# New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations

Authors: Jian Rong Loh; Chang Phang; Abdulnasir Isah
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown functionf(x). This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval [0,1]. This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution f(x). A few numerical examples of FIDE are solved using those expressions derived for Genocchi polynomial approximation. Numerical results show that the Genocchi polynomial approximation adopting the operational matrix of fractional derivative achieves good accuracy comparable to some existing methods. In certain cases, Genocchi polynomial provides better accuracy than the aforementioned methods.


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## Body

## 1. Introduction

Fractional integro-differential equation (FIDE) is an equation that contains a fractional derivative term0Dxαf(x) and an integral kernel operator term K~f(x)=∫K(x,t,f(t))dt. An example is given as follows:(1)D0xαfx=h1xfx+∫0xK1x,tftdt+∫01K2x,tftdt+h2x.Here, f(x) is the unknown function to be solved and K1(x,t), K2(x,t) are the Volterra integral kernel and Fredholm integral kernel, respectively. D0xα is Caputo’s fractional derivative and h1(x), h2(x) are coefficient functions.In the past few years, various wavelet operational matrices and polynomial operational matrices for both fractional differentiation and fractional integration have been derived one after another to solve various kinds of fractional differential equations (FDEs) and fractional integro-differential equations (FIDEs). Examples of polynomial operational matrices that have been proposed in the past are Shifted Legendre polynomials [1], shifted Chebyshev polynomials [2], and shifted Jacobi polynomials [3] to name a few. The operational matrix method is usually combined with methods such as collocation method, spectral tau method, and tau method to solve FDE and FIDE. In [4], Zhu and Fan have proposed Chebyshev wavelet operational matrix of fractional integration and applied it to solve a certain type of nonlinear fractional integro-differential equations (FIDEs). Apart from the operational matrix approach, there exist various approaches such as Adomian decomposition method [5], spectral tau method [6], Taylor expansion method [7], and Bernoulli matrix method [8]. In [9], Alshbool et al. have proposed a new Bernstein function to solve fractional order differential equations.In this paper, different from the existing methods in numerically solving FIDE, we explore the application of a relatively new polynomial, namely, Genocchi polynomials, in the numerical solution of FIDE. Genocchi polynomial belongs to a larger family class of polynomials which is the Appell polynomial family. Throughout this paper, we denote Genocchi polynomials byGn(x). We apply these polynomials to solve the FIDEs given the advantage of Genocchi polynomials having smaller coefficients of each individual term and relatively lesser terms compared to classical orthogonal polynomials. Thus, this is expected to provide us with smaller computational errors.The rest of the paper is organized as follows: Section2 introduces preliminary definitions and properties of the Caputo fractional derivative. Section 3 gives basic definitions and properties of Gn(x), function approximation by Gn(x), and the analytical expression of T(n,m)=∫abGn(x)Gm(x)dx which is the integral of the product of two Genocchi polynomials. Section 4.1 derives the Genocchi polynomial operational matrix of Caputo’s fractional derivative. Section 4.2 shows the way to approximate the integral kernel in terms of Genocchi polynomials and derives the analytical expression of the kernel matrix. Section 5 presents the general approach of approximating (1) into a system of algebraic equations and using collocation methods to solve (1). Section 6 describes the general procedure of using Genocchi polynomials approximation to solve (1). Section 7 shows results of the proposed method. Section 8 sums up the findings of this paper.


## 2. Preliminaries

### 2.1. Fractional Calculus: Definitions

Fractional differentiation comes in different versions [11, 12]. In this paper, we consider Caputo’s fractional differentiation which provides a more realistic physical interpretation in real-life applications. Caputo’s fractional derivative operator Dα of a function f(t) is defined as follows:Definition 1.
(2) D x α  0 f x = 1 Γ n - α ∫ 0 x x - t n - α - 1 f n t d t ,  n - 1 < α ≤ n ,   n ∈ N , n = α .Below are some properties of Caputo’s fractional derivatives:(3)Dxα0k=0,kisconstantDxα0xβ=0,β∈N∪0,β<αΓβ+1Γβ+1-αxβ-α,β∈N∪0,β≥αorβ∉N,β>α,where ⌈α⌉ denotes the smallest integer greater than or equal to α and ⌊α⌋ denotes the largest integer less than or equal to α.Caputo’s fractional differential operator is linear:(4)Dxα0λfx+μhx=λDxα0fx+μDxα0hx,where λ and μ are arbitrary constants.


## 2.1. Fractional Calculus: Definitions

Fractional differentiation comes in different versions [11, 12]. In this paper, we consider Caputo’s fractional differentiation which provides a more realistic physical interpretation in real-life applications. Caputo’s fractional derivative operator Dα of a function f(t) is defined as follows:Definition 1.
(2) D x α  0 f x = 1 Γ n - α ∫ 0 x x - t n - α - 1 f n t d t ,  n - 1 < α ≤ n ,   n ∈ N , n = α .Below are some properties of Caputo’s fractional derivatives:(3)Dxα0k=0,kisconstantDxα0xβ=0,β∈N∪0,β<αΓβ+1Γβ+1-αxβ-α,β∈N∪0,β≥αorβ∉N,β>α,where ⌈α⌉ denotes the smallest integer greater than or equal to α and ⌊α⌋ denotes the largest integer less than or equal to α.Caputo’s fractional differential operator is linear:(4)Dxα0λfx+μhx=λDxα0fx+μDxα0hx,where λ and μ are arbitrary constants.


## 3. Genocchi Polynomials and Function Approximation

### 3.1. Genocchi Polynomials: Definitions and Basic Properties

The Genocchi polynomialsGn(x) are defined as follows [13–15].Definition 2.
The Genocchi polynomialGn(x) is defined by the generating function Q(x,t):(5)Qx,t=2textet+1=∑n=0∞Gnxtnn!,Gnx=∑k=0nnkgn-kxk=2Bnx-2n+1Bnx,where gk=2Bk-2k+1Bk is the Genocchi number. Bn,Bn(x) are the Bernoulli number and Bernoulli polynomial, respectively.Below are some of the important properties of Genocchi polynomials:(6)dGnxdx=nGn-1x,n≥1,dkGnxdxk=0,n≤kk!nkGn-kx,n>kk,n∈N∪0,Gn1+Gn0=0,n>1.


