatives are replaced by conformable derivative, a large error will occur. To overcome this difficulty, we propose a kind of modified conformable fractional derivative in Section 2. This modified conformable fractional derivative is a local operator on the one hand and approximates the Riemann-Liouville and Caputo fractional derivative better on the other hand. So it is a better choice to replace the classical Riemann-Liouville and Caputo fractional derivative with the improved conformable fractional derivative, especially when α∈n−1,n and close to n−1,n=1,2,⋯.Figure 1
0.3th derivative ofy=sinx.Figure 2
0.2th derivative ofy=ex.


## 2. Improvement on Conformable Fractional Derivative

First, we give the definition for the improved Caputo-type conformable fractional derivativeaCT~αt and the improved Riemann-Liouville-type conformable fractional derivative T~αaRLt for 0≤α≤1.Definition 2.
Given a functionf:R⟶R, the improved Caputo-type conformable fractional derivative of f of order α is defined by
(11)T~αaCt=limε→01−αft−fa+αft+εt−a1−α−ftε,where −∞<a<t<+∞, a is a given number.The improved Riemann-Liouville-type conformable fractional derivative off of order α is defined by
(12)T~αaRLt=limε→01−αft+αft+εt−a1−α−ftε,where −∞<a<t<+∞, a is a given number.It is easy to see(13)limα→0aCT~αt=ft−fa=limα→0aCDαt,limα→0aRLT~αt=ft=limα→0aRLDαt.Ifα=1, both T~aaCt and T~αaRLt coincide with f′t. In Definition 2, we introduce a to let T~aaRLt and T~αaRLt have a kind of historical memory as the Caputo and Riemann-Liouville fractional derivative have.Forn<α≤n+1,n=1,2,⋯, we give the following.Definition 3.
Given a functionf:R→R, the improved Caputo-type conformable fractional derivative of f of order α is defined by
(14)T~αaCt=limε→0n+1−αfnt−fna+α−nfnt+εt−an+1−α−fntε,where −∞<a<t<+∞.The improved Riemann-Liouville-type conformable fractional derivative is defined by(15)T~aaRLt=limε→0n+1−αfnx+α−nfnt+εt−an+1−α−fntε,where −∞<a<t<+∞.It is easy to see(16)limα→naCT~αt=fnt−fna=limα→naCDαt,limα→naRLT~αt=fnt=limα→naRLDαt.Ifα=n+1, both T~aaCt and T~αaRLt coincide with fn+1t.Remark 4.
As a result of Definitions2 and 3, we can easily show that if f is 1-differentiable at t, then
(17)T~aaCft=1−αft−fa+αt−a1−αf′t,T~aaRLft=1−αft+αt−a1−αf′t,for α∈0,1. If f is n+1-differentiable at t, then
(18)T~aaCft=n+1−αfnt−fna+α−nt−an+1−αfn+1t,for α∈n,n+1,n=1,2,⋯.In fact, compared with the conformable fractional derivative, the improved conformable fractional derivative can be a better replacement of the Caputo and Riemann-Liouville fractional derivative. One can see in Figures3–9 that the improved conformable fractional derivative approximates the Caputo and Riemann-Liouville fractional derivative in a better way for most elementary functions especially when α is larger than and close to n and even when α is close to n+0.5,n=0,1,2,⋯.Figure 3
0.2th fractional derivative ofy=x2.Figure 4
0.5th fractional derivative ofy=x2.Figure 5
0.8th fractional derivative ofy=x2.Figure 6
0.2th fractional derivative ofy=sinx.Figure 7
0.5th fractional derivative ofy=sinx.Figure 8
0.8th fractional derivative ofy=sinx.Figure 9
0.2th fractional derivative ofy=ex.Based on the results in [12], one can prove the following theorem.Theorem 5.
We can easily show that the improved conformable fractional derivative satisfies the following properties if0≤α≤1.
(1)
T~αaCmf+ng=maCT~αf+naCT~αg(2)
T~αaRLmf+ng=maRLT~αf+naRLT~αg(3)
T~αaRLfg=1−αaRLT~αfg+faRLT~αg−1−αfg(4)
T~αaRLfgx=1−αfgx+αf′gxTαgxTheorem 6.
We get the following improved conformable fractional derivatives of certain functions for0≤α≤1.
(1)
T~α0Ctp=0RLT~αtp=1−αtp+αptp−α(2)
T~α0Cλ=0,foranyconstantλ(3)
T~α0RLλ=1−α,foranyconstantλ(4)
T~α0Cet=1−αet−1+αt1−αet(5)
T~α0RLet=1−αet+αt1−αet(6)
T~α0Csint=0RLT~αsint=1−αsint+αt1−αcost(7)
T~α0Ccost=1−αcost−1−αt1−αsint(8)
T~α0RLcost=1−αcost−αt1−αsint(9)
T~α0RLe1/ataProof.
It is very easy to verify properties (1)–(8) if we take into account the conclusions in [12]. We only prove property (9) here. From [12], we have
(19)Tαe1/ata=e1/ata.
Therefore,(20)T~α0RLe1/ata=1−αe1/ata+αTαe1/ata=1−αe1/ata+αe1/ata=e1/ata.Now, we give the following definition forα fractional integral.Definition 7.
Forα∈0,1 and continuous function f, let
(21)Iαf=1α∫0xftt1−αe1−α/α2tα−xαdt.Whenα=1,I1f=∫0xftdt and coincides with the usual Riemann integral.Theorem 8.
Supposey0=0, we have
(1)
Iα·0RLT~αyt=0RLT~α·Iαyt=yt(2)
Iα·0CT~αyt=0CT~α·Iαyt=ytProof.
We only prove (1); (2) can be proved in the same way.(22)Iα·0RLT~αfx=1−αα∫0xftt1−αe1−α/α2tα−xαdt+∫0xf′te1−α/α2tα−xαdt=1−αα∫0xftt1−αe1−α/α2tα−xαdt+∫0xe1−α/α2tα−xαdft=fte1−α/α2tα−xα0x−1−αα∫0xftt1−αe1−α/α2tα−xαdt+1−αα∫0xftt1−αe1−α/α2tα−xαdt=fx.Let1−αyx+αt1−αy′x=fx; by using a variation of the constant method, we can get
(23)yx=1α∫0xftt1−αe1−α/α2tα−xαdt.Let(24)Iαf=1α∫0xftt1−αe1−α/α2tα−xαdt,we have 0RLT~α·Iαyt=yt.


