On a Generalized Laguerre Operational Matrix of Fractional Integration

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.

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 Section 3 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 𝐽^𝜈𝑓(𝑥)=1Γ(𝜈)^𝑥₀(𝑥−𝑡)^𝜈−1𝑓(𝑡)𝑑𝑡,𝜈>0,𝑥>0,and𝐽⁰𝑓(𝑥)=𝑓(𝑥). (1)

The operator 𝐽^𝜈 has the property: 𝐽^𝜈𝑥^𝛽=Γ𝛽+1Γ𝑥𝛽+1+𝜈^𝛽+𝜈. (2)

The next equation defines the Riemann-Liouville fractional derivative of order 𝜈: 𝐷^𝜈𝑓(𝑥)=𝑑^𝑚𝑑𝑥^𝑚𝐽^𝑚−𝜈𝑓(𝑥), (3) where 𝑚−1<𝜈≤𝑚, 𝑚∈𝑁, and 𝑚 is the smallest integer greater than 𝜈.

If 𝑚−1<𝜈≤𝑚, 𝑚∈𝑁, then 𝐷^𝜈𝐽^𝜈𝑓(𝑥)(𝑥),𝐽=𝑓_𝜈𝐷^𝜈𝑓(𝑥)=𝑓(𝑥)−^𝑚−1_𝑖=0𝑓^(𝑖)0⁺𝑥^𝑖𝑖!,𝑥>0. (4)

Now, let Λ=(0,∞) and 𝑤^(𝛼)(𝑥)=𝑥^𝛼𝑒^−𝑥 be a weight function on Λ in the usual sense. Define the following: 𝐿²_𝑤(𝛼)(Λ)={𝑣|‖𝑣‖𝑣ismeasurableonΛand_𝑤(𝛼)<∞, (5) equipped with the following inner product and norm: (𝑢,𝑣)_𝑤(𝛼)=_Λ𝑢(𝑥)𝑣(𝑥)𝑤^(𝛼)(𝑥)‖𝑣‖𝑑𝑥,_𝑤(𝛼)=(𝑢,𝑣)_𝑤^1/2_(𝛼). (6)

Next, let 𝐿_𝑖^(𝛼)(𝑥) be the generalized Laguerre polynomials of degree 𝑖. We know from [39] that, for 𝛼>−1, 𝐿^(𝛼)_𝑖+1(𝑥)=1)𝐿𝑖+1(2𝑖+𝛼+1−𝑥_𝑖^(𝛼)(𝑥)−(𝑖+𝛼)𝐿^(𝛼)_𝑖−1(𝑥),𝑖=1,2,…, (7) where 𝐿₀^(𝛼)(𝑥)=1 and 𝐿₁^(𝛼)(𝑥)=1+𝛼−𝑥. The set of generalized Laguerre polynomials is the 𝐿²_𝑤(𝛼)(Λ)-orthogonal system, namely, ^∞₀𝐿_𝑗^(𝛼)(𝑥)𝐿_𝑘^(𝛼)(𝑥)𝑤^(𝛼)(𝑥)𝑑𝑥=ℎ_𝑘𝛿_𝑗𝑘, (8) where 𝛿_𝑗𝑘 is the Kronecher function and ℎ_𝑘=(Γ(𝑖+𝛼+1))/𝑖!.

The generalized Laguerre polynomials of degree 𝑖, on the interval Λ, are given by 𝐿_𝑖^(𝛼)(𝑥)=^𝑖_𝑘=0(−1)^𝑘Γ(𝑖+𝛼+1)Γ()𝑥𝑘+𝛼+1)(𝑖−𝑘!𝑘!^𝑘,𝑖=0,1,…. (9)

The special value 𝐷^𝑞𝐿_𝑖^(𝛼)(0)=(−1)^{𝑞𝑖−𝑞}_𝑗=0𝑖−𝑗−1!!!𝐿𝑞−1𝑖−𝑗−𝑞^𝑗(𝛼)(0),𝑖≥𝑞, (10) where 𝐿_𝑗^(𝛼)(0)=(Γ(𝑗+𝛼+1))/(Γ(𝛼+1)𝑗!), will be of important use later.