### 3.2. Function Approximation by Genocchi Polynomials

We may approximate a functionf(x) in terms of Genocchi polynomials Gn(x) by the following infinite series:(7)fx=∑n=1∞cnGnx,where Gn(x) are the Genocchi polynomials and the Genocchi coefficients cn.In practice, we truncate the infinite series up toN number of Genocchi polynomials according to the desired accuracy of the problem into the following truncated Genocchi series:(8)fx≃∑n=1NcnGnx,where in matrix notation,(9)fx=CTGx,where C=c1c2⋯cNT is the Genocchi coefficient vector.G ( x ) = G 1 ( x ) G 2 ( x ) ⋯ G N ( x ) T is the Genocchi vector.In this case, the Genocchi coefficientscn may be computed as follows:(10)cn=12n!fn-10+fn-11,n=1,2,…,N.


### 3.3. Computing Genocchi Coefficients by Matrix Approach

Equation (10) fails to work for functions that are not (n-1) differentiable at the points x=0 and x=1. An example is shown below.LetN=3; x3/2≈∑n=13cnGn(x)=c1G1(x)+c2G2(x)+c3G3(x); (11)c3=123!d2dx2x3/2x=0+d2dx2x3/2x=1=123!34xx=0+34xx=1. To avoid this problem that occurs in the case of differentiation, we may evaluate the Genocchi coefficients cn using Theorem 4. To prove Theorem 4, we first prove the following Theorem 3 which gives the analytical expression of the integral of the product of two Genocchi polynomials over an arbitrary interval [a,b], 0≤a≤b which will be used in the subsequent part of the paper.Theorem 3.
Given any two Genocchi polynomialsGn(x), Gm(x), for x≥0(12)γn,mx=∫GnxGmxdx=∫0xGnxGmxdx=∑r=0n-1-1rnrm+1r+1Gn-rxGm+1+rx-gn-rgm+1+r,where gn-r=Gn-r(0) is the Genocchi number, n(r), (m+1)(r+1) are the falling and rising factorial, respectively. In particular, we have the following for [a,b], 0≤a≤b(13)γn,ma,b=∫abGnxGmxdx=γn,mb-γn,ma=∑r=0n-1-1rnrm+1r+1Gn-rbGm+1+rb-Gn-raGm+1+ra,γn,m0,1=∑r=0n-1-1rnrm+1r+1Gn-rbGm+1+rb-gn-rgm+1+r.Proof.
Using the following expression:(14)dGmxdx=mGm-1x,∫abGmxdx=Gm+1b-Gm+1am+1,∫0xGmxdx=Gm+1x-gm+1m+1.We obtain (15)
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.Continuing this relation recursively for n times, we arrive at (16)=∑r=0n-1-1rnn-1⋯n-r+1m+1⋯m+rGn-rxGm+1+rx-gn-rgm+1+r=∑r=0n-1-1rnrm+1r+1Gn-rxGm+1+rx-gn-rgm+1+r.Adopting the techniques used in [16] for function approximation, we prove the following theorem of function approximation using Genocchi polynomials.Theorem 4.
Letf(x)∈L2[0,1] and {Gi(x),i=1,…,N} be the set of Genocchi polynomials up to order N. Let Y=span⁡{G1(x),…,GN(x)}. Since Y is a finite dimensional closed subspace of L2[0,1], then ∃f∗(x)∈Y is the unique best approximation in Genocchi polynomials such that f(x) can be approximated by unique coefficients cn:(17)fx≈f∗x=∑n=1NcnGnx=CTGx.The Genocchi coefficient matrix C consisting of the unique coefficients cn is given by the following:(18)CT=FTT0,1-1,where F=∫01f(x)Gm(x)dxN×N and T(0,1)=∫01Gn(x)Gm(x)dxN×N as derived in Theorem 3.Proof.
(19) f x ≈ ∑ n = 1 N c n G n x , ∫ 0 1 f x G m x d x = ∑ n = 1 N c n ∫ 0 1 G n x G m x d x .Let fm=∫01f(x)Gm(x)dx. Then, (20)fm=∑n=1Ncn∫01GnxGmxdx=∑n=1Ncnγn,m0,1m=1,…,N. Thus, we have a system of N equations which is written in matrix representation as follows: (21)f1⋮fN=c1,…,cNγ1,10,1⋯γ1,N0,1γ2,10,1⋯γ2,N0,1⋮⋱⋮γN,10,1⋯γN,N0,1,FT=CTT0,1.Therefore, the Genocchi coefficients matrix C is (22)CT=FTT0,1-1, where γi,j can be obtained from Theorem 3.


## 3.1. Genocchi Polynomials: Definitions and Basic Properties

The Genocchi polynomialsGn(x) are defined as follows [13–15].Definition 2.
The Genocchi polynomialGn(x) is defined by the generating function Q(x,t):(5)Qx,t=2textet+1=∑n=0∞Gnxtnn!,Gnx=∑k=0nnkgn-kxk=2Bnx-2n+1Bnx,where gk=2Bk-2k+1Bk is the Genocchi number. Bn,Bn(x) are the Bernoulli number and Bernoulli polynomial, respectively.Below are some of the important properties of Genocchi polynomials:(6)dGnxdx=nGn-1x,n≥1,dkGnxdxk=0,n≤kk!nkGn-kx,n>kk,n∈N∪0,Gn1+Gn0=0,n>1.


## 3.2. Function Approximation by Genocchi Polynomials

We may approximate a functionf(x) in terms of Genocchi polynomials Gn(x) by the following infinite series:(7)fx=∑n=1∞cnGnx,where Gn(x) are the Genocchi polynomials and the Genocchi coefficients cn.In practice, we truncate the infinite series up toN number of Genocchi polynomials according to the desired accuracy of the problem into the following truncated Genocchi series:(8)fx≃∑n=1NcnGnx,where in matrix notation,(9)fx=CTGx,where C=c1c2⋯cNT is the Genocchi coefficient vector.G ( x ) = G 1 ( x ) G 2 ( x ) ⋯ G N ( x ) T is the Genocchi vector.In this case, the Genocchi coefficientscn may be computed as follows:(10)cn=12n!fn-10+fn-11,n=1,2,…,N.


