# Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations

Authors: Abdulnasir Isah; Chang Phang; Piau Phang
Journal: International Journal of Differential Equations
 (2017)

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## Abstract

An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations.


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## Body

## 1. Introduction

Fractional calculus, the calculus of derivative and integral of any order, is used as a powerful tool in science and engineering to study the behaviors of real world phenomena especially the ones that cannot be fully described by the classical methods and techniques [1]. Differential equations with proportional delays are usually referred to as pantograph equations or generalized pantograph equations. The name pantograph was originated from the study work of Ockendon and Tayler [2]. Many researchers have studied different applications of these equations in applied sciences such as biology, physics, economics, and electrodynamics [3–5]. Solutions of pantograph equations were also studied by many authors numerically and analytically. Bhrawy et al. proposed a new generalized Laguerre-Gauss collocation method for numerical solution of generalized fractional pantograph equations [1]. Tohidi et al. in [6] proposed a new collocation scheme based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Yusufoglu [7] proposed an efficient algorithm for solving generalized pantograph equations with linear functional argument. In [8], Yang and Huang presented a spectral-collocation method for fractional pantograph delay integrodifferential equations and in [9] Yüzbasi and Sezer presented an exponential approximation for solutions of generalized pantograph delay differential equations. Chebyshev and Bessel polynomials are, respectively, used in [10, 11] to obtain the solutions of generalized pantograph equations. Operational matrices of fractional derivatives and integration have become very important tool in the field of numerical solution of fractional differential equations. In this paper, a member of Appell polynomials called Genocchi polynomials is used; although this polynomial is not based on orthogonal functions, it possesses operational matrices of derivatives with high accuracy. It is very important to note that this polynomial shares some great advantages with Bernoulli and Euler polynomials for approximating an arbitrary function over some classical orthogonal polynomials; we refer the reader to [6] for these advantages. On top of that, we had successfully applied the operational matrix via Genocchi polynomials for solving integer-order delay differential equations [12] and fractional optimal control problems [13], and the numerical solutions obtained are comparable or even more accurate compared to some existing well-known methods. Motivated by these advantages, in this paper, we intend to extend the result for integer-order delay differential equations in [12] to fractional delay differential equations or so-called generalized fractional pantograph equations. To the best of our knowledge, this is the first time that the operational matrix based on Genocchi polynomials is applied to solve the fractional pantograph equations. On the other hand, some other types of polynomials were employed to solve some special type of fractional calculus problems; for example, Bessel polynomials were used for the solution of fractional-order logistic population model [14]; Bernstein polynomials were also used for the solution of Riccati type differential equations [15].In this paper, we use the new operational matrix of fractional-order derivative via Genocchi polynomials to provide approximate solutions of the generalized fractional pantograph equations of the following form [1]:(1)Dαyt=∑j=0J∑n=0m-1pj,ntDβnyλj,nt+μj,n+gt,0≤t≤1subject to the following conditions: (2)∑n=0m-1an,iyn0=di,i=0,1,…,m-1,where an,i,λj,n, and μj,n are real or complex coefficients; m-1<α<m,0<β0<β1<⋯<βm-1<α, while pj,n(t) and g(t) are given continuous functions in the interval [0,1].The rest of the paper is organized as follows: Section2 introduces some mathematical preliminaries of fractional calculus. In Section 3, we discuss some important properties of Genocchi polynomials. In Section 4, we derive the Genocchi delay operational matrix and we apply the collocation method for solving fractional pantograph equation (1) using the Genocchi operational matrix of fractional derivative and the delay operational matrix in Section 5. In Section 6, the proposed method is applied to several examples and conclusion is given in Section 7.