## 3. Applications of Improved Conformable Fractional Derivative

We solve fractional differential equations by using the improved conformable fractional derivatives.Example 1.
Consider the following fractional differential equation:(25)y0.5=2Γ2.5x3/2,y0=0.Ify0.5 is understood as the Riemann-Liouville or Caputo fractional derivative, the solution to this problem is
(26)y=x2.Now, we takey0.5 as the improved conformable fractional derivative; the problem reduces to
(27)12y+12xy′=2Γ2.5x3/2,y0=0.We can solve this problem through the variation constant method and get the following solution:(28)y=4Γ2.5x3/2−32x+32x−34+3Γ2.5e−2x.Ify0.5 is understood as the conformable fractional derivative, the problem reduces to
(29)xy′=2Γ2.5x3/2,y0=0,and the solution to this problem is
(30)y=1Γ2.5x2.We compare the three solutions in Figure10. We can see that the improved conformable fractional derivative solution is much better than the solution obtained by using the conformable fractional derivative. That means if we prefer a replacement of the Riemann-Liouville or Caputo fractional derivative, the improved conformable fractional derivative is a better choice.Example 2.
In general, we consider the problem(31)yα=2Γ3−αx2−α,y0=0.
We can get the following three solutions:(i)
Ifyα is the Riemann-Liouville or Caputo fractional derivative, the solution is y1x=x2, and(32)limα→0y1x=limα→0x2=x2(ii)
Ifyα is the conformable fractional derivative, the problem can be reduced to(33)x1−αy′=2Γ3−αx2−α,y0=0.
The solution is(34)y2x=1Γ3−αx2.
We can see that(35)limα→0y2x=limα→01Γ3−α=12x2(iii)
Ifyα is the improved conformable fractional derivative, the problem can be reduced to(36)1−αy+αx1−αy′=2Γ3−αx2−α,y0=0
Its solutiony3x can be obtained through the variation of the constant method.Figure 10
Solution to Example1.Since(37)1−αy3+αx1−αy3′=2Γ3−αx2−α,we have
(38)y3=αy3−αx1−αy3′+2Γ3−αx2−α.Therefore,
(39)limα→0y3=limα→0αy3−αx1−αy3′+2Γ3−αx2−α=limα→02Γ3−αx2−α=x2=limα→0y1.The result for the generalα equation shows that the improved conformable derivative has advantages over other fractional derivatives. First, it is a good approximation to the classical Riemann-Liouville or Caputo fractional derivative so it has a similar physical meaning with the Riemann-Liouville and Caputo fractional derivative. Second, it is a local derivative operator so it is easy for numerical computing.Example 3.
Consider the following fractional differential equation:(40)y1.5=3Γ2.5x3/2,y0=0,y′0=0.Ify1.5 is the Caputo fractional derivative, the solution to this problem is
(41)y=x3.Ify1.5 is the conformable fractional derivative, the problem reduces to
(42)x0.5y′′=3Γ2.5x3/2,y0=0,y′0=0.The solution to this problem is(43)y=12Γ2.5x3.Ify1.5′ is the improved conformable fractional derivative, the problem reduces to
(44)12y′+12x0.5y′′=3Γ2.5x3/2,y0=0,y′0=0.Let y′=px, the problem further reduces to
(45)12p+12x0.5p′=3Γ2.5x3/2,p0=0.Solve this problem by using the method of constant variation, we get(46)p=6Γ2.5x3/2−32x+32x−34+34e−2x,(47)y=6Γ2.525x5/2−34x2+x3/2−34x−38e−2x−34xe−2x+94Γ2.5.The solutions to Example3 can be seen in Figure 11.Figure 11
Solution of Example3.


## 4. Conclusion

We propose a kind of improved conformable fractional derivative in this paper. This improved conformable fractional derivative is also local by its definition, and meanwhile, a kind of historical memory parameter is introduced to its definition. The advantage of the improved conformable derivative is that its physical behavior approximates the Riemann-Liouville and Caputo fractional derivative better than the conformable fractional derivative. So this improved conformable fractional derivative has much potential in modelling many physical problems where the Riemann-Liouville and Caputo fractional derivative is usually used.



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