Abstract and Applied Analysis

Volume 2013, Article ID 546502, 10 pages

http://dx.doi.org/10.1155/2013/546502

## Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems

^{1}Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Eskisehir Yolu 29.Km, 06810 Ankara, Turkey^{2}Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia^{3}Institute of Space Sciences, Magurele-Bucharest, Romania^{4}Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia^{5}Department of Mathematics, Faculty of Science, Beni-Suef University, Beni Suef, Egypt

Received 15 April 2013; Accepted 21 May 2013

Academic Editor: Soheil Salahshour

Copyright © 2013 D. Baleanu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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

*Definition 2. *The Caputo fractional derivatives is given by
where is th order differential operator.

The Caputo fractional derivative operator satisfies where and are the ceiling and floor functions, respectively, while and .

The Caputo’s derivative operator is a linear operation: where and are constants.

We recall below some relevant properties of the generalized Laguerre polynomials (Szego [27] and Funaro [28]). let and let be a weight function on in the usual sense. Define with the inner product and norm For , the generalized Laguerre polynomials are given by According to [29] for , we get where and .

The generalized Laguerre polynomials are the -orthogonal system; where .

The generalized Laguerre polynomials on are obtained from The special value where , will be of important use later, for treating the initial conditions of the given FDEs.

Let , then may be expressed in terms of generalized Laguerre polynomials as In particular applications, the generalized Laguerre polynomials up to degree 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 , and

#### 3. GLOM of Fractional Derivatives

Let , and then may be expanded using generalized Laguerre polynomials as

In particular applications, the generalized Laguerre polynomials up to degree are considered. Then we have where the vector and vector are given by Then the derivative of the vector can be expressed by where is the operational matrix of the derivative given by

By using (21), it is clear that where and the superscript in denotes matrix powers. Thus

Lemma 3. *Let be a generalized Laguerre polynomial Then
*

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

Theorem 4. *Suppose , the fractional derivative of order of is given by
**
where is the operational matrix of Caputo fractional derivative and is given by
**
where
*

*Proof. *Applying (4) to (11) gives

Now, can be approximated by terms of the generalized Laguerre polynomials to get
where is directly obtained from (18), and
Employing (29)–(31) yields
where
Accordingly, (32) can be written in a vector form as follows:
Also according to Lemma 3, we can write
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).

#### 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 with initial conditions where are constants and ,??. Also is order of Caputo fractional derivative for , are the initial values, and is a source function.

To solve the fractional FDE (36)-(37), we approximate and by generalized Laguerre polynomials as where vector is known.

By using Theorem 4 (relation equations (26), and (38)) we have Making use of (38)–(40), the residual for (36) can be given from

The use of generalized Laguerre tau approximation generates system of linear equations 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 .

##### 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 .

Let us consider the nonlinear multi-term FDE subject to the nonhomogeneous initial conditions (37), where 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 , , and , for by using (38), (40), respectively, and then the operational matrices formulation of (44) can be expressed as Also, making use of (38) and (23) in (44) yields Collocating the operational matrix equation (45) at nodes of the generalized Laguerre-Gauss quadrature on , Combining algebraic equations (47) with initial conditions (46) generates a system of nonlinear algebraic equations. This system may be evaluated by implementing Newton’s iterative method to find the spectral solution .

#### 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 reveals that the present algorithms are very convenient for all choices of and produces accurate solutions to multi-term FDEs on semi-infinite interval.

*Example 1. *Consider the linear FDE
where is the exact solution.

If we apply the operational matrix formulation, the generalized Laguerre spectral tau method with , we get Here, we have where and are defined in (18) and (27).

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 , , and with different choices of (). Indeed, we can achieve the exact solution of this problem with all choices of the generalized Laguerre parameters .

*Example 2. *Consider linear FED
The analytical solution is .

The use of technique described in Section 4.1 with 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 , and with different choices of ().

*Example 3. *Consider linear initial value problem of fractional order (see [31])
whose exact solution is given by .

If we apply the operational matrix formulation, the generalized Laguerre spectral tau method with , we get Also, Therefore using (42), we obtain Accordingly, we get Thus we can write which is the exact solution. The 4 unknown coefficients with various choices of are listed in Table 3.

*Example 4. *Consider the following FDE:
where
and the exact solution is .

Now, if we use the spectral tau approximation based on with and , then we obtain which is the exact solution.

*Example 5. *Consider the nonhomogeneous fractional initial value problem
where
and the exact solution is given by .

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

*Example 6. *Let us consider the nonlinear fractional initial value problem
where
and the exact solution is given by .

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 , and is given in Figure 3.

*Example 7. *We consider the nonlinear fractional initial value problem
where
and the exact solution is .

The solution of this problem is obtained by applying the generalized Laguerre-Gauss collocation method based on generalized Laguerre operational matrix for , , and . The exact solution and approximate solutions obtained by the proposed method for and two choices of are shown in Figure 4.

