Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems

Abstract

We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs) on semi-infinite interval, using generalized Laguerre polynomials. We derive the operational matrix of Caputo fractional derivative of the generalized Laguerre polynomials which is applied together with generalized Laguerre tau approximation for implementing a spectral solution of linear multiterm FDEs on semi-infinite interval subject to initial conditions. The generalized Laguerre pseudo-spectral approximation based on the generalized Laguerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem. Through several numerical examples, we confirm the accuracy and performance of the proposed spectral algorithms. Indeed, the methods yield accurate results, and the exact solutions are achieved for some tested problems.

1. Introduction

Fractional calculus has been used to develop accurate models of many phenomena of science, engineering, economics, and applied mathematics. These models are found to be best described by FDEs [1–4].

One of the best methods, in terms of the accuracy, for investigating the numerical solution of various kinds of differential equations is spectral method (see, for instance, [5–8]). Because all types of spectral methods are global and numerical computational methods, they are very convenient for approximating linear and nonlinear FDEs [6, 7, 9]. We refer also to recent numerical and analytical methods for solving FEEs [10–16].

In the last few years, theory and numerical solution of FDEs by using spectral methods have received an increasing attention. In this direction, Doha et al. [17] proposed an effective way to approximate solutions of linear and nonlinear multiterm FDEs with constant and variable coefficients using Jacobi spectral approximation, in which they generalized the Chebyshev spectral methods [6] and quadrature Legendre tau method [9]; moreover, other very important cases can be obtained for that approach. Maleki et al. [18] proposed an efficient and accurate spectral collocation method based on shifted Legendre-Gauss quadrature nodes for solving fractional boundary value problems in finite interval. The authors of [19] used the spline functions methods for tackling the linear and nonlinear FDEs. Recently, Bhrawy et al. [20] investigated the fractional integrals of modified generalized Laguerre operational matrix to implement a numerical solution of the integrated form of the linear FDEs on semi-infinite interval. Furthermore, Yuzbasi [21] proposed a new collocation method based on Bessel functions to introduce an approximate solution of a class of FDEs. We refer also to the recent papers [22–26] where operational matrices of several orthogonal polynomials are developed for solving linear and nonlinear ODEs and FDEs.

In this paper, the Caputo fractional derivative of generalized Laguerre operational matrix (GLOM) is stated and proved. The main aim of this paper is to extend the application of generalized Laguerre spectral tau method based on GLOM to develop a direct solution technique for the numerical solution of linear multi-term FDEs on a semi-infinite interval. Moreover, we develop the generalized Laguerre pseudo-spectral approximation based on the GLOM for reducing the nonlinear multi-term FDEs subject to nonhomogeneous initial conditions to a system of nonlinear algebraic equations. Finally, the accuracy of the proposed algorithms is demonstrated by test problems. The numerical results are given to show that the proposed spectral algorithms based on generalized Laguerre operational matrix of Caputo fractional derivatives are very effective for linear and nonlinear FDEs.

The outline of the paper is as follows. In Section 2, we present some preliminaries. Section is devoted to drive the GLOM of Caputo fractional derivative. In Section 4, we extend the generalized Laguerre spectral tau and pseudo-spectral approximations based on the GLOM of fractional derivative for solving multiorder linear and nonlinear FDEs. Some numerical experiments are presented in Section 5. Finally, we conclude the paper with some remarks.

2. Some Basic Preliminaries

The two most commonly used definitions are the Riemann-Liouville operator and the Caputo operator. We give some definitions and properties of fractional derivatives and generalized Laguerre polynomials.

Definition 1. The fractional integral operator of Riemann-Liouville sense is defined as 𝐽^𝜈𝑓(𝑥)=1Γ(𝜈)^𝑥₀(𝑥−𝑡)^𝜈−1𝑓(𝑡)𝐽𝑑𝑡,𝜈>0,𝑥>0,₀𝑓(𝑥)=𝑓(𝑥). (1)

Definition 2. The Caputo fractional derivatives is given by 𝐷^𝜈𝑓(𝑥)=𝐽^𝑚−𝜈𝐷^𝑚𝑓(𝑥)=1Γ(𝑚−𝜈)^𝑥₀(𝑥−𝑡)^{𝑚−𝜈−1}𝑑^𝑚𝑑𝑡^𝑚𝑓(𝑡)𝑑𝑡,𝑚−1<𝜈<𝑚,𝑥>0, (2) where 𝐷^𝑚 is 𝑚th order differential operator.