## 2. Preliminaries

### 2.1. Fractional Derivative and Integration

We recall some basic definitions and properties of fractional calculus that we will use. There are various competing definitions for fractional derivatives [16, 17]. The Riemann-Liouville definition played a vital role in the development of the theory of fractional calculus. However, there are certain disadvantages of using this definition when modeling real world phenomena. To cope with these disadvantages, Caputo definition was introduced which is found to be more reliable in application. So we use this definition of fractional derivatives. We begin with the definition of Riemann-Liouville integral, in which the fractional integral operator I of a function f(t) is defined as follows.Definition 1.
The Riemann-Liouville integralI of fractional-order α of f(t) is given by (3)Iαft=1Γα∫0tt-τα-1fτdτ,t>0,α∈R+,whereΓ(·) is the Gamma function. The fractional derivative of order α>0 due to Riemann-Liouville is defined by (4)Dlαft=ddtmIm-αft,α>0,m-1<α<m.The following are important properties of Riemann-Liouville fractional integral Iα: (5)IαIβft=Iα+βft,α>0,β>0,Iαtβ=Γβ+1Γβ+α+1tβ+α.Definition 2.
The Caputo fractional derivativeDα of a function f(t) is defined as (6)Dαft=1Γn-α∫0tfnτt-τα-n+1dτ,n-1<α≤n,n∈N.Some properties of Caputo fractional derivatives are as follows:(7)DαC=0,Cis  constant,Dαtβ=0,β∈N∪0,β<αΓβ+1Γβ+1-αtβ-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.Similar to the integer-order differentiation, the Caputo fractional differential operator is a linear operator; that is,(8)Dαλft+μgt=λDαft+μDαgtfor λ and μ constants.


## 2.1. Fractional Derivative and Integration

We recall some basic definitions and properties of fractional calculus that we will use. There are various competing definitions for fractional derivatives [16, 17]. The Riemann-Liouville definition played a vital role in the development of the theory of fractional calculus. However, there are certain disadvantages of using this definition when modeling real world phenomena. To cope with these disadvantages, Caputo definition was introduced which is found to be more reliable in application. So we use this definition of fractional derivatives. We begin with the definition of Riemann-Liouville integral, in which the fractional integral operator I of a function f(t) is defined as follows.Definition 1.
The Riemann-Liouville integralI of fractional-order α of f(t) is given by (3)Iαft=1Γα∫0tt-τα-1fτdτ,t>0,α∈R+,whereΓ(·) is the Gamma function. The fractional derivative of order α>0 due to Riemann-Liouville is defined by (4)Dlαft=ddtmIm-αft,α>0,m-1<α<m.The following are important properties of Riemann-Liouville fractional integral Iα: (5)IαIβft=Iα+βft,α>0,β>0,Iαtβ=Γβ+1Γβ+α+1tβ+α.Definition 2.
The Caputo fractional derivativeDα of a function f(t) is defined as (6)Dαft=1Γn-α∫0tfnτt-τα-n+1dτ,n-1<α≤n,n∈N.Some properties of Caputo fractional derivatives are as follows:(7)DαC=0,Cis  constant,Dαtβ=0,β∈N∪0,β<αΓβ+1Γβ+1-αtβ-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.Similar to the integer-order differentiation, the Caputo fractional differential operator is a linear operator; that is,(8)Dαλft+μgt=λDαft+μDαgtfor λ and μ constants.


## 3. Genocchi Polynomials and Some Properties

Genocchi polynomials and numbers have been extensively studied in many different contexts in branches of mathematics such as elementary number theory, complex analytic number theory, homotopy theory (stable homotopy groups of spheres), differential topology (differential structures on spheres), theory of modular forms (Eisenstein series), and quantum physics (quantum groups). The classical Genocchi polynomialGn(x) is usually defined by means of the exponential generating functions [18–20].(9)2textet+1=∑n=0∞Gnxtnn!,t<π,where Gn(x) is the Genocchi polynomial of degree n and is given by (10)Gnx=∑k=0nnkGn-kxk.Gn-k here is the Genocchi number.Some of the important properties of these polynomials include(11)∫01GnxGmxdx=2-1nn!m!m+n!Gm+nn,m≥1,(12)dGnxdx=nGn-1x,n≥1,(13)Gn1+Gn0=0,n>1.Before we move to the next level, we need the following linear independence on which the rest of theoretical results are based.Lemma 3.
The setA=G1t,G2t,…,GNt⊂L2[0,1] is a linearly independent set in L2[0,1].Proof.
To show thatA is the set of linearly independent elements of L2[0,1], it is enough to show that the Gram determinant is not zero. That is,(14)GramG1,G2,…,GN≠0,where(15)GramG1,G2,…,GN=G1,G1G1,G2⋯G1,GNG2,G1G2,G2⋯G2,GN⋮⋮⋯⋮Gn,G1Gn,G2⋯Gn,GN.Now, to prove that this determinant is not equal to zero, we first reduce the Gram matrix to an upper triangular matrix by Gaussian elimination and it is not difficult to see that the elements of the diagonal of the reduced matrix are given by (16)an=n!n+1!22n!2n+1!,n∈N.Clearly, one can see that, for any ∈N, a(n)≠0. Consequently, the determinant given by (17)∏n=1Nanis not equal to zero. Therefore, the set A is the set of linearly independent sets.


