# On a  Generalized Laguerre Operational Matrix of Fractional Integration

Authors: A. H. Bhrawy; D. Baleanu; L. M. Assas; J. A. Tenreiro Machado
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

A new operational matrix of fractional integration of arbitrary order for generalized Laguerre polynomials is derived. The fractional integration is described in the Riemann-Liouville sense. This operational matrix is applied together with generalized Laguerre  tau method for solving general linear multiterm fractional differential equations (FDEs). The  method has the advantage of obtaining the solution in terms of the generalized Laguerre parameter. In addition, only a small dimension of generalized Laguerre operational matrix is needed to obtain a satisfactory result. Illustrative examples reveal that the proposed method is very effective and convenient for linear multiterm FDEs on a semi-infinite interval.


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## Body

## 1. Introduction

The problems of FDEs arise in various areas of science and engineering. In particular, multiterm fractional differential equations have been used to model various types of viscoelastic damping (see, e.g., [1–13] and the references therein). In the last few decades both theory and numerical analysis of FDEs have received an increasing attention (see, e.g., [1–4, 14–17] and references therein).Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve some differential equations. The main idea is to write the solution of the differential equation as a sum of certain orthogonal polynomial and then obtain the coefficients in the sum in order to satisfy the differential equation. Due to high-order accuracy, spectral methods have gained increasing popularity for several decades, particularly in the field of computational fluid dynamics (see, e.g., [18–24] and the references therein).The usual spectral methods are only available for bounded domains for solving FDEs; see [25–28]. However, it is also interesting to consider spectral methods for FDEs on the half line. Several authors developed the generalized Laguerre spectral method for the half line for ordinary, partial, and delay differential equations; see [29–31]. Recently, Saadatmandi and Dehghan [25] have proposed an operational Legendre-tau technique for the numerical solution of multiterm FDEs. The same technique based on operational matrix of Chebyshev polynomials has been used for the same problem (see [32]). In [33], Doha et al. derived the Jacobi operational matrix of fractional derivatives which applied together with spectral tau method for numerical solution of general linear multiterm fractional differential equations. Bhrawy et al. [27] used a quadrature shifted Legendre-tau method for treating multiterm linear FDEs with variable coefficients. More recently, Bhrawy and Alofi [34] proposed the operational Chebyshev matrix of fractional integration in the Riemann-Liouville sense which was applied together with spectral tau method for solving linear FDEs.The operational matrix of integer integration has been determined for several types of orthogonal polynomials, such as Chebyshev polynomials [35], Legendre polynomials [36], and Laguerre and Hermite [37]. Recently, Singh et al. [38] derived the Bernstein operational matrix of integration. Till now, and to the best of our knowledge, most of formulae corresponding to those mentioned previously are unknown and are traceless in the literature for fractional integration for generalized Laguerre polynomials in the Riemann-Liouville sense. This partially motivates our interest in operational matrix of fractional integration for generalized Laguerre polynomials. Another motivation is concerned with the direct solution techniques for solving the integrated forms of FDEs on the half line using generalized Laguerre tau method based on operational matrix of fractional integration in the Riemann-Liouville sense. Finally, the accuracy of the proposed algorithm is demonstrated by test problems.The paper is organized as follows. In the next section, we introduce some necessary definitions. In Section3 the generalized Laguerre operational matrix of fractional integration is derived. In Section 4 we develop the generalized Laguerre operational matrix of fractional integration for solving linear multiorder FDEs. In Section 5 the proposed method is applied to two examples.