The Caputo fractional derivative operator satisfies 𝐷^𝜈(),𝐷𝐶=0,𝐶isaconstant_𝜈𝑥^𝛽=0,for𝛽∈𝑁₀⌈𝜈⌉,Γ,𝛽<𝛽+1Γ𝑥𝛽+1−𝜈^𝛽−𝜈,for𝛽∈𝑁₀⌈𝜈⌉⌊𝜈⌋,,𝛽≥or𝛽∉𝑁,𝛽> (3)(4) where ⌈𝜈⌉ and ⌊𝜈⌋ are the ceiling and floor functions, respectively, while 𝑁={1,2,…} and 𝑁₀={0,1,2,…}.

The Caputo’s derivative operator is a linear operation: 𝐷^𝜈𝜆𝑓(𝑥)+𝜇𝑔(𝑥)=𝜆𝐷^𝜈𝑓(𝑥)+𝜇𝐷^𝜈𝑔(𝑥), (5) where 𝜆 and 𝜇 are constants.

We recall below some relevant properties of the generalized Laguerre polynomials (Szego [27] and Funaro [28]). let Λ=(0,∞) and let 𝑤^(𝛼)(𝑥)=𝑥^𝛼𝑒^−𝑥 be a weight function on Λ in the usual sense. Define 𝐿²_𝑤(𝛼)(Λ)=𝑣∣𝑣ismeasurableonΛand‖𝑣‖_𝑤(𝛼)<∞, (6) with the inner product and norm (𝑢,𝑣)_𝑤(𝛼)=_Λ𝑢(𝑥)𝑣(𝑥)𝑤^(𝛼)(𝑥)‖𝑣‖𝑑𝑥,_𝑤(𝛼)=(𝑣,𝑣)_𝑤^1/2_(𝛼). (7) For 𝛼>−1, the generalized Laguerre polynomials are given by 𝐿_𝑖^(𝛼)(𝑥)=1𝑥𝑖!^−𝛼𝑒^𝑥𝜕^𝑖_𝑥𝑥^𝑖+𝛼𝑒^−𝑥,𝑖=1,2,…. (8) According to [29] for 𝛼>−1, we get 𝜕_𝑥𝐿_𝑖^(𝛼)(𝑥)=−𝐿^(𝛼+1)_𝑖−1(𝑥),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,…, (9) where 𝐿₀^(𝛼)(𝑥)=1 and 𝐿₁^(𝛼)(𝑥)=1+𝛼−𝑥.

The generalized Laguerre polynomials are the 𝐿²_𝑤(𝛼)(Λ)-orthogonal system; ^∞₀𝐿_𝑗^(𝛼)(𝑥)𝐿_𝑘^(𝛼)(𝑥)𝑤^(𝛼)(𝑥)𝑑𝑥=ℎ_𝑘𝛿_𝑗𝑘, (10) where ℎ_𝑘=Γ(𝑘+𝛼+1)/𝑘!.

The generalized Laguerre polynomials on Λ are obtained from 𝐿_𝑖^(𝛼)(𝑥)=^𝑖_𝑘=0(−1)^𝑘Γ(𝑖+𝛼+1)Γ()𝑥𝑘+𝛼+1)(𝑖−𝑘!𝑘!^𝑘,𝑖=0,1,…. (11) The special value 𝐷^𝑞𝐿_𝑖^(𝛼)(0)=(−1)^{𝑞𝑖−𝑞}_𝑗=0𝑖−𝑗−1!!!𝐿𝑞−1𝑖−𝑗−𝑞^𝑗(𝛼)(0),𝑖⩾𝑞, (12) where 𝐿_𝑗^(𝛼)(0)=Γ(𝑗+𝛼+1)/Γ(𝛼+1)𝑗!, will be of important use later, for treating the initial conditions of the given FDEs.