A function 𝑢(𝑥)∈𝐿²_𝑤(𝛼)(Λ) may be expressed in terms of generalized Laguerre polynomials as 𝑢(𝑥)=^∞_𝑗=0𝑎_𝑗𝐿_𝑗^(𝛼)(𝑥),𝑎_𝑗=1ℎ_𝑘^∞₀𝑢(𝑥)𝐿_𝑗^(𝛼)(𝑥)𝑤^(𝛼)(𝑥)𝑑𝑥,𝑗=0,1,2,…. (11)

In practice, only the first (𝑁+1) terms of generalized Laguerre polynomials are considered. Then we have 𝑢_𝑁(𝑥)=^𝑁_𝑗=0𝑎_𝑗𝐿_𝑗^(𝛼)(𝑥)=𝐶^𝑇𝜙(𝑥), (12) where the generalized Laguerre coefficient vector 𝐶 and the generalized Laguerre vector 𝜙(𝑥) are given by 𝐶^𝑇=𝑐₀,𝑐₁,…,𝑐_𝑁,𝜙(𝑥)=𝐿₀^(𝛼)(𝑥),𝐿₁^(𝛼)(𝑥),…,𝐿_𝑁^(𝛼)(𝑥)^𝑇. (13)

If we define the 𝑞 times repeated integration of generalized Laguerre vector 𝜙(𝑥) by 𝐽^𝑞𝜙(𝑥), then (cf. Paraskevopoulos [36]) 𝐽^𝑞𝜙(𝑥)≃𝑃^(𝑞)𝜙(𝑥), (14) 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. Let 𝜙(𝑥) be the generalized Laguerre vector and 𝜈>0, then 𝐽^𝜈𝜙(𝑥)≃𝐏^(𝜈)𝜙(𝑥), (15) 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: 𝐏^(𝜈)=Θ_𝜈(0,0)Ω_𝜈(0,1)Θ_𝜈(0,2)⋯Θ_𝜈()Θ0,𝑁_𝜈(1,0)Θ_𝜈(1,1)Θ_𝜈(1,2)⋯Θ_𝜈()Θ1,𝑁⋮⋮⋮⋯⋮_𝜈(𝑖,0)Θ_𝜈(𝑖,1)Θ_𝜈(𝑖,2)⋯Θ_𝜈()Θ𝑖,𝑁⋮⋮⋮⋯⋮_𝜈(𝑁,0)Θ_𝜈(𝑁,1)Θ_𝜈(𝑁,2)⋯Θ_𝜈(),𝑁,𝑁 (16) where Θ_𝜈=𝑖,𝑗_{𝑖𝑗𝑘=0𝑟=0}(−1)^𝑘+𝑟𝑗!Γ(𝑖+𝛼+1)Γ(𝑘+𝜈+𝛼+𝑟+1)()!()Γ()Γ().𝑖−𝑘𝑗−𝑟!𝑟!Γ𝑘+𝜈+1𝑘+𝛼+1𝛼+𝑟+1 (17)

