Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems

and Applied Analysis 3 where L(α) j (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


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][2][3][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][6][7][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][11][12][13][14][15][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 semiinfinite 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][23][24][25][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 pseudospectral 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.

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 Definition 2. The Caputo fractional derivatives is given by where   is th order differential operator.
The generalized Laguerre polynomials on Λ are obtained from The special value Abstract and Applied Analysis 3 where  ()  (0) = Γ( +  + 1)/Γ( + 1)!, will be of important use later, for treating the initial conditions of the given FDEs.
Let () ∈  2  () (Λ), then () may be expressed in terms of generalized Laguerre polynomials as In particular applications, the generalized Laguerre polynomials up to degree  + 1 are considered.Then we have We will present the Laguerre-Gauss quadrature.Let { ()   ,  ()  } be the set of generalized Laguerre-Gauss quadrature nodes and weights: For the generalized Laguerre-Gauss quadrature, { ()  } are the zeros of  ()  +1 (), and
A combination of (34) and (35) leads to the desired result.
Remark 5.In the case of ] =  ∈ , Theorem 4 gives the same result as (23).

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.

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.
(42) Substituting ( 23) and ( 38) into (37) generates  set of linear equations 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  by any direct solver technique to find the spectral solution ().

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   ().

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  reveals that the present algorithms are very convenient for all choices of  and produces accurate solutions to multi-term FDEs on semiinfinite interval.
Example 1.Consider the linear FDE where () = 1 +  is the exact solution.
Applying variational formulation of the tau method of (48) yields The treatment of initial conditions using (43) gives Solving the resulted system of algebraic equations ( 51)-(52) provides the unknown coefficients in terms of the parameter : Accordingly, the approximate solution can be written as which is the exact solution.
Table 1 lists the values of  0 ,  1 , and  2 with different choices of ().Indeed, we can achieve the exact solution of this problem with all choices of the generalized Laguerre parameters .
The use of technique described in Section 4.1 with  = 2 enables one to approximate the solution as Here, we have Therefore using (42), we obtain Now, making use of (43) yields Finally by solving (58)-( 59), then, we get Thus we can write which is the exact solution.
In Table 2, we exhibit the values of  0 ,  1 and  2 with different choices of ().
Example 3. Consider linear initial value problem of fractional order (see [31]) whose exact solution is given by () =  3 .
where and the exact solution is () =  9 .
Now, if we use the spectral tau approximation based on with  = 9 and  ∈ Λ, then we obtain which is the exact solution. where and the exact solution is given by () =  ((1/] 3 )) − sin((1/] 3 )).
Table 4 introduces the maximum absolute errors, using the tau method based on GLOM of fractional derivative, at ] = 1.5 and with different choices of the parameters  and .The curves of exact solutions and approximate solutions obtained by the proposed method for  = 1,  = 10, and ] = 1.5, 1.7, 1.9 are shown in Figures 1 and 2. From these figures, the exact and approximate solutions are completely conciet.x Exact and approximate solutions u N (x, 1.5) u(x, 1.5) u N (x, 1.7) u(x, 1.7) u N (x, 1.9) u(x, 1.9) Example 6.Let us consider the nonlinear fractional initial value problem where and the exact solution is given by () = cos().
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  = 0.01,  = 1 and  = 30 is given in Figure 3.The solution of this problem is obtained by applying the generalized Laguerre-Gauss collocation method based on generalized Laguerre operational matrix for  = 2.001,  = 1.001, and  = 0.001.The exact solution and approximate  solutions obtained by the proposed method for  = 1 and two choices of  are shown in Figure 4.

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.
x) at N = 5 u N (x) at N = 10

Figure 4 :
Figure 4: Comparing the exact solution and the approximate solutions at  = 5, 10 and  = 1.

Table 4 :
Maximum absolute error for various choices of  and  for Example 5.