On a Generalized Laguerre Operational Matrix of Fractional Integration

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt 3 Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Eskisehir Yolu 29.km, 06810 Yenimahalle Ankara, Turkey 4Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia 5 Institute of Space Sciences, RO 76900, Magurele-Bucharest, Romania 6Department of Mathematics, Faculty of Science, Umm Al-Qura University, Mecca 21955, Saudi Arabia 7 Department of Electrical Engineering, Polytechnic of Porto, Institute of Engineering, 4314200-072 Porto, Portugal

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][19][20][21][22][23][24] and the references therein).
The usual spectral methods are only available for bounded domains for solving FDEs; see [25][26][27][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][30][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.

Some Basic Preliminaries
The most used definition of fractional integration is due to Riemann-Liouville, which is defined as > 0,  > 0, and  0  () =  () . ( The operator   has the property: The next equation defines the Riemann-Liouville fractional derivative of order : where  − 1 <  ≤ ,  ∈ , and  is the smallest integer greater than .
A function () ∈  2  () (Λ) may be expressed in terms of generalized Laguerre polynomials as In practice, only the first ( + 1) terms of generalized Laguerre polynomials are considered.Then we have where the generalized Laguerre coefficient vector  and the generalized Laguerre vector () are given by If we define the  times repeated integration of generalized Laguerre vector () by   (), then (cf.Paraskevopoulos [36]) where  is an integer value and P () is the operational matrix of integration of ().For more details see [36].

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.
Equation ( 23) leads to the desired result.

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.
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  on (24), after making use of (4), we get the integrated form of (24), namely, where where In order to use the tau method with Laquerre operational matrix for solving the fully integrated problem (27) with initial conditions (25), we approximate () and () by the Laguerre polynomials: where the vector  = [ 0 , . . .,   ]  is given but  = [ 0 , . . .,   ]  is an unknown vector.After making use of Theorem 1 (relation ( 15)) the Riemann-Liouville integral of orders  and ( −   ) of the approximate solution (29) can be written as respectively, where P () is the ( + 1) × ( + 1) operational matrix of fractional integration of order .Employing ( 29)-( 32) the residual   () for ( 27) can be written as As in a typical tau method, we generate  −  + 1 linear algebraic equations by applying Also by substituting Eqs. ( 11) and (29) in Eq (25), we get Equations ( 34) and ( 35) generate −+1 and  set of linear equations, respectively.These linear equations can be solved for unknown coefficients of the vector .Consequently,   () given in (29) can be calculated, which leads to the solution of ( 24) with the initial conditions (25).

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
whose exact solution is given by () =  2 .
If we apply the technique described in Section 4 with  = 2, then the approximate solution can be written as ) , ) .
Example 3. As the first example, we consider the following fractional initial value problem: whose exact solution is given by () =  3 .If we apply the technique described in Section 4 with  = 3, then the approximate solution can be written as ) .
(44) Using (34) we obtain Now, applying (35) we get By solving the linear system (45)-(49) we have the 4 unknown coefficients with various choices of  in Table 2, and we get (47) Thereby we can write Numerical results will not be presented since the exact solution is obtained.which is the exact solution.

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 semiinfinite interval.