## 3.3. Computing Genocchi Coefficients by Matrix Approach

Equation (10) fails to work for functions that are not (n-1) differentiable at the points x=0 and x=1. An example is shown below.LetN=3; x3/2≈∑n=13cnGn(x)=c1G1(x)+c2G2(x)+c3G3(x); (11)c3=123!d2dx2x3/2x=0+d2dx2x3/2x=1=123!34xx=0+34xx=1. To avoid this problem that occurs in the case of differentiation, we may evaluate the Genocchi coefficients cn using Theorem 4. To prove Theorem 4, we first prove the following Theorem 3 which gives the analytical expression of the integral of the product of two Genocchi polynomials over an arbitrary interval [a,b], 0≤a≤b which will be used in the subsequent part of the paper.Theorem 3.
Given any two Genocchi polynomialsGn(x), Gm(x), for x≥0(12)γn,mx=∫GnxGmxdx=∫0xGnxGmxdx=∑r=0n-1-1rnrm+1r+1Gn-rxGm+1+rx-gn-rgm+1+r,where gn-r=Gn-r(0) is the Genocchi number, n(r), (m+1)(r+1) are the falling and rising factorial, respectively. In particular, we have the following for [a,b], 0≤a≤b(13)γn,ma,b=∫abGnxGmxdx=γn,mb-γn,ma=∑r=0n-1-1rnrm+1r+1Gn-rbGm+1+rb-Gn-raGm+1+ra,γn,m0,1=∑r=0n-1-1rnrm+1r+1Gn-rbGm+1+rb-gn-rgm+1+r.Proof.
Using the following expression:(14)dGmxdx=mGm-1x,∫abGmxdx=Gm+1b-Gm+1am+1,∫0xGmxdx=Gm+1x-gm+1m+1.We obtain (15)
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.Continuing this relation recursively for n times, we arrive at (16)=∑r=0n-1-1rnn-1⋯n-r+1m+1⋯m+rGn-rxGm+1+rx-gn-rgm+1+r=∑r=0n-1-1rnrm+1r+1Gn-rxGm+1+rx-gn-rgm+1+r.Adopting the techniques used in [16] for function approximation, we prove the following theorem of function approximation using Genocchi polynomials.Theorem 4.
Letf(x)∈L2[0,1] and {Gi(x),i=1,…,N} be the set of Genocchi polynomials up to order N. Let Y=span⁡{G1(x),…,GN(x)}. Since Y is a finite dimensional closed subspace of L2[0,1], then ∃f∗(x)∈Y is the unique best approximation in Genocchi polynomials such that f(x) can be approximated by unique coefficients cn:(17)fx≈f∗x=∑n=1NcnGnx=CTGx.The Genocchi coefficient matrix C consisting of the unique coefficients cn is given by the following:(18)CT=FTT0,1-1,where F=∫01f(x)Gm(x)dxN×N and T(0,1)=∫01Gn(x)Gm(x)dxN×N as derived in Theorem 3.Proof.
(19) f x ≈ ∑ n = 1 N c n G n x , ∫ 0 1 f x G m x d x = ∑ n = 1 N c n ∫ 0 1 G n x G m x d x .Let fm=∫01f(x)Gm(x)dx. Then, (20)fm=∑n=1Ncn∫01GnxGmxdx=∑n=1Ncnγn,m0,1m=1,…,N. Thus, we have a system of N equations which is written in matrix representation as follows: (21)f1⋮fN=c1,…,cNγ1,10,1⋯γ1,N0,1γ2,10,1⋯γ2,N0,1⋮⋱⋮γN,10,1⋯γN,N0,1,FT=CTT0,1.Therefore, the Genocchi coefficients matrix C is (22)CT=FTT0,1-1, where γi,j can be obtained from Theorem 3.