## 2. Some Basic Preliminaries

The most used definition of fractional integration is due to Riemann-Liouville, which is defined as(1)Jνf(x)=1Γ(ν)∫0x‍(x-t)ν-1f(t)dt,ν>0,x>0,andJ0f(x)=f(x).The operatorJν has the property:
(2)Jνxβ=Γ(β+1)Γ(β+1+ν)xβ+ν.The next equation defines the Riemann-Liouville fractional derivative of orderν:
(3)Dνf(x)=dmdxm(Jm-νf(x)),
where m-1<ν≤m, m∈N, and m is the smallest integer greater than ν.Ifm-1<ν≤m, m∈N, then
(4)DνJνf(x)=f(x),JνDνf(x)=f(x)-∑i=0m-1‍f(i)(0+)xii!,x>0.Now, letΛ=(0,∞) and w(α)(x)=xαe-x be a weight function on Λ in the usual sense. Define the following:
(5)Lw(α)2(Λ)={v|vismeasurableonΛand∥v∥w(α)<∞},
equipped with the following inner product and norm:
(6)(u,v)w(α)=∫Λ‍u(x)v(x)w(α)(x)dx,∥v∥w(α)=(u,v)w(α)1/2.Next, letLi(α)(x) be the generalized Laguerre polynomials of degree i. We know from [39] that, for α>-1,
(7)Li+1(α)(x)=1i+1[(2i+α+1-x)Li(α)(x)-(i+α)Li-1(α)(x)],i=1,2,…,
where L0(α)(x)=1 and L1(α)(x)=1+α-x. The set of generalized Laguerre polynomials is the Lw(α)2(Λ)-orthogonal system, namely,
(8)∫0∞‍Lj(α)(x)Lk(α)(x)w(α)(x)dx=hkδjk,
where δjk is the Kronecher function and hk=(Γ(i+α+1))/i!.The generalized Laguerre polynomials of degreei, on the interval Λ, are given by
(9)Li(α)(x)=∑k=0i‍(-1)kΓ(i+α+1)Γ(k+α+1)(i-k)!k!xk,i=0,1,….The special value(10)DqLi(α)(0)=(-1)q∑j=0i-q‍(i-j-1)!(q-1)!(i-j-q)!Lj(α)(0),i≥q,
where Lj(α)(0)=(Γ(j+α+1))/(Γ(α+1)j!), will be of important use later.A functionu(x)∈Lw(α)2(Λ) may be expressed in terms of generalized Laguerre polynomials as
(11)u(x)=∑j=0∞‍ajLj(α)(x),aj=1hk∫0∞‍u(x)Lj(α)(x)w(α)(x)dx,j=0,1,2,….In practice, only the first(N+1) terms of generalized Laguerre polynomials are considered. Then we have
(12)uN(x)=∑j=0N‍ajLj(α)(x)=CTϕ(x),
where the generalized Laguerre coefficient vector C and the generalized Laguerre vector ϕ(x) are given by
(13)CT=[c0,c1,…,cN],ϕ(x)=[L0(α)(x),L1(α)(x),…,LN(α)(x)]T.If we define theq times repeated integration of generalized Laguerre vector ϕ(x) by Jqϕ(x), then (cf. Paraskevopoulos [36])
(14)Jqϕ(x)≃P(q)ϕ(x),
where q is an integer value and P(q) is the operational matrix of integration of ϕ(x). For more details see [36].