Let 𝑢(𝑥)∈𝐿²_𝑤(𝛼)(Λ), then 𝑢(𝑥) 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,…. (13)(14) In particular applications, the generalized Laguerre polynomials up to degree 𝑁+1 are considered. Then we have 𝑢_𝑁(𝑥)=^𝑁_𝑗=0𝑎_𝑗𝐿_𝑗^(𝛼)(𝑥). (15)

We will present the Laguerre-Gauss quadrature. Let {𝑥_𝑗^(𝛼),𝜛_𝑗^(𝛼)} be the set of generalized Laguerre-Gauss quadrature nodes and weights: _Λ𝜙(𝑥)𝑤^(𝛼)(𝑥)𝑑𝑥=^𝑁_𝑗=0𝜙𝑥_𝑗^(𝛼)𝜛_𝑗^(𝛼). (16) For the generalized Laguerre-Gauss quadrature, {𝑥_𝑗^(𝛼)} are the zeros of 𝐿^(𝛼)_𝑖+1(𝑥), and 𝜛_𝑗^(𝛼)Γ()=−𝑖+𝛼+1(𝑖+1)!𝐿_𝑖^(𝛼)𝑥_𝑗^(𝛼)𝜕_𝑥𝐿^(𝛼)_𝑖+1𝑥_𝑗^(𝛼)=Γ(𝑖+𝛼+1)𝑥_𝑗^(𝛼)(𝑖+𝛼+1)(𝑖+1)!𝐿_𝑖^(𝛼)𝑥_𝑗^(𝛼)2,0≤𝑗≤𝑖. (17)

3. GLOM of Fractional Derivatives

Let 𝑢(𝑥)∈𝐿²_𝑤(𝛼)(Λ), and then 𝑢(𝑥) may be expanded using 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,…. (18)

In particular applications, the generalized Laguerre polynomials up to degree 𝑁+1 are considered. Then we have 𝑢_𝑁(𝑥)=^𝑁_𝑗=0𝑎_𝑗𝐿_𝑗^(𝛼)(𝑥)=𝐶^𝑇𝜙(𝑥), (19) where the vector 𝐶 and 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. (20) Then the derivative of the vector 𝜙(𝑥) can be expressed by 𝑑𝜙(𝑥)𝑑𝑥=𝐃⁽¹⁾𝜙(𝑥), (21) where 𝐃⁽¹⁾ is the (𝑁+1)×(𝑁+1) operational matrix of the derivative given by 𝐃⁽¹⁾=−00000⋯0010000⋯0011000⋯0011100⋯0011110⋯00⋮⋮⋮⋮⋮⋯⋮⋮11111⋯10. (22)

By using (21), it is clear that 𝑑^𝑛𝜙(𝑥)𝑑𝑥^𝑛=𝐃^(1)𝑛𝜙(𝑥), (23) where 𝑛∈𝑁 and the superscript in 𝐃⁽¹⁾ denotes matrix powers. Thus 𝐃^(𝑛)=𝐃^(1)𝑛,𝑛=1,2,…. (24)

Lemma 3. Let 𝐿_𝑖^(𝛼)(𝑥) be a generalized Laguerre polynomial Then 𝐷^𝜈𝐿_𝑖^(𝛼)(𝑥)=0,𝑖=0,1,…,⌈𝜈⌉−1,𝜈>0. (25)

In the following theorem we prove the operational matrix of Caputo fractional derivative for the generalized Laguerre vector (20).

Theorem 4. Suppose 𝜈>0, the fractional derivative of order 𝜈 of 𝜙(𝑥) is given by 𝐷^𝜈𝜙(𝑥)≃𝐃^(𝜈)𝜙(𝑥), (26) where 𝐃^(𝜈) is the (𝑁+1)×(𝑁+1) operational matrix of Caputo fractional derivative and is given by 𝐃^(𝜈)=^{𝑆000⋯0⋮⋮⋮⋯⋮000⋯0𝜈𝜈⌉)𝑆(⌈,0𝜈𝜈⌉)𝑆(⌈,1𝜈𝜈⌉)(⌈,2⋯𝑆𝜈𝜈⌉)}_𝑆^(⌈,𝑁_{⋮⋮⋮⋯⋮𝜈()𝑆𝑖,0𝜈()𝑆𝑖,1𝜈()𝑖,2⋯𝑆𝜈()𝑆𝑖,𝑁⋮⋮⋮⋯⋮𝜈()𝑆𝑁,0𝜈()𝑆𝑁,1𝜈()𝑁,2⋯𝑆𝜈()𝑁,𝑁}, (27) where 𝑆_𝜈=𝑖,𝑗_{𝑖⌈𝜈⌉𝑗𝑘=ℓ=0}(−1)^𝑘+ℓ()(×()!()()Γ()𝑗!Γ𝑖+𝛼+1×Γ𝑘−𝜈+𝛼+ℓ+1)𝑖−𝑘𝑗−ℓ!ℓ!Γ𝑘−𝜈+1×Γ𝑘+𝛼+1𝛼+ℓ+1⁻¹. (28)