Proof. Using the analytic form of the generalized Laguerre polynomials 𝐿_𝑖^(𝛼)(𝑥) of degree 𝑖 (9) and (2), then 𝐽^𝜈𝐿_𝑖^(𝛼)(𝑥)=^𝑖_𝑘=0(−1)^𝑘Γ(𝑖+𝛼+1)()()𝐽𝑖−𝑘!𝑘!Γ𝑘+𝛼+1^𝜈𝑥^𝑘=^𝑖_𝑘=0(−1)^𝑘Γ(𝑖+𝛼+1)(𝑖−𝑘)!Γ(𝑘+𝜈+1)Γ(𝑘+𝛼+1)×𝑥^𝑘+𝜈,𝑖=0,1,⋯,𝑁. (18)
Now, approximate 𝑥^𝑘+𝜈 by 𝑁+1 terms of generalized Laguerre series, we have 𝑥^𝑘+𝜈=^𝑁_𝑗=0𝑐_𝑗𝐿_𝑗^(𝛼)(𝑥), (19) where 𝑐_𝑗 is given from (11) with 𝑢(𝑥)=𝑥^𝑘+𝜈; that is, 𝑐_𝑗=^𝑗_𝑟=0(−1)^𝑟𝑗!Γ(𝑘+𝜈+𝛼+𝑟+1)𝑗−𝑟!𝑟!Γ(𝑟+𝛼+1),𝑗=1,2,…,𝑁. (20)
In virtue of (18) and (19), we get 𝐽^𝜈𝐿_𝑖^(𝛼)(𝑥)=^𝑁_𝑗=0Θ_𝜈𝑖,𝑗𝐿_𝑗^(𝛼)(𝑥),𝑖=0,1,…,𝑁, (21) where Θ_𝜈=𝑖,𝑗_{𝑖𝑗𝑘=0𝑟=0}(−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. (22)
Accordingly, (21) can be written in a vector form as follows: 𝐽^𝜈𝐿_𝑖(𝑥)≃Θ_𝜈(𝑖,0),Θ_𝜈(𝑖,1),Θ_𝜈()Θ𝑖,2,⋯,_𝜈(𝑖,𝑁)𝜙(𝑥),𝑖=0,1,…,𝑁. (23)
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: 𝐷^𝜈𝑢(𝑥)=^𝑘_𝑖=1𝛾_𝑗𝐷^𝛽𝑖𝑢(𝑥)+𝛾_𝑘+1𝑢(𝑥)(𝑥),(),+𝑓inΛ=0,∞ (24) with initial conditions 𝑢^(𝑖)(0)=𝑑_𝑖,𝑖=0,…,𝑚−1, (25) where 𝛾_𝑖 (𝑖=1,…,𝑘+1) are real constant coefficients, 𝑚−1<𝜈≤𝑚, 0<𝛽₁<𝛽₂<…<𝛽_𝑘<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 order 𝜈 on (24), after making use of (4), we get the integrated form of (24), namely, 𝑢(𝑥)−^𝑚−1_𝑗=0𝑢^(𝑗)0⁺𝑥^𝑗=𝑗!_𝑘𝑖=1𝛾_𝑖𝐽^{𝜈−𝛽𝑖}𝑢(𝑥)−^𝑚𝑖−1_𝑗=0𝑢^(𝑗)0⁺𝑥^𝑗𝑗!+𝛾_𝑘+1𝐽^𝜈𝑢(𝑥)+𝐽^𝜈𝑓(𝑥),𝑢^(𝑖)(0)=𝑑_𝑖,𝑖=0,…,𝑚−1, (26) where 𝑚_𝑖−1<𝛽_𝑖≤𝑚_𝑖, 𝑚_𝑖∈𝑁, implies that 𝑢(𝑥)=^𝑘_𝑖=1𝛾_𝑖𝐽^{𝜈−𝛽𝑖}𝑢(𝑥)+𝛾_𝑘+1𝐽^𝜈𝑢(𝑥)(𝑥),𝑢+𝑔_(𝑖)(0)=𝑑_𝑖,𝑖=0,…,𝑚−1, (27) where 𝑔(𝑥)=𝐽^𝜈𝑓(𝑥)+^𝑚−1_𝑗=0𝑑_𝑗𝑥^𝑗+𝑗!^𝑘_𝑖=1𝛾_𝑖𝐽^{𝜈−𝛽𝑖𝑚𝑖−1}_𝑗=0𝑑_𝑗𝑥^𝑗.𝑗! (28)

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: 𝑢_𝑁(𝑥)≃^𝑁_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(𝑥)=𝐶^𝑇𝜙(𝑥),𝑔(𝑥)≃^𝑁_𝑖=0𝑔_𝑖𝐿_𝑖^(𝛼)(𝑥)=𝐺^𝑇𝜙(𝑥), (29)(30) where the vector 𝐺=[𝑔₀,…,𝑔_𝑁]^𝑇 is given but 𝐶=[𝑐₀,…,𝑐_𝑁]^𝑇 is an unknown vector.