## 4. Main Result

### 4.1. Genocchi Polynomial Operational Matrix of Caputo’s Fractional Differentiation

In this section, we derive the analytical expression of the Genocchi polynomial operational matrix of Caputo’s fractional derivative, which is theN×N matrix PGα0, where (23)Dxα0Gx=PGα0Gx,Dxα0G1G2⋮GN=ρ11ρ12⋯ρ1Nρ21ρ22⋯ρ2N⋮⋯⋯⋮ρN1ρN2⋯ρNNG1G2⋮GN.To derive PGα0, we first prove the following Lemmas 5 and 6.Lemma 5.
Caputo’s fractional derivative of fractional orderα of a Genocchi polynomial of order i is given by(24)Dxα0Gix=∑r=αii!gi-ri-r!Γr-α+1xr-α,n-1<α≤n,n=α∈N,i≥α0,i<α.Proof.
Forn-1<α≤n, n=α(25)Dxα0Gix=1Γn-α∫0xx-tn-α-1dndtnGitdt=1Γα-α∫0xx-tn-α-1dndtn∑r=0iirgi-rtrdt=1Γα-α∑r=0iirgi-r∫0xx-tα-α-1dαdtαtrdt=1Γα-α∑r=αiirgi-r∫0xx-tα-α-1α!rαtr-αdt=1Γα-α∑r=αiirα!rαgi-r∫0xx-tα-α-1tr-αdt,where i≥⌈α⌉. Substituting t=xu(26)Dxα0Gix=1Γα-α∑r=αiirα!rαgi-rxr-α∫011-uα-α-1ur-αdu=1Γα-α∑r=αiirα!rαgi-rxr-αBr-α+1,α-α=∑r=αii!gi-ri-r!Γr-α+1xr-α,where B(x,y)=∫01ux-1(1-u)y-1du is the Beta function for Re(x),Re(y)>0.Lemma 6.
The matrixΘN defined as ΘN=∫01Dxα0G(x)∗GT(x)dx is given by the following:(27)ΘN=θikN×N=∫01Dxα0GixGkxdxN×N,where(28)θik=∑r=αi∑p=0ki!kpgi-rgk-pi-r!r-α+p+1Γr-α+1.Proof.
From Lemma5, (29)Dxα0Gix=∑r=αii!gi-ri-r!Γr-α+1xr-α,Dxα0GixGkx=∑r=αii!gi-ri-r!Γr-α+1xr-α∑p=0kkpgk-pxp=∑r=αi∑p=0ki!gi-ri-r!Γr-α+1kpgk-pxr-α+p,∫01Dxα0GixGkxdx=∑r=αi∑p=0ki!gi-ri-r!Γr-α+1kpgk-p∫01xr-α+pdx=∑r=αi∑p=0ki!kpgi-rgk-pi-r!r-α+p+1Γr-α+1.Now, we can prove the following theorem that computes the operational matrixPGα0.Theorem 7.
Given a set ofGi(x), i=1,…,N, of N Genocchi polynomials, the Genocchi polynomial operational matrix of Caputo’s fractional derivative of order α over the interval [0,1] is the N×N matrix PGα0 and is given by(30)PGα0=ΘNT0,1-1,where ΘN is given in Lemma 6 and T(0,1)=[γn,m(0,1)]N×N with elements γn,m(0,1) given in Theorem 3.Proof.
From Lemma5, (31)Dxα0Gx=PGα0Gx,Dxα0Gx∗GTx=PGα0GxGTx,∫01Dxα0GxGTxdx=PGα0∫01GxGTxdx,ΘN=PGα0T0,1,PGα0=ΘNT0,1-1.Here, we justify the better accuracy of the new operational matrix of Caputo’s fractional derivative due to Genocchi polynomials by comparing its errors and that obtained when using Shifted Legendre operational matrix. This may be due to the fact that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. We define the absolute error and relative error respectively, as follows:(32)DαYix-Pα∗Yi,DαYix-Pα∗YiDαYix,where Yi(x) is either the Genocchi polynomial Gi(x) or Shifted Legendre polynomial L~i(x). In Figure 1, we compare the relative error obtained due to Genocchi polynomial operational matrix of fractional derivative (GPOMFD) and the Shifted Legendre operational matrix of fractional derivative (SLOMFD) derived in [17]. One clearly sees from the figure that the errors for GPOMFD become smaller than that of SLOMFD for N≥3 and their difference gets larger as N increases over the interval [0,1]. Further, we also show the absolute error and relative error in Tables 1 and 2, respectively, for α=0.5 at different points.Table 1
Comparison of absolute error between SLOMFD and GPOMFD withα=0.5 at points x=0,0.5,1.
SLOMFD  GPOMFD N x = 0 x = 0.5 x = 1 N x = 0 x = 0.5 x = 1 1 0.113121 0.001562 0.005656 2 0.113121 0.001306 0.005656 3 0.696638 0.010167 0.036679 4 0.003583 0.000429 0.000549 5 1.836044 0.030079 0.107732 6 0.008628 0.000272 0.001283 7 3.773422 0.077279 0.273317 8 0.048392 0.004996 0.007218 9 7.468363 0.257570 0.899849 10 0.440694 0.045400 0.065718Table 2
Comparison of relative error between SLOMFD and GPOMFD withα=0.5 at points x=0.1,0.6,1.
SLOMFD  GPOMFD N x = 0.1 x = 0.6 x = 1 N x = 0.1 x = 0.6 x = 1 1 0.0061268 0.0009516 0.0025063 2 0.0061268 0.0009516 0.0025063 3 0.0095486 0.0256030 0.0081265 4 0.0016094 0.0000698 0.0003038 5 0.0235837 0.0171498 0.0159125 6 0.0014129 0.0000535 0.0002389 7 0.1314007 0.5765935 0.0302776 8 0.0014150 0.0000529 0.0002372 9 8.7021303 0.1871596 0.0797470 10 0.0014147 0.0000528 0.0002369Figure 1
Comparison of relative error of operational matrix of fractional derivative (OMFD) withα=0.5 between N=9 Shifted Legendre polynomials and N=10 Genocchi polynomials.


### 4.2. Approximation of Integral Kernel by Genocchi Polynomials

The integral kernel functionK(x,t) will be approximated by Genocchi polynomials for each variable Gi(x),Gj(t) as follows:(33)Kx,t=∑i=1∞∑j=1∞kijGixGjt≈∑i=1N∑j=1NkijGixGjt=GTxKGGt,where we refer to KG as thekernel matrix by Genocchi polynomial. Theorem 8 provides the formula for obtaining the kernel matrix KG.Theorem 8.
LetK(x,t) be a two-variable continuous function in CN-1([0,1]).
Then,K(x,t) can be approximated in terms of Genocchi polynomials up to order N, that is, K(x,t)≈∑i=1N∑j=1NkijGixGjt=GTxKGGt,N∈N, where G is the Genocchi polynomial basis matrix and KG is the N×N integral kernel matrix in Genocchi basis given by(34)KG=T0,1-1NGT0,1-1,where(35)NG=ηpqN×N=∫01∫01Kx,tGpxGqtdxdtN×N,T0,1-1=γn,m0,1N×N-1=∑r=0n-1-1rnrm+1r+1Gn-rbGm+1+rb-gn-rgm+1+rN×N-1.Proof.
FromK(x,t)≈GT(x)KGG(t), (36)GxKx,t=GxGTxKGGt,∫01GxKx,tdx=∫01GxGTxdxKGGt=T0,1KGGt,∫01GxKx,tdxGTt=T0,1KGGtGTt,∫01∫01Kx,tGxGTtdxdt=T0,1KG∫01GtGTtdt,where T(0,1)=∫01G(x)GT(x)dx=∫01G(t)GT(t)dt.
DefineNG=∫01∫01Kx,tGxGTtdxdt. Thus, (37)NG=T0,1KGT0,1,KG=T0,1-1NGT0,1-1.