Proof. Applying (4) to (11) gives 𝐷^𝜈𝐿_𝑖^(𝛼)(𝑥)=^𝑖_𝑘=0(−1)^𝑘Γ(𝑖+𝛼+1)()()𝐷𝑖−𝑘!𝑘!Γ𝑘+𝛼+1^𝜈𝑥^𝑘=^𝑖_{⌈𝜈⌉𝑘=}(−1)^𝑘Γ(𝑖+𝛼+1)()()Γ()𝑥𝑖−𝑘!Γ𝑘−𝜈+1𝑘+𝛼+1^𝑘−𝜈,𝑖=⌈𝜈⌉,…,𝑁. (29)
Now, 𝑥^𝑘−𝜈 can be approximated by 𝑁+1 terms of the generalized Laguerre polynomials to get 𝑥^𝑘−𝜈=^𝑁_𝑗=0𝑏_𝑘𝑗𝐿_𝑗^(𝛼)(𝑥), (30) where 𝑏_𝑗 is directly obtained from (18), and 𝑏_𝑗=^𝑗_ℓ=0(−1)^ℓ𝑗!Γ(𝑘−𝜈+𝛼+ℓ+1)!(ℓ)().𝑗−ℓ!Γℓ+𝛼+1 (31) Employing (29)–(31) yields 𝐷^𝜈𝐿_𝑖^(𝛼)(𝑥)=^𝑁_𝑗=0𝑆_𝜈𝑖,𝑗𝐿_𝑗^(𝛼)(𝑥),𝑖=⌈𝜈⌉,…,𝑁, (32) where 𝑆_𝜈𝑖,𝑗=∑^𝑖_⌈𝜈⌉^𝑗_𝑘=∑_ℓ=0(−1)^𝑘+ℓ()()()!()Γ()()𝑗!Γ𝑖+𝛼+1×Γ𝑘−𝜈+𝛼+ℓ+1𝑖−𝑘𝑗−ℓ!ℓ!×Γ𝑘−𝜈+1𝑘+𝛼+1×Γ𝛼+ℓ+1⁻¹. (33) Accordingly, (32) can be written in a vector form as follows: 𝐷^𝜈𝐿_𝑖^(𝛼)(𝑥)≃𝑆_𝜈(𝑖,0),𝑆_𝜈(𝑖,1),𝑆_𝜈(𝑖,2),…,𝑆_𝜈(𝜙(𝑥),⌈𝜈⌉𝑖,𝑁)𝑖=,…,𝑁. (34) Also according to Lemma 3, we can write 𝐷^𝜈𝐿_𝑖^(𝛼)(𝑥)≃[0,0,0,…,0]𝜙(𝑥),𝑖=0,1,…,⌈𝜈⌉−1. (35) A combination of (34) and (35) leads to the desired result.

Remark 5. In the case of 𝜈=𝑛∈𝑁, Theorem 4 gives the same result as (23).

4. Applications of GLOM for Multiterm FDEs

In this section, we are interested in using GLOM in combination with two types of spectral methods for solving linear and nonlinear FDEs.

4.1. Linear Multiorder FDEs

Here, we propose a direct solution technique to approximate linear multi-term FDEs with constant coefficients using the generalized Laguerre tau method in combination with GLOM.

Consider the linear multi-order FDE 𝐷^𝜈𝑢(𝑥)=^𝑘_𝑗=1𝛾_𝑗𝐷^𝛽𝑗𝑢(𝑥)+𝛾_𝑘+1𝑢(𝑥)+𝑔(𝑥),inΛ, (36) with initial conditions 𝑢^(𝑖)(0)=𝑑_𝑖,𝑖=0,…,𝑚−1, (37) where 𝛾_𝑗(𝑗=1,…,𝑘+1) are constants and 𝑚−1<𝜈≤𝑚,??0<𝛽₁<𝛽₂<⋯<𝛽_𝑘<𝜈. Also 𝐷^𝜈𝑢(𝑥)≡𝑢^(𝜈)(𝑥) is 𝜈 order of Caputo fractional derivative for 𝑢(𝑥), 𝑑_𝑖(𝑖=0,…,𝑚−1) are the initial values, and 𝑔(𝑥) is a source function.

