We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs) on semiinfinite 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 semiinfinite interval subject to initial conditions. The generalized Laguerre pseudospectral 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.
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 [
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, [
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. [
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 multiterm FDEs on a semiinfinite interval. Moreover, we develop the generalized Laguerre pseudospectral approximation based on the GLOM for reducing the nonlinear multiterm 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
The two most commonly used definitions are the RiemannLiouville operator and the Caputo operator. We give some definitions and properties of fractional derivatives and generalized Laguerre polynomials.
The fractional integral operator of RiemannLiouville sense is defined as
The Caputo fractional derivatives is given by
The Caputo fractional derivative operator satisfies
The Caputo’s derivative operator is a linear operation:
We recall below some relevant properties of the generalized Laguerre polynomials (Szego [
The generalized Laguerre polynomials are the
The generalized Laguerre polynomials on
Let
We will present the LaguerreGauss quadrature. Let
Let
In particular applications, the generalized Laguerre polynomials up to degree
By using (
Let
In the following theorem we prove the operational matrix of Caputo fractional derivative for the generalized Laguerre vector (
Suppose
Applying (
Now,
In the case of
In this section, we are interested in using GLOM in combination with two types of spectral methods for solving linear and nonlinear FDEs.
Here, we propose a direct solution technique to approximate linear multiterm FDEs with constant coefficients using the generalized Laguerre tau method in combination with GLOM.
Consider the linear multiorder FDE
To solve the fractional FDE (
By using Theorem
The use of generalized Laguerre tau approximation generates
In this section, we present the generalized Laguerre pseudospectral approximation in combination with GLOM of fractional derivative to find the approximate solution
Let us consider the nonlinear multiterm FDE
Now, we will implement the generalized Laguerre operational matrix for treating this nonlinear problem. To do this, firstly, we approximate
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
Consider the linear FDE
If we apply the operational matrix formulation, the generalized Laguerre spectral tau method with
Applying variational formulation of the tau method of (
The treatment of initial conditions using (
Solving the resulted system of algebraic equations (
Table





























Consider linear FED
The use of technique described in Section
In Table





























Consider linear initial value problem of fractional order (see [
If we apply the operational matrix formulation, the generalized Laguerre spectral tau method with




































Consider the following FDE:
Now, if we use the spectral tau approximation based on with
Consider the nonhomogeneous fractional initial value problem
Table
Maximum absolute error for various choices of



















































Comparing the exact solution and the approximate solutions at
Comparing the exact solution and the approximate solutions at
Let us consider the nonlinear fractional initial value problem
The solution of this problem is obtained by applying the generalized LaguerreGauss collocation method based on generalized Laguerre operational matrix. The absolute error between the exact and the approximate solution obtained by the proposed method
The absolute error for
We consider the nonlinear fractional initial value problem
The solution of this problem is obtained by applying the generalized LaguerreGauss collocation method based on generalized Laguerre operational matrix for
Comparing the exact solution and the approximate solutions at
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 pseudospectral approximation and the GLOM of fractional derivative was applied to develop an accurate approximate solution of nonlinear FDEs. The generalized LaguerreGauss 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.