## 4.1. Genocchi Polynomial Operational Matrix of Caputo’s Fractional Differentiation

In this section, we derive the analytical expression of the Genocchi polynomial operational matrix of Caputo’s fractional derivative, which is theN×N matrix PGα0, where (23)Dxα0Gx=PGα0Gx,Dxα0G1G2⋮GN=ρ11ρ12⋯ρ1Nρ21ρ22⋯ρ2N⋮⋯⋯⋮ρN1ρN2⋯ρNNG1G2⋮GN.To derive PGα0, we first prove the following Lemmas 5 and 6.Lemma 5.
Caputo’s fractional derivative of fractional orderα of a Genocchi polynomial of order i is given by(24)Dxα0Gix=∑r=αii!gi-ri-r!Γr-α+1xr-α,n-1<α≤n,n=α∈N,i≥α0,i<α.Proof.
Forn-1<α≤n, n=α(25)Dxα0Gix=1Γn-α∫0xx-tn-α-1dndtnGitdt=1Γα-α∫0xx-tn-α-1dndtn∑r=0iirgi-rtrdt=1Γα-α∑r=0iirgi-r∫0xx-tα-α-1dαdtαtrdt=1Γα-α∑r=αiirgi-r∫0xx-tα-α-1α!rαtr-αdt=1Γα-α∑r=αiirα!rαgi-r∫0xx-tα-α-1tr-αdt,where i≥⌈α⌉. Substituting t=xu(26)Dxα0Gix=1Γα-α∑r=αiirα!rαgi-rxr-α∫011-uα-α-1ur-αdu=1Γα-α∑r=αiirα!rαgi-rxr-αBr-α+1,α-α=∑r=αii!gi-ri-r!Γr-α+1xr-α,where B(x,y)=∫01ux-1(1-u)y-1du is the Beta function for Re(x),Re(y)>0.Lemma 6.
The matrixΘN defined as ΘN=∫01Dxα0G(x)∗GT(x)dx is given by the following:(27)ΘN=θikN×N=∫01Dxα0GixGkxdxN×N,where(28)θik=∑r=αi∑p=0ki!kpgi-rgk-pi-r!r-α+p+1Γr-α+1.Proof.
From Lemma5, (29)Dxα0Gix=∑r=αii!gi-ri-r!Γr-α+1xr-α,Dxα0GixGkx=∑r=αii!gi-ri-r!Γr-α+1xr-α∑p=0kkpgk-pxp=∑r=αi∑p=0ki!gi-ri-r!Γr-α+1kpgk-pxr-α+p,∫01Dxα0GixGkxdx=∑r=αi∑p=0ki!gi-ri-r!Γr-α+1kpgk-p∫01xr-α+pdx=∑r=αi∑p=0ki!kpgi-rgk-pi-r!r-α+p+1Γr-α+1.Now, we can prove the following theorem that computes the operational matrixPGα0.Theorem 7.
Given a set ofGi(x), i=1,…,N, of N Genocchi polynomials, the Genocchi polynomial operational matrix of Caputo’s fractional derivative of order α over the interval [0,1] is the N×N matrix PGα0 and is given by(30)PGα0=ΘNT0,1-1,where ΘN is given in Lemma 6 and T(0,1)=[γn,m(0,1)]N×N with elements γn,m(0,1) given in Theorem 3.Proof.
From Lemma5, (31)Dxα0Gx=PGα0Gx,Dxα0Gx∗GTx=PGα0GxGTx,∫01Dxα0GxGTxdx=PGα0∫01GxGTxdx,ΘN=PGα0T0,1,PGα0=ΘNT0,1-1.Here, we justify the better accuracy of the new operational matrix of Caputo’s fractional derivative due to Genocchi polynomials by comparing its errors and that obtained when using Shifted Legendre operational matrix. This may be due to the fact that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. We define the absolute error and relative error respectively, as follows:(32)DαYix-Pα∗Yi,DαYix-Pα∗YiDαYix,where Yi(x) is either the Genocchi polynomial Gi(x) or Shifted Legendre polynomial L~i(x). In Figure 1, we compare the relative error obtained due to Genocchi polynomial operational matrix of fractional derivative (GPOMFD) and the Shifted Legendre operational matrix of fractional derivative (SLOMFD) derived in [17]. One clearly sees from the figure that the errors for GPOMFD become smaller than that of SLOMFD for N≥3 and their difference gets larger as N increases over the interval [0,1]. Further, we also show the absolute error and relative error in Tables 1 and 2, respectively, for α=0.5 at different points.Table 1
Comparison of absolute error between SLOMFD and GPOMFD withα=0.5 at points x=0,0.5,1.
SLOMFD  GPOMFD N x = 0 x = 0.5 x = 1 N x = 0 x = 0.5 x = 1 1 0.113121 0.001562 0.005656 2 0.113121 0.001306 0.005656 3 0.696638 0.010167 0.036679 4 0.003583 0.000429 0.000549 5 1.836044 0.030079 0.107732 6 0.008628 0.000272 0.001283 7 3.773422 0.077279 0.273317 8 0.048392 0.004996 0.007218 9 7.468363 0.257570 0.899849 10 0.440694 0.045400 0.065718Table 2
Comparison of relative error between SLOMFD and GPOMFD withα=0.5 at points x=0.1,0.6,1.
SLOMFD  GPOMFD N x = 0.1 x = 0.6 x = 1 N x = 0.1 x = 0.6 x = 1 1 0.0061268 0.0009516 0.0025063 2 0.0061268 0.0009516 0.0025063 3 0.0095486 0.0256030 0.0081265 4 0.0016094 0.0000698 0.0003038 5 0.0235837 0.0171498 0.0159125 6 0.0014129 0.0000535 0.0002389 7 0.1314007 0.5765935 0.0302776 8 0.0014150 0.0000529 0.0002372 9 8.7021303 0.1871596 0.0797470 10 0.0014147 0.0000528 0.0002369Figure 1
Comparison of relative error of operational matrix of fractional derivative (OMFD) withα=0.5 between N=9 Shifted Legendre polynomials and N=10 Genocchi polynomials.


## 4.2. Approximation of Integral Kernel by Genocchi Polynomials

The integral kernel functionK(x,t) will be approximated by Genocchi polynomials for each variable Gi(x),Gj(t) as follows:(33)Kx,t=∑i=1∞∑j=1∞kijGixGjt≈∑i=1N∑j=1NkijGixGjt=GTxKGGt,where we refer to KG as thekernel matrix by Genocchi polynomial. Theorem 8 provides the formula for obtaining the kernel matrix KG.Theorem 8.
LetK(x,t) be a two-variable continuous function in CN-1([0,1]).
Then,K(x,t) can be approximated in terms of Genocchi polynomials up to order N, that is, K(x,t)≈∑i=1N∑j=1NkijGixGjt=GTxKGGt,N∈N, where G is the Genocchi polynomial basis matrix and KG is the N×N integral kernel matrix in Genocchi basis given by(34)KG=T0,1-1NGT0,1-1,where(35)NG=ηpqN×N=∫01∫01Kx,tGpxGqtdxdtN×N,T0,1-1=γn,m0,1N×N-1=∑r=0n-1-1rnrm+1r+1Gn-rbGm+1+rb-gn-rgm+1+rN×N-1.Proof.
FromK(x,t)≈GT(x)KGG(t), (36)GxKx,t=GxGTxKGGt,∫01GxKx,tdx=∫01GxGTxdxKGGt=T0,1KGGt,∫01GxKx,tdxGTt=T0,1KGGtGTt,∫01∫01Kx,tGxGTtdxdt=T0,1KG∫01GtGTtdt,where T(0,1)=∫01G(x)GT(x)dx=∫01G(t)GT(t)dt.
DefineNG=∫01∫01Kx,tGxGTtdxdt. Thus, (37)NG=T0,1KGT0,1,KG=T0,1-1NGT0,1-1.