To solve the fractional FDE (36)-(37), we approximate 𝑢(𝑥) and 𝑔(𝑥) by generalized Laguerre polynomials as 𝑢(𝑥)≃^𝑁_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(𝑥)=𝐶^𝑇𝜙(𝑥),𝑔(𝑥)≃^𝑁_𝑖=0𝑔_𝑖𝐿_𝑖^(𝛼)(𝑥)=𝐺^𝑇𝜙(𝑥), (38)(39) where vector 𝐺=[𝑔₀,…,𝑔_𝑁]^𝑇 is known.

By using Theorem 4 (relation equations (26), and (38)) we have 𝐷^𝜈𝑢(𝑥)≃𝐶^𝑇𝐷^𝜈𝜙(𝑥)=𝐶^𝑇𝐷^(𝜈)𝜙(𝑥),𝐷^𝛽𝑗𝑢(𝑥)≃𝐶^𝑇𝐷^𝛽𝑗𝜙(𝑥)=𝐶^𝑇𝐷^(𝛽𝑗)𝜙(𝑥),𝑗=1,…,𝑘. (40) Making use of (38)–(40), the residual 𝑅_𝑁(𝑥) for (36) can be given from 𝑅_𝑁(𝑥)=𝐶^𝑇𝐷^(𝜈)−𝐶^𝑇𝑘_𝑗=1𝛾_𝑗𝐃^(𝛽𝑗)−𝛾_𝑘+1𝐶^𝑇−𝐺^𝑇𝜙(𝑥). (41)

The use of generalized Laguerre tau approximation generates (𝑁−𝑚+1) system of linear equations 𝑅_𝑁(𝑥),𝐿_𝑗^(𝛼)(𝑥=_∞0)𝑤(𝑥)𝑅_𝑁(𝑥)𝐿_𝑗^(𝛼)(𝑥)𝑑𝑥=0,𝑗=0,1,…,𝑁−𝑚. (42) Substituting (23) and (38) into (37) generates 𝑚 set of linear equations 𝑢^(𝑖)(0)=𝐶^𝑇𝐃^(𝑖)𝜙(0)=𝑑_𝑖,𝑖=0,1,…,𝑚−1. (43) The combination of (42) and (43) reduces the solution of (36)-(37) to a linear system of algebraic equations, which can be solved for unknown coefficients of the vector 𝐶 by any direct solver technique to find the spectral solution 𝑢(𝑥).

4.2. Nonlinear Multiorder FDEs

In this section, we present the generalized Laguerre pseudo-spectral approximation in combination with GLOM of fractional derivative to find the approximate solution 𝑢_𝑁(𝑥).

Let us consider the nonlinear multi-term FDE 𝐷^𝜈𝑢(𝑥)=𝐹𝑥,𝑢(𝑥),𝐷^𝛽1𝑢(𝑥),…,𝐷^𝛽𝑘𝑢(𝑥),inΛ, (44) subject to the nonhomogeneous initial conditions (37), where 𝐹 can be nonlinear in general. In [30], the authors studied the existence of solutions of a class of nonlinear FDEs.

Now, we will implement the generalized Laguerre operational matrix for treating this nonlinear problem. To do this, firstly, we approximate 𝑢(𝑥), 𝐷^𝜈𝑢(𝑥), and 𝐷^𝛽𝑗𝑢(𝑥), for 𝑗=1,…,𝑘 by using (38), (40), respectively, and then the operational matrices formulation of (44) can be expressed as 𝐶^𝑇𝐃^(𝜈)𝜙(𝑥)≃𝐹𝑥,𝐶^𝑇𝜙(𝑥),𝐶^𝑇𝐃^(𝛽1)𝜙(𝑥),…,𝐶^𝑇𝐃^(𝛽𝑘)𝜙(𝑥). (45) Also, making use of (38) and (23) in (44) yields 𝑢^(𝑖)(0)=𝐶^𝑇𝐃^(𝑖)𝜙(0)=𝑑_𝑖,𝑖=0,1,…,𝑚−1. (46) Collocating the operational matrix equation (45) at (𝑁−𝑚+1) nodes of the generalized Laguerre-Gauss quadrature on Λ, 𝐶^𝑇𝐃^(𝜈)𝜙𝑥_𝑗^(𝛼)≃𝐹𝑥,𝐶^𝑇𝜙𝑥_𝑗^(𝛼),𝐶^𝑇𝐃^(𝛽1)𝜙𝑥_𝑗^(𝛼),…,𝐶^𝑇𝐃^(𝛽𝑘)𝜙𝑥_𝑗^(𝛼). (47) Combining (𝑁−𝑚+1) algebraic equations (47) with 𝑚 initial conditions (46) generates a system of (𝑁+1) nonlinear algebraic equations. This system may be evaluated by implementing Newton’s iterative method to find the spectral solution 𝑢(𝑥).