## 3. Generalized Laguerre Operational Matrix of Fractional Integration

The main objective of this section is to derive an operational matrix of fractional integration for generalized Laguerre vector.Theorem 1.
Letϕ(x) be the generalized Laguerre vector and ν>0, then
(15)Jνϕ(x)≃P(ν)ϕ(x),
where P(ν) is the (N+1)×(N+1) operational matrix of fractional integration of order ν in the Riemann-Liouville sense and is defined as follows:
(16)P(ν)=(Θν(0,0)Ων(0,1)Θν(0,2)⋯Θν(0,N)Θν(1,0)Θν(1,1)Θν(1,2)⋯Θν(1,N)⋮⋮⋮⋯⋮Θν(i,0)Θν(i,1)Θν(i,2)⋯Θν(i,N)⋮⋮⋮⋯⋮Θν(N,0)Θν(N,1)Θν(N,2)⋯Θν(N,N)),
where
(17)Θν(i,j)=∑k=0i‍∑r=0j‍(-1)k+rj!Γ(i+α+1)Γ(k+ν+α+r+1)(i-k)!(j-r)!r!Γ(k+ν+1)Γ(k+α+1)Γ(α+r+1).Proof.
Using the analytic form of the generalized Laguerre polynomialsLi(α)(x) of degree i (9) and (2), then
(18)JνLi(α)(x)=∑k=0i‍(-1)kΓ(i+α+1)(i-k)!k!Γ(k+α+1)Jνxk=∑k=0i‍(-1)kΓ(i+α+1)(i-k)!Γ(k+ν+1)Γ(k+α+1)×xk+ν,i=0,1,⋯,N.
Now, approximatexk+ν by N+1 terms of generalized Laguerre series, we have
(19)xk+ν=∑j=0N‍cjLj(α)(x),
where cj is given from (11) with u(x)=xk+ν; that is,
(20)cj=∑r=0j‍(-1)rj!Γ(k+ν+α+r+1)(j-r)!r!Γ(r+α+1),j=1,2,…,N.
In virtue of (18) and (19), we get
(21)JνLi(α)(x)=∑j=0N‍Θν(i,j)Lj(α)(x),i=0,1,…,N,
where
(22)Θν(i,j)=∑k=0i‍∑r=0j‍(-1)k+rj!Γ(i+α+1)Γ(k+ν+α+r+1)(i-k)!(j-r)!r!Γ(k+ν+1)Γ(k+α+1)Γ(α+r+1)j=1,2,…N.
Accordingly, (21) can be written in a vector form as follows:
(23)JνLi(x)≃[Θν(i,0),Θν(i,1),Θν(i,2),⋯,Θν(i,N)]ϕ(x),i=0,1,…,N.
Equation (23) leads to the desired result.


## 4. Generalized Laguerre Tau Method Based on Operational Matrix

In this section, the generalized Laguerre tau method based on operational matrix is proposed to numerically solve FDEs. In order to show the fundamental importance of generalized Laguerre operational matrix of fractional integration, we adopt it for solving the following multiorder FDE:(24)Dνu(x)=∑i=1k‍γjDβiu(x)+γk+1u(x)+f(x),inΛ=(0,∞),
with initial conditions
(25)u(i)(0)=di,i=0,…,m-1,