After making use of Theorem 1 (relation (15)) the Riemann-Liouville integral of orders 𝜈 and (𝜈−𝛽_𝑗) of the approximate solution (29) can be written as 𝐽^𝜈𝑢_𝑁(𝑥)≃𝐶^𝑇𝐽^𝜈𝜙(𝑥)≃𝐶^𝑇𝐏^(𝜈)𝜙(𝑥),𝐽^{𝜈−𝛽𝑗}𝑢_𝑁(𝑥)≃𝐶^𝑇𝐽^{𝜈−𝛽𝑗}𝜙(𝑥)≃𝐶^𝑇𝐏^{(𝜈−𝛽𝑗)}𝜙(𝑥),𝑗=1,…,𝑘, (31)(32) respectively, where 𝐏^(𝜈) is the (𝑁+1)×(𝑁+1) operational matrix of fractional integration of order 𝜈. Employing (29)-(32) the residual 𝑅_𝑁(𝑥) for (27) can be written as 𝑅_𝑁(𝑥)=𝐶^𝑇−𝐶^𝑇𝑘_𝑗=1𝛾_𝑗𝐏^{(𝜈−𝛽𝑗)}−𝛾_𝑘+1𝐶^𝑇𝐏^(𝜈)−𝐺^𝑇𝜙(𝑥). (33)

As in a typical tau method, we generate 𝑁−𝑚+1 linear algebraic equations by applying 𝑅_𝑁(𝑥),𝐿_𝑗^(𝛼)(𝑥)=^∞₀𝑅_𝑁(𝑥)𝑤^(𝛼)(𝑥)𝐿_𝑗^(𝛼)(𝑥)𝑑𝑥=0,𝑗=0,1,…,𝑁−𝑚. (34)

Also by substituting Eqs. (11) and (29) in Eq (25), we get 𝑢^(𝑖)(0)=𝐶^𝑇𝐃^(𝑖)𝜙(0)=𝑑_𝑖,𝑖=0,1,…,𝑚−1. (35) 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 vector 𝐶. Consequently, 𝑢_𝑁(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 FDE𝐷²𝑢(𝑥)+𝐷^1/2𝑢(𝑥)+𝑢(𝑥)=𝑥²+2+2.6666666667Γ()𝑥0.5^1.5,𝑢(0)=0,𝑢^′(0)=0,𝑥∈Λ, (36)whose exact solution is given by??𝑢(𝑥)=𝑥².

If we apply the technique described in Section 4 with 𝑁=2, then the approximate solution can be written as 𝑢_𝑁(𝑥)=²_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(𝑥)=𝐶^𝑇𝜙(𝑥),𝑃⁽²⁾=Θ₂(0,0)Θ₂(0,1)Θ₂()Θ0,2₂(1,0)Θ₂(1,1)Θ₂()Θ1,2₂(2,0)Θ₂(2,1)Θ₂(),𝑃2,2_(3/2)=Θ_3/2(0,0)Θ_3/2(0,1)Θ_3/2()Θ0,2_3/2(1,0)Θ_3/2(1,1)Θ_3/2()Θ1,2_3/2(2,0)Θ_3/2(2,1)Θ_3/2(),𝑔2,2𝐺=𝑔⁰₁𝑔₂. (37)

Using (34) we obtain Θ_3/2(0,2)+Θ₂(0,2)𝑐₀+Θ_3/2(1,2)+Θ₂(1,2)𝑐₁+1+Θ_3/2(2,2)+Θ₂(2,2)𝑐₂+𝑔₂=0. (38)

Now, by applying (35), we have 𝑐₀+(𝛼+1)𝑐₁+(𝛼+1)(𝛼+2)2𝑐₂=0.−𝑐₁−(𝛼+2)𝑐₂=0. (39)(40)

Finally by solving (38)–(40), we have the 3 unknown coefficients with various choices of 𝛼 given in Table 1. Then, we get 𝑐₀=𝛼²+3𝛼+2,𝑐₁=−2𝛼−4,𝑐₂=2. (41)

Table 1 

𝑐₀, 𝑐_1, and 𝑐_2, for different values of 𝛼 for Example 2.