5. Numerical Results

This section considers several numerical examples to demonstrate the accuracy and applicability of the proposed spectral algorithms based on operational matrix of fractional derivatives of generalized Laguerre polynomials. A comparison of the results obtained by adopting different choices of the generalized Laguerre parameter 𝛼 reveals that the present algorithms are very convenient for all choices of 𝛼 and produces accurate solutions to multi-term FDEs on semi-infinite interval.

Example 1. Consider the linear FDE 𝐷²𝑢(𝑥)+𝐷^3/2𝑢(𝑥)(𝑥)𝑢(0)+𝑢=1+𝑥,=1,𝑢^′(0)=1,𝑥∈Λ, (48) where 𝑢(𝑥)=1+𝑥 is the exact solution.

If we apply the operational matrix formulation, the generalized Laguerre spectral tau method with 𝑁=2, we get 𝑢(𝑥)=𝑐₀𝐿₀^(𝛼)(𝑥)+𝑐₁𝐿₁^(𝛼)(𝑥)+𝑐₂𝐿₂^(𝛼)(𝑥)=𝐶^𝑇𝜙(𝑥). (49) Here, we have 𝐃⁽¹⁾,𝐃=−000100110₍₂₎=,𝐃000000100_(3/2)=𝑆000000_3/2(2,0)𝑆_3/2(2,1)𝑆_3/2(),𝑔2,2𝐺=𝑔⁰₁𝑔₂, (50) where 𝑔_𝑗 and 𝑆_𝜈(𝑖,𝑗) are defined in (18) and (27).

Applying variational formulation of the tau method of (48) yields 𝑐₀+𝑐₂+𝑆_3/2(2,0)=𝑔₀. (51)

The treatment of initial conditions using (43) gives 𝑐₀+(𝛼+1)𝑐₁+(𝛼+1)(𝛼+2)2𝑐₂=1,−𝑐₁−(𝛼+2)𝑐₂−1=0. (52)

Solving the resulted system of algebraic equations (51)-(52) provides the unknown coefficients in terms of the parameter 𝛼: 𝑐₀=𝛼+2,𝑐₁=−1,𝑐₂=0. (53) Accordingly, the approximate solution can be written as 𝑢(𝑥)=𝑐₀,𝑐₁,𝑐₂𝐿₀^(𝛼)(𝑥)𝐿₁^(𝛼)(𝑥)𝐿₂^(𝛼)(𝑥)=1+𝑥, (54) which is the exact solution.

Table 1 lists the values of 𝑐₀, 𝑐₁, and 𝑐₂ with different choices of (𝛼). Indeed, we can achieve the exact solution of this problem with all choices of the generalized Laguerre parameters 𝛼.

Table 1 

𝑐₀, 𝑐₁, and 𝑐₂ for different values of 𝛼 for Example 1.

Example 2. Consider linear FED 𝐷²𝑢(𝑥)+𝐷^1/2𝑢(𝑥)+𝑢(𝑥)=𝑥²+2+2.6666666667Γ()𝑥0.5^1.5,𝑢(0)=0,𝑢^′(0)=0,𝑥∈Λ. (55) The analytical solution is 𝑢(𝑥)=𝑥².

The use of technique described in Section 4.1 with 𝑁=2 enables one to approximate the solution as 𝑢(𝑥)=𝑐₀𝐿₀^(𝛼)(𝑥)+𝑐₁𝐿₀^(𝛼)(𝑥)+𝑐₂𝐿₂^(𝛼)(𝑥)=𝐶^𝑇𝜙(𝑥). (56) Here, we have 𝐃⁽²⁾=,𝐃000000100_(1/2)=𝑆000_1/2(1,0)𝑆_1/2(1,1)𝑆_1/2()𝑆1,2_1/2(2,0)𝑆_1/2(2,0)𝑆_1/2(),𝑔2,0𝐺=𝑔⁰₁𝑔₂. (57) Therefore using (42), we obtain 𝑐₀+𝑆_1/2(1,0)𝑐₁+𝑆_1/2(2,0)+1𝑐₂=6. (58) Now, making use of (43) yields 𝑐₀+(𝛼+1)𝑐₁+(𝛼+1)(𝛼+2)2𝑐₂=0.−𝑐₁−(𝛼+2)𝑐₂=0. (59) Finally by solving (58)-(59), then, we get 𝑐₀=𝛼²+3𝛼+2,𝑐₁=−2𝛼−4,𝑐₂=2. (60) Thus we can write 𝑢(𝑥)=𝑐₀,𝑐₁,𝑐₂𝐿₀^(𝛼)(𝑥)𝐿₁^(𝛼)(𝑥)𝐿₂^(𝛼)(𝑥)=𝑥², (61) which is the exact solution.