### 3.1. Function Approximation

Assume that{G1(t),G2(t),…,GN(t)}⊂L2[0,1] is the set of Genocchi polynomials and Y=Span⁡{G1(t),G2(t),…,GN(t)}. Let f(t) be arbitrary element of L2[0,1]; since Y is a finite dimensional subspace of L2[0,1] space, f(t) has a unique best approximation in Y, say f∗(t), such that(18)ft-f∗t2≤ft-yt2,∀yt∈Y.This implies that,∀y(t)∈Y, (19)ft-f∗t,yt=0,where 〈·〉 denotes inner product. Since f∗t∈Y, there exist the unique coefficients c1,c2,…,cN such that (20)ft≈f∗t=∑n=1NcnGnt=CTGt,where C=[c1,c2,…,cN]T,G(t)=G1t,G2t,…,GNtT.Using (19), we have (21)ft-CTGt,Git=0i=1,2,…,N;for simplicity, we write (22)CTGt,Gt=ft,Gt,where 〈G(t),G(t)〉 is an N×N matrix.LetW=Gt,Gt=∫01G(t)GT(t)dt.The entries of the matrixW can be calculated from (11). Therefore, any function f(t)∈L2[0,1] can be expanded by Genocchi polynomials as f(t)=CTG(t), where (23)C=W-1ft,Gt.


## 3.1. Function Approximation

Assume that{G1(t),G2(t),…,GN(t)}⊂L2[0,1] is the set of Genocchi polynomials and Y=Span⁡{G1(t),G2(t),…,GN(t)}. Let f(t) be arbitrary element of L2[0,1]; since Y is a finite dimensional subspace of L2[0,1] space, f(t) has a unique best approximation in Y, say f∗(t), such that(18)ft-f∗t2≤ft-yt2,∀yt∈Y.This implies that,∀y(t)∈Y, (19)ft-f∗t,yt=0,where 〈·〉 denotes inner product. Since f∗t∈Y, there exist the unique coefficients c1,c2,…,cN such that (20)ft≈f∗t=∑n=1NcnGnt=CTGt,where C=[c1,c2,…,cN]T,G(t)=G1t,G2t,…,GNtT.Using (19), we have (21)ft-CTGt,Git=0i=1,2,…,N;for simplicity, we write (22)CTGt,Gt=ft,Gt,where 〈G(t),G(t)〉 is an N×N matrix.LetW=Gt,Gt=∫01G(t)GT(t)dt.The entries of the matrixW can be calculated from (11). Therefore, any function f(t)∈L2[0,1] can be expanded by Genocchi polynomials as f(t)=CTG(t), where (23)C=W-1ft,Gt.


## 4. Genocchi Operational Matrix

In this section, we derive the operational matrices for the delay and that of fractional derivative based on Genocchi polynomials for the solution of fractional pantograph equations.


### 4.1. Genocchi Delay Operational Matrix

The Genocchi delay vectorGt-μ can be expressed as (24)Gt-μ=RGt,whereR is the N×N operational delay matrix given by (25)R=W1W-1=10000⋯0-2μ1000⋯03μ2-3μ100⋯0-4μ36μ2-4μ10⋯05μ4-10μ310μ2-5μ1⋯0⋮⋮⋮⋯⋮⋮⋮bn1bn2bn3bn4bn5⋯1,where W1=∫01Gt-μGT(t)dt and bn(i)=(-1)n-iniμn-i,i=1,2,…,n.Also, for any delay functionf(t-μ), we can express it in terms of Genocchi polynomials as shown in (26): (26)ft-μ=∑i=1NciGit-μ=CTRGt,where C is given in (23).The following lemma is also of great importance.Lemma 4.
LetGi(t) be the Genocchi polynomials; then DαGi(t)=0, for i=1,...,α-1,α>0.The proof of this lemma is obvious; one can use (7) and (8) on (10).