## 5. Error Analysis

Following the analysis made in [18], we derive the error estimates of numerical approximation using Genocchi polynomials considering the two following scenarios.(1) If the exact solutionf(x) of (1) is known, and fN∗(x) is the numerical approximate solution using Genocchi polynomials up to order N, then we define the error estimation as follows:(38)δN=fx-fN∗x2,where y(x)2=∫01y2(x)dx. In the following theorem, we derive a bound for the L2 error estimation of our Genocchi polynomial method.Theorem 9.
Letf(x)∈L2[0,1] be an arbitrary function. If f(x) is approximated by truncated Genocchi polynomial series fN∗(x)=∑n=1NcnGn(x) and fn-1xx∈[0,1]≤M (M is finite), then the error in L2 is norm; δN=f(x)-fN∗(x)2 for N order approximation is bounded above by (39)δN≤∑n=N+1∞∑k=1nM22∗n!2nk2gn-k22k+1.Proof.
(40) δ N 2 = ∫ 0 1 f x - f N ∗ x 2 d x = ∫ 0 1 f x - ∑ n = 1 N c n G n x 2 d x = ∫ 0 1 ∑ n = N + 1 ∞ c n G n x 2 d x ≤ ∫ 0 1 ∑ n = N + 1 ∞ c n G n x 2 d x = ∑ n = N + 1 ∞ ∫ 0 1 c n 2 G n x 2 d x , (41) ∫ 0 1 G n x 2 d x = ∫ 0 1 ∑ k = 1 n n k g n - k x k 2 d x ≤ ∑ k = 1 n ∫ 0 1 n k 2 g n - k 2 x k 2 d x = ∑ k = 1 n n k 2 g n - k 2 ∫ 0 1 x 2 k d x = ∑ k = 1 n n k 2 g n - k 2 2 k + 1 .Therefore, (42)δN2≤∑n=N+1∞cn2∑k=1nnk2gn-k22k+1=∑n=N+1∞12∗n!fn-10+fn-112∑k=1nnk2gn-k22k+1≤∑n=N+1∞∑k=1n14∗n!2fn-102+fn-112nk2gn-k22k+1=∑n=N+1∞∑k=1n14∗n!22M2nk2gn-k22k+1=∑n=N+1∞∑k=1nM22∗n!2nk2gn-k22k+1,δN≤∑n=N+1∞∑k=1nM22∗n!2nk2gn-k22k+1.Obviously, the estimation would become more precise if the orderN increases.(2) In usual cases, the exact solution is unknown. So, first we obtain the approximate solution of orderN, fN∗(x). Then, we obtain the approximate solution of order N+1, fN∗(x). Define the error estimation as follows:(43)ϵN=fN∗x-fN+1∗x2.For a given desired error ϵN, one is able to determine the smallest number N which gives the desired error.Theorem 10.
SupposefN∗(x) and fN+1∗(x) are numerical solutions for the unknown solution f(x) of (1) with order N,N+1, respectively. The error estimation defined as ϵN=fN∗(x)-fN+1∗(x)2 is convergent.Proof.
(44) ϵ N = f N ∗ x - f N + 1 ∗ x 2 = f N ∗ x - f x + f x - f N + 1 ∗ x 2 ≤ f N ∗ x - f x 2 + f x - f N + 1 ∗ x 2 = δ N + δ N + 1 . δ N and δN+1 are bounded by upper bound given in Theorem 9 and, in turn, are both convergent. Thus, ϵN is convergent too.


## 6. Solving Fractional Integro-Differential Equation by Genocchi Polynomial Approximation

As in any operational matrix method, we approximate each term in the following fractional integro-differential equation (FIDE) with Genocchi polynomials.(45)Dxα0fx=h1xfx+∫0xK1x,tftdt+∫01K2x,tftdt+h2x,fixk=yk,i,k=0,…,α-1.For the FIDE in (1), we approximate each term using a set of Genocchi polynomials up to order N. The functions h1(x) and h2(x), which are known a priori in the FIDE, will not be approximated since this adds additional computational error to the numerical solution. (46)Dxα0fx≈C0TPGαGx,hx≈HTGx,K1x,t≈GTxK1Gt,K2x,t≈GTxK2Gt.After approximation by Genocchi polynomial, the FIDE written in the matrix representation becomes the following:(47)GTxPGα0TC=h1xGTxC+∫0xGTxK1GtGTtCdt+∫01GTxK2GtGTtCdt+h2x,GTxPGα0TC=h1xGTxC+GTxK1T0,xCdt+GTxK2T0,1Cdt+h2x,where T(0,x)=∫0x‍G(t)GT(t)dt and T(0,1)=∫01‍G(t)GT(t)dt.Then, we selectN+1 equally spaced points as the collocation points in the interval [0,1]; that is,(48)vr=rN,r=0,…,N.After collocating (47), we obtain a system of N+1 algebraic equations consisting of N Genocchi coefficients C=c1⋯cNT. Given ⌈α⌉ initial conditions, fixk=yk, i,k=0,…,⌈α⌉-1, we select N-⌈α⌉ equations out of the N algebraic equations and solve C numerically. Thus, we obtain the approximate solution fN∗(x)=CTG(x) of order N.