In Table 2, we exhibit the values of 𝑐₀,𝑐₁ and 𝑐₂ with different choices of (𝛼).

Table 2 

𝑐₀, 𝑐₁, and 𝑐₂ for different values of 𝛼 for Example 2.

Example 3. Consider linear initial value problem of fractional order (see [31]) 𝐷²𝑢(𝑥)−2𝐷𝑢(𝑥)+𝐷^1/2𝑢(𝑥)+𝑢(𝑥)=𝑥³−6𝑥²+6𝑥+165√𝜋𝑥^2.5,𝑢(0)=0,𝑢^′(0)=0,𝑥∈Λ, (62) whose exact solution is given by 𝑢(𝑥)=𝑥³.

If we apply the operational matrix formulation, the generalized Laguerre spectral tau method with 𝑁=2, we get 𝑢(𝑥)=³_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(𝑥)=𝐶^𝑇𝜙(𝑥). (63) Also, 𝐃⁽²⁾=0000000010002100,𝐃⁽¹⁾,𝐃=−0000100011001110_(1/2)=𝑆0000_1/2(1,0)𝑆_1/2(1,1)𝑆_1/2(1,2)𝑆_1/2()𝑆1,3_1/2(2,0)𝑆_1/2(2,1)𝑆_1/2(2,2)𝑆_1/2()𝑆2,3_1/2(3,0)𝑆_1/2(3,1)𝑆_1/2(3,2)𝑆_1/2(),𝑔3,3𝐺=⁰𝑔₁𝑔₂𝑔₃. (64) Therefore using (42), we obtain 𝑐₀+2𝑐₁+𝑆_1/2(1,0)𝑐₁+3+𝑆_1/2(2,0)𝑐₂+4+𝑆_1/2(3,0)𝑐₃−𝑔₀=0,1+𝑆_1/2(1,1)𝑐₁+2+𝑆_1/2(2,1)𝑐₂+3+𝑆_1/2(3,1)𝑐₃−𝑔₁𝐶=0,_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. (65) Accordingly, we get 𝑐₀=𝛼³𝑐+6𝛼+11𝛼+6,₁=−3𝛼²𝑐−15𝛼−18,₂=6𝛼+18,𝑐₃=−6. (66) Thus we can write 𝑢(𝑥)=𝑐₀,𝑐₁,𝑐₂,𝑐₃𝐿₀^(𝛼)(𝑥)𝐿₁^(𝛼)(𝑥)𝐿₂^(𝛼)(𝑥)𝐿₃^(𝛼)(𝑥)=𝑥³, (67) which is the exact solution. The 4 unknown coefficients with various choices of 𝛼 are listed in Table 3.

Table 3 

𝑐₀, 𝑐₁, 𝑐₂, and 𝑐₃ for different values of 𝛼 for Example 3.

Example 4. Consider the following FDE: 𝐷^5/2𝑢(𝑥)+𝐷²𝑢(𝑥)−2𝐷^1/2𝑢(𝑥)(𝑥)(𝑥),𝑢(0)+4𝑢=𝑔=0,𝑢^′(0)=0,𝑥∈Λ, (68) where 𝑔(𝑥)=4𝑥⁹+72𝑥⁷−2Γ(10)Γ()𝑥19/2^17/2+Γ(10)Γ()𝑥15/2^13/2 (69) and the exact solution is 𝑢(𝑥)=𝑥⁹.

Now, if we use the spectral tau approximation based on with 𝑁=9 and 𝑥∈Λ, then we obtain 𝑢(𝑥)=⁹_𝑖=0𝑐_𝑖𝐿_𝑖^(𝛼)(𝑥)=𝑥⁹, (70) which is the exact solution.