### 4.2. Genocchi Operational Matrix of Fractional Derivative

If we consider the Genocchi vectorG(t) given by G(t)=[G1(t),G2(t),…,GN(t)], then the derivative of G(t) with the aid of (12) can be expressed in the matrix form by (27)dGtTdt=MGtT,where (28)M=000⋯000200⋯000030⋯000004⋯000⋮⋮⋮⋯⋮⋮⋮000⋯N-100000⋯0N0.Thus, M is N×N operational matrix of derivative.It is not difficult to show inductively that thekth derivative of G(t) can be given by (29)dkGtTdtk=GtMTk.In the following theorem, the operational matrix of fractional-order derivative for the Genocchi polynomials is given.Theorem 5 (see [21]).
Suppose thatG(t) is the Genocchi vector given in (20) and let α>0. Then, (30)DαGtT=PαGtT,where Pα is N×N operational matrix of fractional derivative of order α in Caputo sense and is defined as follows:(31)Pα=00⋯0⋮⋮⋯⋮00⋯0∑k=ααρα,k,1∑k=ααρα,k,2⋯∑k=ααρα,k,N⋮⋮⋯⋮∑k=αiρi,k,1∑k=αiρi,k,2⋯∑k=αiρi,k,N⋮⋮⋯⋮∑k=αNρN,k,1∑k=αNρN,k,2⋯∑k=αNρN,k,N,where ρi,k,j is given by (32)ρi,k,j=i!Gi-ki-k!Γk+1-αcj.G
i
-
k is the Genocchi number and cj can be obtained from (23).Proof.
For the proof, see [21].


### 4.3. Upper Bound of the Error for the Operational Matrix of Fractional DerivativePα

We begin here by proving the upper bound of the error of arbitrary function approximation by Genocchi polynomials in the following Lemma.Lemma 6.
Suppose thatf(t)∈Cn+1[0,1] and Y=Span{G1(t),G2(t),…,GN(t)}; if CTG(t) is the best approximation of f(t) out of Y, then (33)ft-CTGt≤Rn+1!2n+3,where R=maxt∈0,1⁡fn+1t.
To see this, we set1,t,…,tn as a basis for the polynomial space of degree n.
Definey1(t)=f(0)+tf′(0)+t2/2!f′′(0)+⋯+(tn/n!)f(n)(0).
From Taylor’s expansion, one hasft-y1t=tn+1/n+1!fn+1ξt, where ξt∈(0,1).
SinceCTG(t) is the best approximation of f(t) out of Y and y1(t)∈Y, from (18), one has(34)ft-CTGt22≤ft-y1t22≤∫01ft-y1t2dt=∫01tn+1n+1!2fn+1ξt2dt≤R2n+1!2∫01t2n+2dt=R2n+1!22n+3.
Taking the square root of both sides, one has(35)ft-CTGt≤Rn+1!2n+3which is the desired error bound.We use the following theorem from [22].Theorem 7 (see [22]).
Suppose thatH is a Hilbert space and Y is a closed subspace of H such that dim⁡Y<∞ and y1,y2,…,yn is a basis for Y. Let f be an arbitrary element in H and let y0 be the unique best approximation of f out of Y. Then, (36)f-y02=Gramf,y1,y2,…,ynGramy1,y2,…,yn,where(37)Gramy1,y2,…,yn=y1,y1y1,y2⋯y1,yny2,y1y2,y2⋯y2,yn⋮⋮⋯⋮yn,y1yn,y2⋯yn,yn.Theorem 8.
Suppose thatf(t)∈L2[0,1] is approximated by fn(t) as (38)fnt=∑i=1nciGit=CTGt;then, (39)limn→∞⁡ft-fnt=0.The proof of this theorem obviously follows from Lemma6.The operational matrix error vectorEα is given by (40)Eα=PαGt-DαGt,where(41)Eα=e1αe2α⋮enα;from Theorem 7, we get (42)f0t-∑j=1NcjGjt=Gramft,G1t,…,GNtGramG1t,…,GNt1/2.Thus, according to equations (29) and (30) in [21], one has (43)eiα=DαGit-∑j=1N∑k=αii!Gi-ki-k!Γk+1-αcjGjt≤∑k=αii!Gi-ki-k!Γk+1-αtk-αf0t-∑j=1NcjGjt≤∑k=αii!Gi-ki-k!Γk+1-αtk-αGramft,G1t,…,GNtGramG1t,…,GNt1/2.By considering Theorem8 and (43), we can conclude that by increasing the number of the Genocchi bases the vector eiα tends to zero.For comparison purpose in Table1, we show below the errors of operational matrix of fractional derivative based on Genocchi polynomials and shifted Legendre polynomials derived in [23, 24] when N=10 and α=0.75 at different points on [0,1]. From this table, it is clear that the accuracy of Genocchi polynomials operational matrix of fractional derivative (GPOMFD) is better than the shifted Legendre polynomials operational matrix of fractional derivatives (SLPOMFD). We believe that this is the case for any value of N because the Genocchi polynomials have smaller coefficients of individual terms compared to shifted Legendre polynomials.Table 1
Comparison of the operational matrix errors for the GPOMFD and SLPOMFD.
Eα
x
=
1
x
=
0
x=0.5
GPOMFD
SLPOMFD
GPOMFD
SLPOMFD
GPOMFD
SLPOMFD
e1
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
e2
0.01288
0.01343
0.51657
0.51648
0.00353
0.00345
e3
0.02204
0.03462
0.77767
1.56518
0.00531
0.01047
e4
0.00004
0.02916
0.01241
3.17608
0.00008
0.02487
e5
0.03944
0.17666
1.28048
5.36557
0.00911
0.02570
e6
0.00292
0.33223
0.02450
8.00571
0.00001
0.05647
e7
0.15747
2.61748
5.38767
15.06874
0.03839
0.04226
e8
0.00748
11.8527
0.15766
11.80719
0.00060
0.92194
e9
1.21090
38.04476
39.19581
54.74287
0.27897
15.64398
e
10
0.04902
806.14232
1.32181
629.08480
0.00519
20.30772