## 7. Numerical Examples

We present some examples of FIDEs which are solved using the proposed method of Genocchi polynomial approximation and Genocchi polynomial operational matrix of fractional derivative (GPOMFD). As in [8], we compute the maximum error function of the approximate function fN∗(x) of order N for each numerical solution obtained as follows:(49)e∞N=fx-fN∗x∞=max⁡fx-fN∗x,a≤x≤b,where f(x) is the exact solution and fN∗(x) is the approximate solution of order N. Also, we compute the absolute error of N order approximation at a particular point x∈[0,1] as follows:(50)eNx=fx-fN∗x.Example 1 (integer case).
Given(51)Dx20fx=-x2fx+∫0xxtftdt+h2xwith initial conditions f(0)=2, f(1)(0)=0. h2(x)=-(1/30)x7+(1/5)x6+(1/4)x5-x4+x3+(2/5)x2-2. The exact solution is f(x)=(1/5)x4-x2-2. Using N=5, that is, f5∗(x)=∑k=15ciGi(x), the Genocchi polynomial method gives us the exact solution f5∗(x)=(1/5)x4-x2-2.Example 2 (integer case).
As for Example 3 in [8], given (52)Dx10fx=xex-ex-x+∫01xftdtwith initial condition f(0)=0, the exact solution is f(x)=xex. Figure 2 shows both f(x) and f11∗(x) over the interval [0,1]. Tables 3 and 4 show the numerical results of the exact solutions and approximate solutions (f11∗(x),f17∗(x)) by Genocchi polynomial for N=11,17 and approximate solutions (y10(x),y16(x)) by Bernoulli matrix [8] together with absolute errors for both methods. For comparison, Genocchi polynomial approximation of N+1 (i.e., degree N) is compared with the Bernoulli matrix method of order N. Therefore, the Genocchi method of order 11 (i.e., degree 10) is compared with the Bernoulli method of N=10.Table 3
Comparison of absolute errors for Example2 using N=11 orders of Genocchi polynomials and Bernoulli method N=10 [8].
x Exactf(x) Genocchif11∗(x) Genocchie11(x) Bernoullie10(x) [8] 0.0 0.0000000000000000 0.0000000000000000 0.00000 E + 00 6.33107E-16 0.1 0.1105170918075647 0.1105170918090843 1.51950E-12 1.50312E-10 0.2 0.2442805516320338 0.2442805516333059 1.27210E-12 6.01255E-10 0.3 0.4049576422728009 0.4049576422741780 1.37710E-12 1.35331E-09 0.4 0.5967298790565080 0.5967298790578598 1.35200E-12 2.41699E-09 0.5 0.8243606353500640 0.8243606353514914 1.42950E-12 3.89875E-09 0.6 1.0932712802343040 1.0932712802357070 1.40858E-12 6.46841E-09 0.7 1.4096268952293330 1.4096268952309410 1.61360E-12 1.31824E-08 0.8 1.7804327427939730 1.7804327427938810 8.72000E-14 3.51353E-08 0.9 2.2136428000412540 2.2136427999488390 9.24140E-11 1.06298E-07 1.0 2.7182818284590450 2.7182818284628360 3.79650E-12 3.17917E-07Table 4
Comparison of absolute errors for Example2 using N=17 orders of Genocchi polynomials and Bernoulli method N=16 [8].
x Exactf(x) Genocchif17∗(x) Genocchie17(x) Bernoullie16(x) [8] 0.0 0.0000000000000000 0.0000000000000000 5.80000 E - 17 3.00738 E - 15 0.1 0.1105170918075647 0.1105170918075627 2.00000 E - 15 2.34124 E - 15 0.2 0.2442805516320338 0.2442805516320308 3.00000 E - 15 1.23102 E - 15 0.3 0.4049576422728009 0.4049576422727964 4.50000 E - 15 4.89821 E - 16 0.4 0.5967298790565080 0.5967298790565028 5.20000 E - 15 2.87680 E - 15 0.5 0.8243606353500640 0.8243606353500572 6.80000 E - 15 5.76338 E - 15 0.6 1.0932712802343040 1.0932712802342920 1.20000 E - 14 8.76098 E - 15 0.7 1.4096268952293330 1.4096268952293210 1.20000 E - 14 1.25357 E - 14 0.8 1.7804327427939730 1.7804327427939550 1.80000 E - 14 1.65325 E - 14 0.9 2.2136428000412540 2.2136428000412350 1.90000 E - 14 2.76347 E - 14 1.0 2.7182818284590450 2.7182818284590240 2.10000 E - 14 7.29318 E - 14Figure 2
Comparison of exact solutionf(x) and approximate solution f∗(x) for Example 2.Example 3 (integer case).
We consider the following problem as in [8]: (53)Dx10fx=fx+∫01extftdt+1-ex+1x+1with the initial condition f(0)=1. The exact solution is f(x)=ex. Table 5 shows the L∞ error of the Genocchi approximate solutions for order N=7,…,10 and that of Bernoulli method [8] for the corresponding order of N=7,…,9.Table 5
Comparison ofL∞ error for Example 3 using N=7,8,9 orders of Genocchi polynomials and that of Bernoulli method [8].
N Genocchie∞(N) N Bernoullie∞(N) [8] 7 2.685084410123100 E - 07 — — 8 4.733793870463971 E - 08 7 1.11973 E - 04 9 8.708190200000000 E - 08 8 1.14161 E - 05Example 4 (fractional case).
We consider the following problem in [10]. (54)Dx1.70fx=∫0xx-tftdt+∫01x+tftdt+gxwith the initial conditions f(1)(0)=f(0)=0 and g(x)=(Γ3/Γ1.3)x0.3+(Γ4/Γ2.3)x1.3-x4/12-x5/20-7x/12-9/20. The exact solution is f(x)=x2+x3. Table 6 shows the numerical results of the Genocchi approximate solutions, f8∗(x) for N=8, and Legendre wavelet method (LWM) approximate solution for LWMk=4,M=2(x) [10]. Though the exact solution is a polynomial, one is unable to obtain the exact solution. This is because the operational matrix does not give exact approximation of fractional derivative Dx1.70f(x), that is,(55)Dx1.70fx-C0TPG1.7GxL2≠0.Table 6
Comparison of approximate solutions for Example4 using N=8 order Genocchi polynomials and Legendre wavelet method (LWM) [10] for k=4 and M=2.
x Exact Genocchi Genocchi LWM LWM f ( x ) f 8 ∗ ( x ) e 8 ( x ) f k = 4 , M = 2 ∗ ( x ) e k = 4 , M = 2 ( x ) [10] 0.000 0.000000 0.000003 3.50000 E - 06 0.000024 2.40000 E - 05 0.125 0.017578 0.015382 2.19540 E - 03 0.015822 1.75600 E - 03 0.250 0.078125 0.074814 3.31010 E - 03 0.075531 2.59400 E - 03 0.375 0.193359 0.189006 4.35300 E - 03 0.192880 4.79000 E - 04 0.500 0.375000 0.364240 1.07600 E - 02 0.361498 1.35020 E - 02 0.625 0.634765 0.608990 2.57750 E - 02 0.622950 1.18150 E - 02 0.750 0.984375 0.936805 4.75700 E - 02 0.930901 5.34740 E - 02 0.875 1.435540 1.365460 7.00800 E - 02 1.391340 4.42000 E - 02Example 5 (fractional case).
We consider the following problem in [10]. (56)Dx2.30fx=14∫0xx-tftdt+12∫01xtftdt+gxwith the initial conditions f(2)(0)=f(1)(0)=f(0)=0 and g(x)=(Γ(4.5)/Γ(2.2))x1.2-x5.5/99-x/11. The exact solution is f(x)=x7/2. Table 7 shows the numerical results of the Genocchi approximate solutions f8∗(x) for N=8 and f9∗(x) for N=9 and Legendre wavelet method (LWM) approximate solution (LWMk=3,M=2(x)) in [10]. We manage to obtain comparable errors for Genocchi polynomials with that of Legendre wavelets which are known to be a more powerful tool with a higher accuracy than its counterpart, Legendre polynomials.Table 7
Comparison of absolute errors for Example5 using N=8 order of Genocchi polynomials with respect to Legendre wavelet method (LWM k=3, M=2) of [10].
x Genocchie8(x) Genocchie9(x) LWMek=3,M=2(x) [10] 0.000 1.10000 E - 25 1.20000 E - 25 0 0.125 1.60950 E - 04 1.54175 E - 04 2.2475 E - 04 0.250 6.47020 E - 04 6.30863 E - 04 5.6882 E - 03 0.375 1.26848 E - 03 1.27497 E - 03 8.0634 E - 03 0.500 1.94127 E - 03 2.01243 E - 03 2.3712 E - 04 0.625 2.66168 E - 03 2.82528 E - 03 2.8084 E - 02 0.750 3.43505 E - 03 3.70353 E - 03 3.1612 E - 02 0.875 4.25649 E - 03 4.63810 E - 03 3.3599 E - 02Example 6 (fractional case and solution is unknown).
We consider the following problem:(57)Dx1.50fx=14∫0xx-tftdt+12∫01xtftdt+gxwith the initial conditions f(1)(0)=f(0)=0 and g(x)=(1/60)x6-(1/48)x4+(1/24)x3-(3/8)x2-(128/5π)x5/2-(13/24)x+4(x/π). The solution is unknown. Using Theorem 10 which follows the concept given in [18], we calculate ϵN=fN∗(x)-fN+1∗(x)2=∫01fN∗x-fN+1∗x2dx for N=4,5,6,7,8 shown in Table 8. Table 9 shows ΔN=fN∗x-fN+1∗x, the so-called Nth level error estimate [18].
Based on the results of Table8 which is calculated using Theorem 10, one should choose N≥5 if ϵN<0.005 is to be achieved, where 0.005 is called the tolerance.Table 8
ϵ N = ∫ 0 1 f N ∗ x - f N + 1 ∗ x 2 d x of Example 6 for N=4,5,6,7,8 of Genocchi polynomials.
ϵ 4 ϵ 5 ϵ 6 ϵ 7 ϵ 8 0.295460 0.000466 0.000461 0.000337 0.000318Table 9
Δ N = f N ∗ x - f N + 1 ∗ x of Example 6 for N=4,5,6,7,8 of Genocchi polynomials.
x Δ 4 Δ 5 Δ 6 Δ 7 Δ 8 0.00 0.000000 0.000000 0.000000 0.000000 0.000000 0.10 0.027424 0.000156 0.000099 0.000047 0.000004 0.20 0.092740 0.000364 0.000091 0.000073 0.000142 0.30 0.174125 0.000393 0.000120 0.000267 0.000237 0.40 0.254577 0.000185 0.000379 0.000340 0.000223 0.50 0.321910 0.000189 0.000507 0.000290 0.000243 0.60 0.368759 0.000577 0.000435 0.000276 0.000349 0.70 0.392579 0.000786 0.000268 0.000416 0.000406 0.80 0.395643 0.000630 0.000251 0.000605 0.000322 0.90 0.385042 0.000020 0.000653 0.000503 0.000363 1.00 0.372688 0.001202 0.001572 0.000118 0.000917