where γi (i=1,…,k+1) are real constant coefficients, m-1<ν≤m, 0<β1<β2<…<βk<ν, and g(x) is a given source function.The proposed technique, based on the FDE (24), is converted to a fully integrated form via fractional integration in the Riemann-Liouville sense. Subsequently, the integrated form equations are approximated by representing them as linear combinations of generalized Laguerre polynomials. Finally, the integrated form equation is converted to an algebraic equation by introducing the operational matrix of fractional integration of the generalized Laguerre polynomials.If we apply the Riemann-Liouville integral of orderν on (24), after making use of (4), we get the integrated form of (24), namely,
(26)u(x)-∑j=0m-1‍u(j)(0+)xjj!=∑i=1k‍γiJν-βi[u(x)-∑j=0mi-1‍u(j)(0+)xjj!]+γk+1Jνu(x)+Jνf(x),u(i)(0)=di,i=0,…,m-1,
where mi-1<βi≤mi, mi∈N, implies that
(27)u(x)=∑i=1k‍γiJν-βiu(x)+γk+1Jνu(x)+g(x),u(i)(0)=di,i=0,…,m-1,
where
(28)g(x)=Jνf(x)+∑j=0m-1‍djxjj!+∑i=1k‍γiJν-βi(∑j=0mi-1‍djxjj!).In order to use the tau method with Laquerre operational matrix for solving the fully integrated problem (27) with initial conditions (25), we approximate u(x) and g(x) by the Laguerre polynomials:
(29)uN(x)≃∑i=0N‍ciLi(α)(x)=CTϕ(x),(30)g(x)≃∑i=0N‍giLi(α)(x)=GTϕ(x),
where the vector G=[g0,…,gN]T is given but C=[c0,…,cN]T is an unknown vector.After making use of Theorem1 (relation (15)) the Riemann-Liouville integral of orders ν and (ν-βj) of the approximate solution (29) can be written as
(31)JνuN(x)≃CTJνϕ(x)≃CTP(ν)ϕ(x),(32)Jν-βjuN(x)≃CTJν-βjϕ(x)≃CTP(ν-βj)ϕ(x),j=1,…,k,
respectively, where P(ν) is the (N+1)×(N+1) operational matrix of fractional integration of order ν. Employing (29)-(32) the residual RN(x) for (27) can be written as
(33)RN(x)=(CT-CT∑j=1k‍γjP(ν-βj)-γk+1CTP(ν)-GT)ϕ(x).As in a typical tau method, we generateN-m+1 linear algebraic equations by applying
(34)〈RN(x),Lj(α)(x)〉=∫0∞‍RN(x)w(α)(x)Lj(α)(x)dx=0,j=0,1,…,N-m.Also by substituting Eqs. (11) and (29) in Eq (25), we get
(35)u(i)(0)=CTD(i)ϕ(0)=di,i=0,1,…,m-1.
Equations (34) and (35) generate N-m+1 and m set of linear equations, respectively.These linear equations can be solved for unknown coefficients of the vectorC. Consequently, uN(x) given in (29) can be calculated, which leads to the solution of (24) with the initial conditions (25).