## 4.1. Genocchi Delay Operational Matrix

The Genocchi delay vectorGt-μ can be expressed as (24)Gt-μ=RGt,whereR is the N×N operational delay matrix given by (25)R=W1W-1=10000⋯0-2μ1000⋯03μ2-3μ100⋯0-4μ36μ2-4μ10⋯05μ4-10μ310μ2-5μ1⋯0⋮⋮⋮⋯⋮⋮⋮bn1bn2bn3bn4bn5⋯1,where W1=∫01Gt-μGT(t)dt and bn(i)=(-1)n-iniμn-i,i=1,2,…,n.Also, for any delay functionf(t-μ), we can express it in terms of Genocchi polynomials as shown in (26): (26)ft-μ=∑i=1NciGit-μ=CTRGt,where C is given in (23).The following lemma is also of great importance.Lemma 4.
LetGi(t) be the Genocchi polynomials; then DαGi(t)=0, for i=1,...,α-1,α>0.The proof of this lemma is obvious; one can use (7) and (8) on (10).


## 4.2. Genocchi Operational Matrix of Fractional Derivative

If we consider the Genocchi vectorG(t) given by G(t)=[G1(t),G2(t),…,GN(t)], then the derivative of G(t) with the aid of (12) can be expressed in the matrix form by (27)dGtTdt=MGtT,where (28)M=000⋯000200⋯000030⋯000004⋯000⋮⋮⋮⋯⋮⋮⋮000⋯N-100000⋯0N0.Thus, M is N×N operational matrix of derivative.It is not difficult to show inductively that thekth derivative of G(t) can be given by (29)dkGtTdtk=GtMTk.In the following theorem, the operational matrix of fractional-order derivative for the Genocchi polynomials is given.Theorem 5 (see [21]).
Suppose thatG(t) is the Genocchi vector given in (20) and let α>0. Then, (30)DαGtT=PαGtT,where Pα is N×N operational matrix of fractional derivative of order α in Caputo sense and is defined as follows:(31)Pα=00⋯0⋮⋮⋯⋮00⋯0∑k=ααρα,k,1∑k=ααρα,k,2⋯∑k=ααρα,k,N⋮⋮⋯⋮∑k=αiρi,k,1∑k=αiρi,k,2⋯∑k=αiρi,k,N⋮⋮⋯⋮∑k=αNρN,k,1∑k=αNρN,k,2⋯∑k=αNρN,k,N,where ρi,k,j is given by (32)ρi,k,j=i!Gi-ki-k!Γk+1-αcj.G
i
-
k is the Genocchi number and cj can be obtained from (23).Proof.
For the proof, see [21].