#### 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.

#### References

- D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus: Models and Numerical Methods*, vol. 3 of*Series on Complexity, Nonlinearity and Chaos*, World Scientific Publishing, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet - M. D. Ortigueira, “Introduction to fractional linear systems. Part 1: continuous-time case,”
*IEE Proceedings: Vision, Image and Signal Processing*, vol. 147, no. 1, pp. 62–70, 2000. View at Publisher · View at Google Scholar · View at Scopus - S. Das,
*Functional Fractional Calculus for System Identification and Controls*, Springer, New York, NY, USA, 2008. View at MathSciNet - C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang,
*Spectral Methods in Fluid Dynamics*, Springer, New York, NY, USA, 1988. View at MathSciNet - E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order,”
*Computers & Mathematics with Applications*, vol. 62, no. 5, pp. 2364–2373, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A new Jacobi operational matrix: an application for solving fractional differential equations,”
*Applied Mathematical Modelling*, vol. 36, no. 10, pp. 4931–4943, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - D. Funaro, “Estimates of Laguerre spectral projectors in Sobolev spaces,” in
*Orthogonal Polynomials and Their Applications*, C. Brezinski, L. Gori, and A. Ronveaux, Eds., vol. 9, pp. 263–266, Scientific Publishing, Singapore, 1991. View at Google Scholar · View at MathSciNet - A. H. Bhrawy, A. S. Alofi, and S. S. Ezz-Eldien, “A quadrature tau method for fractional differential equations with variable coefficients,”
*Applied Mathematics Letters*, vol. 24, no. 12, pp. 2146–2152, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - J. Song, F. Yin, X. Cao, and F. Lu, “Fractional variational iteration method versus Adomian’s decomposition method in some fractional partial differential equations,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 392567, 10 pages, 2013. View at Publisher · View at Google Scholar - H. Jafari, H. Tajadodi, and D. Baleanu, “A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials,”
*Fractional Calculus and Applied Analysis*, vol. 16, no. 1, pp. 109–122, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - A. A. Hemeda, “New iterative method: an application for solving fractional physical differential equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 617010, 9 pages, 2013. View at Publisher · View at Google Scholar - G.-C. Wu and D. Baleanu, “Variational iteration method for fractional calculus: a universal approach by Laplace transform,”
*Advances in Difference Equations*, vol. 2013, article 18, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - C. Yang and J. Hou, “An approximate solution of nonlinear fractional differential equation by Laplace transform and adomian polynomials,”
*Journal of Information and Computational Science*, vol. 10, no. 1, pp. 213–222, 2013. View at Google Scholar - M. Gulsua, Y. Ozturka, and A. Anapalia, “Numerical approach for solving fractional Fredholm integro-differential equation,”
*International Journal of Computer Mathematics*, 2013. View at Publisher · View at Google Scholar - H. S. Nik, S. Effati, S. S. Motsa, and S. Shateyi, “A new piecewise-spectral homotopy analysis method for solving chaotic systems of initial value problems,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 583193, 13 pages, 2013. View at Publisher · View at Google Scholar - E. H. Doha, A. H. Bhrawy, D. Baleanu, and S. S. Ezz-Eldien, “On shifted Jacobi spectral approximations for solving fractional differential equations,”
*Applied Mathematics and Computation*, vol. 219, no. 15, pp. 8042–8056, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. Maleki, I. Hashim, M. Tavassoli Kajani, and S. Abbasbandy, “An adaptive pseudospectral method for fractional order boundary value problems,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 381708, 19 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - A. Al-Rabtah, S. Momani, and M. A. Ramadan, “Solving linear and nonlinear fractional differential equations using spline functions,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 426514, 9 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - A. H. Bhrawy, M. M. Alghamdi, and T. M. Taha, “A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line,”
*Advances in Difference Equations*, vol. 2012, article 179, 2012. View at Publisher · View at Google Scholar - S. Yuzbasi, “Numerical solution of the Bagley-Torvik equation by the Bessel collocation method,”
*Mathematical Methods in the Applied Sciences*, vol. 36, no. 3, pp. 300–312, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. Shaban, S. Kazem, and J. A. Rad, “A modification of the homotopy analysis method based on Chebyshev operational matrices,”
*Mathematical and Computer Modelling*, vol. 57, pp. 1227–1239, 2013. View at Google Scholar - K. Parand and S. A. Kaviani, “Application of the exact operational matrices based on the Bernstein polynomials,”
*Journal of Mathematics and Computer Science*, vol. 6, pp. 36–59, 2013. View at Google Scholar - E. Tohidi, F. Soleymani, and A. Kilicman, “Robustness of operational matrices of differentiation for solving state-space analysis and optimal control problems,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 535979, 9 pages, 2013. View at Publisher · View at Google Scholar - A. H. Bhrawy, D. Baleanu, L. M. Assas, and J. A. Tenreiro Machado, “On a generalized laguerre operational matrix of fractional integration,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 569286, 7 pages, 2013. View at Publisher · View at Google Scholar - M. El-Kady and A. El-Sayed, “Fractional differentiation matrices for solving fractional orders differential equations,”
*International Journal of Pure and Applied Mathematics*, vol. 84, no. 2, pp. 1–13, 2013. View at Google Scholar - G. Szego,
*Orthogonal Polynomials*, American Mathematical Society, Providence, RI, USA, 4th edition, 1975. - D. Funaro,
*Polynomial Approximation of Differential Equations*, vol. 8, Springer, Berlin, Germany, 1992. View at MathSciNet - D. K. Dimitrov, F. Marcellán, and F. R. Rafaeli, “Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials,”
*Journal of Mathematical Analysis and Applications*, vol. 368, no. 1, pp. 80–89, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - D. Baleanu, H. Mohammadi, and S. Rezapour, “Positive solutions of an initial value problem for nonlinear fractional differential equations,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 837437, 7 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - E. A. Rawashdeh, “Numerical solution of fractional integro-differential equations by collocation method,”
*Applied Mathematics and Computation*, vol. 176, no. 1, pp. 1–6, 2006. View at Publisher · View at Google Scholar · View at MathSciNet