## 8. Conclusion

In this paper, FIDE is solved with a numerical approximation scheme using Genocchi polynomial and Genocchi polynomial operational matrix of fractional derivative (GPOMFD). To overcome the problem that occurs for the classical formula of Genocchi coefficients in the case of nondifferentiable function, we use the matrix approach to determine the Genocchi coefficients of an arbitrary integrable function. Also, we derive the analytical expression ofT which is the integral of the product of Genocchi polynomials, KG which is the integral kernel matrix, and 0PGα which is the Genocchi polynomial operational matrix of Caputo’s fractional derivatives. The approximation of Genocchi polynomial operational matrix of fractional derivative (GPOMFD) is shown to achieve smaller error than that of Shifted Legendre polynomial operational matrix of fractional derivative (SLOMFD). This is due to the fact that Genocchi polynomials have smaller coefficients for each xn-term than the Shifted Legendre polynomial. Applying Genocchi polynomial approximation, the FIDE is transformed into a system of algebraic equations of Genocchi coefficients C of the unknown function f(x). Using collocation method of equally spaced points over the interval [0,1] together with the initial conditions given, we are able to solve FIDE with good accuracy. Comparison with the Bernoulli matrix method [8] and the Legendre wavelet method [10] indicates that Genocchi polynomial can achieve comparable results as those of the aforementioned methods which are known to have better accuracy than Legendre polynomial. Estimation errors occur when one tries to solve the FIDE in the case when the derivative 0Dxα is of fractional order.



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