## 4.3. Upper Bound of the Error for the Operational Matrix of Fractional DerivativePα

We begin here by proving the upper bound of the error of arbitrary function approximation by Genocchi polynomials in the following Lemma.Lemma 6.
Suppose thatf(t)∈Cn+1[0,1] and Y=Span{G1(t),G2(t),…,GN(t)}; if CTG(t) is the best approximation of f(t) out of Y, then (33)ft-CTGt≤Rn+1!2n+3,where R=maxt∈0,1⁡fn+1t.
To see this, we set1,t,…,tn as a basis for the polynomial space of degree n.
Definey1(t)=f(0)+tf′(0)+t2/2!f′′(0)+⋯+(tn/n!)f(n)(0).
From Taylor’s expansion, one hasft-y1t=tn+1/n+1!fn+1ξt, where ξt∈(0,1).
SinceCTG(t) is the best approximation of f(t) out of Y and y1(t)∈Y, from (18), one has(34)ft-CTGt22≤ft-y1t22≤∫01ft-y1t2dt=∫01tn+1n+1!2fn+1ξt2dt≤R2n+1!2∫01t2n+2dt=R2n+1!22n+3.
Taking the square root of both sides, one has(35)ft-CTGt≤Rn+1!2n+3which is the desired error bound.We use the following theorem from [22].Theorem 7 (see [22]).
Suppose thatH is a Hilbert space and Y is a closed subspace of H such that dim⁡Y<∞ and y1,y2,…,yn is a basis for Y. Let f be an arbitrary element in H and let y0 be the unique best approximation of f out of Y. Then, (36)f-y02=Gramf,y1,y2,…,ynGramy1,y2,…,yn,where(37)Gramy1,y2,…,yn=y1,y1y1,y2⋯y1,yny2,y1y2,y2⋯y2,yn⋮⋮⋯⋮yn,y1yn,y2⋯yn,yn.Theorem 8.
Suppose thatf(t)∈L2[0,1] is approximated by fn(t) as (38)fnt=∑i=1nciGit=CTGt;then, (39)limn→∞⁡ft-fnt=0.The proof of this theorem obviously follows from Lemma6.The operational matrix error vectorEα is given by (40)Eα=PαGt-DαGt,where(41)Eα=e1αe2α⋮enα;from Theorem 7, we get (42)f0t-∑j=1NcjGjt=Gramft,G1t,…,GNtGramG1t,…,GNt1/2.Thus, according to equations (29) and (30) in [21], one has (43)eiα=DαGit-∑j=1N∑k=αii!Gi-ki-k!Γk+1-αcjGjt≤∑k=αii!Gi-ki-k!Γk+1-αtk-αf0t-∑j=1NcjGjt≤∑k=αii!Gi-ki-k!Γk+1-αtk-αGramft,G1t,…,GNtGramG1t,…,GNt1/2.By considering Theorem8 and (43), we can conclude that by increasing the number of the Genocchi bases the vector eiα tends to zero.For comparison purpose in Table1, we show below the errors of operational matrix of fractional derivative based on Genocchi polynomials and shifted Legendre polynomials derived in [23, 24] when N=10 and α=0.75 at different points on [0,1]. From this table, it is clear that the accuracy of Genocchi polynomials operational matrix of fractional derivative (GPOMFD) is better than the shifted Legendre polynomials operational matrix of fractional derivatives (SLPOMFD). We believe that this is the case for any value of N because the Genocchi polynomials have smaller coefficients of individual terms compared to shifted Legendre polynomials.Table 1
Comparison of the operational matrix errors for the GPOMFD and SLPOMFD.
Eα
x
=
1
x
=
0
x=0.5
GPOMFD
SLPOMFD
GPOMFD
SLPOMFD
GPOMFD
SLPOMFD
e1
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
e2
0.01288
0.01343
0.51657
0.51648
0.00353
0.00345
e3
0.02204
0.03462
0.77767
1.56518
0.00531
0.01047
e4
0.00004
0.02916
0.01241
3.17608
0.00008
0.02487
e5
0.03944
0.17666
1.28048
5.36557
0.00911
0.02570
e6
0.00292
0.33223
0.02450
8.00571
0.00001
0.05647
e7
0.15747
2.61748
5.38767
15.06874
0.03839
0.04226
e8
0.00748
11.8527
0.15766
11.80719
0.00060
0.92194
e9
1.21090
38.04476
39.19581
54.74287
0.27897
15.64398
e
10
0.04902
806.14232
1.32181
629.08480
0.00519
20.30772


