Fractional and Time-Scales Differential EquationsView this Special Issue
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Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems
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.
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.  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  and quadrature Legendre tau method ; moreover, other very important cases can be obtained for that approach. Maleki et al.  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  used the spline functions methods for tackling the linear and nonlinear FDEs. Recently, Bhrawy et al.  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  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  and Funaro ). 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  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.
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.
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 , 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.
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 ) 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.
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.
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