Thus we can write 𝑢(𝑥)=𝑐₀,𝑐₁,𝑐₂𝐿₀^(𝛼)(𝑥)𝐿₁^(𝛼)(𝑥)𝐿₂^(𝛼)(𝑥)=𝑥², (42) which is the exact solution.

Example 3. As the first example, we consider the following fractional initial value problem:𝐷^3/2𝑢(𝑥)+3𝑢(𝑥)=3𝑥³+8Γ()𝑥0.5^1.5,𝑢(0)=0,𝑢^′(0)=0,𝑥∈Λ, (43)whose exact solution is given by 𝑢(𝑥)=𝑥³.
If we apply the technique described in Section 4 with 𝑁=3, then the approximate solution can be written as 𝑢_𝑁(𝑥)=³_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(𝑥)=𝐶^𝑇𝜙(𝑥),𝑃^(3/2)=Θ_3/2(0,0)Θ_3/2(0,1)Θ_3/2(0,2)Θ_3/2()Θ0,3_3/2(1,0)Θ_3/2(1,1)Θ_3/2(1,2)Θ_3/2()Θ1,3_3/2(2,0)Θ_3/2(2,1)Θ_3/2(2,2)Θ_3/2()Θ2,3_3/2(3,0)Θ_3/2(3,1)Θ_3/2(3,2)Θ_3/2(),𝑔3,3𝐺=⁰𝑔₁𝑔₂𝑔₃. (44)
Using (34) we obtain 3Θ_3/2(0,2)𝑐₀+3Θ_3/2(1,2)𝑐₁+1+3Θ_3/2(2,2)𝑐₂+3Θ_3/2(3,2)𝑐₃+𝑔₂=0,3Θ_3/2(0,3)𝑐₀+3Θ_3/2(1,3)𝑐₁+3Θ_3/2(2,3)𝑐₂+1+3Θ_3/2(3,3)𝑐₃+𝑔₃=0. (45)
Now, applying (35) we get 𝐶^𝑇𝜙(0)=𝑐₀+(𝛼+1)𝑐₁+(𝛼+1)(𝛼+2)2𝑐₂+(𝛼+1)(𝛼+2)(𝛼+3)6𝑐₃𝐶=0,_𝑇𝐷⁽¹⁾𝜙(0)=−𝑐₁−(𝛼+2)𝑐₂−(𝛼+3)(𝛼+2)2𝑐₃=0. (46)
By solving the linear system (45)–(49) we have the 4 unknown coefficients with various choices of 𝛼 in Table 2, and we get 𝑐₀=𝛼³𝑐+6𝛼+11𝛼+6,₁=−3𝛼²𝑐−15𝛼−18,₂𝑐=6𝛼+18,₃=−6. (47)
Thereby we can write 𝑢_𝑁(𝑥)=³_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(𝑥)=𝑥³. (48)
Numerical results will not be presented since the exact solution is obtained.

Table 2 

𝑐₀, 𝑐₁, 𝑐_2, and 𝑐₃ for different values of 𝛼 for Example 3.

Example 4. Consider the following equation:𝐷²𝑢(𝑥)−2𝐷𝑢(𝑥)+𝐷^1/2𝑢(𝑥)+𝑢(𝑥)=𝑥⁷+2048429√𝜋𝑥^6.5−14𝑥⁶+42𝑥⁵−𝑥²−83√𝜋𝑥^1.5𝑢(0)+4𝑥−2,=0,𝑢^′(0)=0,𝑥∈Λ, (49)whose exact solution is given by??𝑢(𝑥)=𝑥⁷−𝑥².

Now, we can apply the technique described in Examples 2 and 3, with 𝛼=0 and 𝑁=7, then we have 𝑐₀=5038,𝑐₁𝑐=−35276,₂=105838,𝑐₃𝑐=−176400,₄=176400,𝑐₅𝑐=−105840,₆=35280,𝑐₇=−5040. (50) Thus we can write 𝑢_𝑁(𝑥)=⁷_𝑖=0𝑐_𝑖𝐿_𝑖(𝑥)=𝑥⁷−𝑥², (51) 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.