## 5. Collocation Method Based on Genocchi Operational Matrices

In this section, we use the collocation method based on Genocchi operational matrix of fractional derivatives and Genocchi delay operational matrix to solve numerically the generalized fractional pantograph equation. We now derive an algorithm for solving (1). To do this, let the solution of (1) be approximated by the first N terms Genocchi polynomials. Thus, we write (44)yNt≈∑n=1NcnGnt=GtC,where the Genocchi coefficient vector C and the Genocchi vector G(t) are given by (45)CT=c1,c2,…,cN,Gt=G1t,G2t,…,GNt;thus, DαyN(t) and DβnyN(t),n=0,1,…,m-1, can be expressed, respectively, as follows: (46)DαyNt=GtPTαC,DβnyNt=GtPTβnC,n=0,1,…,m-1.Substituting (44) and (46) in (1), we have (47)GtPTαC=∑j=0J∑n=0m-1pj,ntGλj,nt+μj,nPTβnC+gt,where G(λj,nt+μj,n)=[G1(λj,nt+μj,n),G2(λj,nt+μj,n),…,GN(λj,nt+μj,n)].Also the initial condition will producem other equations: (48)∑n=0m-1an,iG0PTiC=di,i=0,1,…,m-1.To find the solution yN(t) we collocate (47) at the collocation points tj=j/(N-m), j=1,2,…,N-m, to obtain (49)GtjPTαC=∑j=0J∑n=0m-1pj,ntjGλj,ntj+μj,nPTβnC+gtjfor j=1,2,…,N-m. Additionally, one can also use both the operational matrix of fractional derivative and delay operational matrix to solve problem (1). According to (44), we can approximate the delay function y(λj,ntj+μj,n) and its fractional derivative using the operational matrices P and R as follows: (50)yNλj,ntj+μj,n=RCTGt,DβnyNλj,ntj+μj,n=RCTPβnGt.Putting this approximation together with (44) in (1), we have(51)GtPTαC=∑j=0J∑n=0m-1pj,ntRCTPβnGt+gt.Thus, collocating (51) at the same collocation point as that in (47), we get (52)GtjPTαC=∑j=0J∑n=0m-1pj,ntjRCTPβnGtj+gtj.Hence, (49) or (52) is N-m nonlinear algebraic equation. Any of these equations together with (48) makes N algebraic equations which can be solved using Newton’s iterative method. Consequently, yN(x) given in (44) can be calculated.