## 5. Illustrative Examples

To illustrate the effectiveness of the proposed method in the present paper, two test examples are carried out in this section. The results obtained by the present methods reveal that the present method is very effective and convenient for linear FDEs on the half line.Example 2.
Consider the FDE
(36)
D
2
u
(
x
)
+
D
1
/
2
u
(
x
)
+
u
(
x
)
=
x
2
+
2
+
2.6666666667
Γ
(
0.5
)
x
1.5
,
u
(
0
)
=
0
,
u
′
(
0
)
=
0
,
x
∈
Λ
,
whose exact solution is given by  u(x)=x2.If we apply the technique described in Section4 with N=2, then the approximate solution can be written as
(37)uN(x)=∑i=02‍ciLi(α)(x)=CTϕ(x),P(2)=(Θ2(0,0)Θ2(0,1)Θ2(0,2)Θ2(1,0)Θ2(1,1)Θ2(1,2)Θ2(2,0)Θ2(2,1)Θ2(2,2)),P(3/2)=(Θ3/2(0,0)Θ3/2(0,1)Θ3/2(0,2)Θ3/2(1,0)Θ3/2(1,1)Θ3/2(1,2)Θ3/2(2,0)Θ3/2(2,1)Θ3/2(2,2)),G=(g0g1g2).Using (34) we obtain
(38)(Θ3/2(0,2)+Θ2(0,2))c0+(Θ3/2(1,2)+Θ2(1,2))c1+(1+Θ3/2(2,2)+Θ2(2,2))c2+g2=0.Now, by applying (35), we have
(39)c0+(α+1)c1+(α+1)(α+2)2c2=0.(40)-c1-(α+2)c2=0.Finally by solving (38)–(40), we have the 3 unknown coefficients with various choices of α given in Table 1. Then, we get
(41)c0=α2+3α+2,c1=-2α-4,c2=2.Table 1
c
0, c1, and c2, for different values of α for Example 2.
α
c
0
c
1
c
2
-
0.5
0.75
-
3
2
0
2
-
4
2
0.5
3.75
-
5
2
1
6
-
6
2
2
12
-
8
2
3
20
-
10
2Thus we can write(42)u(x)=(c0,c1,c2)(L0(α)(x)L1(α)(x)L2(α)(x))=x2,
which is the exact solution.Example 3.
As the first example, we consider the following fractional initial value problem:(43)D3/2u(x)+3u(x)=3x3+8Γ(0.5)x1.5,u(0)=0,u′(0)=0,x∈Λ,whose exact solution is given byu(x)=x3.
If we apply the technique described in Section4 with N=3, then the approximate solution can be written as
(44)uN(x)=∑i=03‍ciLi(α)(x)=CTϕ(x),P(3/2)=(Θ3/2(0,0)Θ3/2(0,1)Θ3/2(0,2)Θ3/2(0,3)Θ3/2(1,0)Θ3/2(1,1)Θ3/2(1,2)Θ3/2(1,3)Θ3/2(2,0)Θ3/2(2,1)Θ3/2(2,2)Θ3/2(2,3)Θ3/2(3,0)Θ3/2(3,1)Θ3/2(3,2)Θ3/2(3,3)),G=(g0g1g2g3).
Using (34) we obtain
(45)3Θ3/2(0,2)c0+3Θ3/2(1,2)c1+(1+3Θ3/2(2,2))c2+3Θ3/2(3,2)c3+g2=0,3Θ3/2(0,3)c0+3Θ3/2(1,3)c1+3Θ3/2(2,3)c2+(1+3Θ3/2(3,3))c3+g3=0.
Now, applying (35) we get
(46)CTϕ(0)=c0+(α+1)c1+(α+1)(α+2)2c2+(α+1)(α+2)(α+3)6c3=0,CTD(1)ϕ(0)=-c1-(α+2)c2-(α+3)(α+2)2c3=0.
By solving the linear system (45)–(49) we have the 4 unknown coefficients with various choices of α in Table 2, and we get
(47)c0=α3+6α+11α+6,c1=-3α2-15α-18,c2=6α+18,c3=-6.
Thereby we can write(48)uN(x)=∑i=03‍ciLi(α)(x)=x3.
Numerical results will not be presented since the exact solution is obtained.Table 2
c
0, c1, c2, and c3 for different values of α for Example 3.
α
c
0
c
1
c
2
c
3
-
0.5
15
/
8
-
45
/
4
15
-
6
0
6
-
18
18
-
6
0.5
105
/
8
-
105
/
4
21
-
6
1
24
-
36
24
-
6
2
60
-
60
30
-
6
3
120
-
90
36
-
6Example 4.
Consider the following equation:
(49)
D
2
u
(
x
)
-
2
D
u
(
x
)
+
D
1
/
2
u
(
x
)
+
u
(
x
)
=
x
7
+
2048
429
π
x
6.5
-
14
x
6
+
42
x
5
-
x
2
-
8
3
π
x
1.5
+
4
x
-
2
,
u
(
0
)
=
0
,
u
′
(
0
)
=
0
,
x
∈
Λ
,
whose exact solution is given by  u(x)=x7-x2.Now, we can apply the technique described in Examples2 and 3, with α=0 and N=7, then we have
(50)c0=5038,c1=-35276,c2=105838,c3=-176400,c4=176400,c5=-105840,c6=35280,c7=-5040.
Thus we can write
(51)uN(x)=∑i=07‍ciLi(x)=x7-x2,
which is the exact solution.


## 6. Conclusions

In this paper, we have presented the operational matrix of fractional integration of the generalized Laguerre polynomials, and, as an important application, we describe how to use the operational tau technique to numerically solve the FDEs. The basic idea of this technique is as follows.(i)
The FDE is converted to a fully integrated form via multiple integration in the Riemann-Liouville sense.(ii)
Subsequently, the various signals involved in the integrated form equation are approximated by representing them as linear combinations of generalized Laguerre polynomials.(iii)
Finally, the integrated form equation is converted into an algebraic equation by introducing the operational matrix of fractional integration of the generalized Laguerre polynomials.To the best of our knowledge, the presented theoretical formula for generalized Laguerre is completely new, and we do believe that this formula may be used to solve some other kinds of fractional-order initial value problems on a semi-infinite interval.



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