Example 5. Consider the nonhomogeneous fractional initial value problem 𝐷²𝑢(𝑥)+𝐷^𝜈𝑢(𝑥)(𝑥)(𝑥),𝑢(0)+𝑢=𝑔=1,𝑢^′(0)=0,𝑥∈(0,10), (71) where 𝑔(𝑥)=𝜈⁶𝑒^{((1/𝜈3)𝑥)}1+sin𝜈³𝑥+𝑒^{((1/𝜈3)𝑥)}1−sin𝜈³𝑥+1Γ(−𝜈)^𝑥₀(𝑥−𝑡)^−𝜈−1𝑒^{((1/𝜈3)𝑥)}1−sin𝜈³𝑥𝑑𝑡 (72) and the exact solution is given by 𝑢(𝑥)=𝑒^{((1/𝜈3)𝑥)}−sin((1/𝜈³)𝑥).

Table 4 introduces the maximum absolute errors, using the tau method based on GLOM of fractional derivative, at 𝜈=1.5 and with different choices of the parameters 𝛼 and 𝑁. The curves of exact solutions and approximate solutions obtained by the proposed method for 𝛼=1, 𝑁=10, and 𝜈=1.5,1.7,1.9 are shown in Figures 1 and 2. From these figures, the exact and approximate solutions are completely conciet.

Table 4 

Maximum absolute error for various choices of 𝛼 and 𝑁 for Example 5.

Figure 1 

Comparing the exact solution and the approximate solutions at 𝑁=10, 𝛼=1, and 𝜈=1.5,1.7,1.9.

Figure 2 

Comparing the exact solution and the approximate solutions at 𝑁=10, 𝛼=1, and 𝜈=1.5,1.7,1.9.

Example 6. Let us consider the nonlinear fractional initial value problem 𝐷²𝑢(𝑥)+𝐷^1/2𝑢(𝑥)+𝑢²(𝑥)(𝑥),𝑢(0)=𝑔=1,𝑢^′(0)=0,𝑥∈(0,20), (73) where 𝑔(𝑥)=cos²𝛾𝑥−𝛾²+1cos𝛾𝑥Γ(−1/2)^𝑥₀(𝑥−𝑡)^−𝜈−1cos𝛾𝑡𝑑𝑡 (74) and the exact solution is given by 𝑢(𝑥)=cos(𝛾𝑥).

The solution of this problem is obtained by applying the generalized Laguerre-Gauss collocation method based on generalized Laguerre operational matrix. The absolute error between the exact and the approximate solution obtained by the proposed method 𝛾=0.01, 𝛼=1 and 𝑁=30 is given in Figure 3.

Figure 3 

The absolute error for 𝛾=1/100, 𝛼=1 at 𝑁=30.

Example 7. We consider the nonlinear fractional initial value problem 𝐷^𝜉𝑢(𝑥)+𝐷^𝜂𝑢(𝑥)𝐷^𝜃𝑢(𝑥)+𝑢²(𝑥)(𝑥),𝑢(0)=𝑔=0,𝑢^′(0)=0,𝑢^′′(0)(),()()(),=0,𝑥∈0,1𝜁∈2,3,𝜂∈1,2,𝜃∈0,1 (75) where 𝑔(𝑥)=𝑥⁶+6𝑥^3−𝜉Γ()+4−𝜉36𝑥^{6−𝜂−𝜃}Γ4−𝜂Γ(4−𝜃) (76) and the exact solution is 𝑢(𝑥)=𝑥³.

The solution of this problem is obtained by applying the generalized Laguerre-Gauss collocation method based on generalized Laguerre operational matrix for 𝜉=2.001, 𝜂=1.001, and 𝜃=0.001. The exact solution and approximate solutions obtained by the proposed method for 𝛼=1 and two choices of 𝑁 are shown in Figure 4.

Figure 4 

Comparing the exact solution and the approximate solutions at 𝑁=5,10 and 𝛼=1.

6. Conclusion

In this paper, the generalized Laguerre operational matrix of Caputo fractional derivative was derived. Furthermore, we have implemented the generalized Laguerre tau approximation in combination with the GLOM with the generalized Laguerre family to solve the linear FDEs. In addition, combining the pseudo-spectral approximation and the GLOM of fractional derivative was applied to develop an accurate approximate solution of nonlinear FDEs. The generalized Laguerre-Gauss quadrature points were used as a collocation nodes. This method reduces the nonlinear FDEs to a system of algebraic equation in the expansion coefficients which can be solved by any standard technique. The numerical results demonstrate that the proposed spectral algorithms are accurate and efficient.