## 6. Numerical Examples

In this section, some numerical examples are given to illustrate the applicability and accuracy of the proposed method. All the numerical computations have been done using Maple 18.Example 1.
Consider the following example solved in [25]: (53)Dαyt=2Γ3-αy1-α/2t+yt-τ-yt+2τyt-τ2subject to (54)yt=0,t≤0.The exact solution for this example is given byy(t)=t2. We solve the example when α=0.6 and τ=0.3. In Table 2, we compare the errors obtained by our method with those obtained using FAM and new approach in [25]. As reported in [25], the time required for the new method is 104.343750 seconds and for the FAM the time taken is 215.031250 seconds for completing the same task, whereas in our method we only need 38.080 seconds to complete the computations.Table 2
Comparison errors obtained by the present method and those obtained in [25] when α=0.6 and τ=0.3 for Example 1.
t
Error (new method) [25]
Error (FAM) [25]
Error (present method)
0.2
0.0781197
0.078155
3.37154E-03
0.4
0.129928
0.129978
6.52102E-03
0.6
0.190687
0.19076
9.77309E-03
0.8
0.248601
0.248694
1.30349E-02
1.0
0.307649
0.307763
1.64103E-02Example 2 (see [1]).
Consider the following generalized fractional pantograph equation:(55)D5/2yt=-yt-yt-0.5+gt,t∈0,1subject to (56)y0=0,y′0=0,y′′0=0,where (57)gt=Γ4Γ3/2t1/2+t3+t-0.53.The exact solution of this problem is known to bey(t)=t3. This problem is solved in [1] using generalized Laguerre-Gauss collocation scheme. We apply our technique with N=4. Approximating (55) with Genocchi polynomials, we have (58)GtPT5/2C=-GtC+Gt-0.5C+gt.Also from the initial conditions we have (59)G0C=0,G0PTC=0,G0PT2C=0.Thus, collocating (58) at t=0.267339, we get (60)-3.507078326+15.27479180c4+2c1+0.2727646328c3-1.930640400c2=0,and (59) gives (61)c1-c2+c4=0,2c1-3c2=0,6c3-12c4=0.Solving these equations, we have (62)c1=0.4999969148,c2=0.7499953722,c3=0.4999969148,c4=0.2499984574.Thus,y(t)=G(x)C is calculated and we have 0.9999938296t3 which is almost the exact solution. In Table 3, we compare the absolute errors obtained by our method (with only few terms N=4) and the absolute errors obtained in [1] when N=22 with different Laguerre parameters β.Table 3
Comparison of the absolute errors obtained by the present method and those obtained in [1] for Example 2.
t
N
=
22 [1]
N=4
β=2
β=3
β=5
Present method
0.1
1.030
E
-
0
4
1.019
E
-
0
5
6.273
E
-
0
6
6.17040
E
-
09
0.2
6.510
E
-
0
4
6.051
E
-
0
5
3.892
E
-
0
5
4.93630
E
-
08
0.3
1.740
E
-
0
3
1.495
E
-
0
4
1.023
E
-
0
4
1.66600
E
-
07
0.4
3.283
E
-
0
3
2.559
E
-
0
4
1.901
E
-
0
4
3.94910
E
-
07
0.5
5.138
E
-
0
3
3.546
E
-
0
4
2.944
E
-
0
4
7.71300
E
-
07
0.6
7.175
E
-
0
3
4.261
E
-
0
4
4.088
E
-
0
4
1.33280
E
-
06
0.7
9.303
E
-
0
3
4.592
E
-
0
4
5.306
E
-
0
4
2.11640
E
-
06
0.8
1.147
E
-
0
2
4.510
E
-
0
4
6.597
E
-
0
4
3.15920
E
-
06
0.9
1.367
E
-
0
2
4.055
E
-
0
4
7.977
E
-
0
4
4.49820
E
-
06
1.0
1.589
E
-
0
2
3.311
E
-
0
4
9.468
E
-
0
4
6.17040
E
-
06Example 3.
Consider the following fractional pantograph equation:(63)D1/2yt=2y3t2+8t3/23π-9t22,t∈0,1subject to (64)y0=0,y1=1.The exact solution of this problem is known to bey(t)=t2. We solve (63) using our technique with N=3 only. As in Example 2, we obtained the values of the coefficients to be (65)c1=12,c2=12,c3=13.Thus, yN(t)=G(t)C is calculated to be t2 which is the exact solution and so there is nothing to compare for the error is zero.Example 4.
Consider the following fractional pantograph equation solved in [26]: (66)D2yt+D3/2yt+yt=yt2+3t24+4tπ+2,t∈0,1subject to (67)y0=0,y1=1.The exact solution of this problem is known to bey(t)=t2. As in Example 3, we solve (66) using our technique with N=3 and the values of the coefficients obtained are (68)c1=0.5000000011,c2=0.5000000011,c3=0.3333333384.Thus, yN(t)=G(t)C is calculated and compared with the exact solution. This problem is solved using Taylor collocation method in [26] when N=4,5, and 6. In Table 4, we compare the absolute errors obtained by present method when N=3 with the errors obtained when N=4 in [26].Table 4
Comparison of the absolute errors obtained by the present method and those in [26] for Example 4.
t
Absolute error [26]N=4
Absolute error (present method)N=3
0.1
0.100
E
-
0
7
1.15000
E
-
09
0.2
0.115
E
-
0
7
2.00000
E
-
09
0.3
0.115
E
-
0
7
2.55000
E
-
09
0.4
0.107
E
-
0
7
2.80000
E
-
09
0.5
0.967
E
-
0
8
2.70000
E
-
09
0.6
0.811
E
-
0
8
2.40000
E
-
09
0.7
0.641
E
-
0
8
1.70000
E
-
09
0.8
0.440
E
-
0
8
8.00000
E
-
10
0.9
0.223
E
-
0
8
5.00000
E
-
10
1.0
0.372
E
-
0
9
2.00000
E
-
09


## 7. Conclusion

In this paper, a collocation method based on the Genocchi delay operational matrix and the operational matrix of fractional derivative for solving generalized fractional pantograph equations is presented. The comparison of the results shows that the present method is an excellent mathematical tool for finding the numerical solutions delay equation. The advantage of the method over others is that only few terms are needed and every operational matrix involves more numbers of zeroes; as such the method has less computational complexity and provides the solution at high accuracy.



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*Source: 2097317-2017-06-13.xml*