Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials ()
1. Introduction
The subject of fractional calculus was planted over 300 year ago. The theory of derivative and integrals of non- integer order goes back to Liouville, Leibnitz, Grunwald-Letnikov, Reimann and Letnikov. In the recent years, fractional calculus has played a very significant role in many areas in fluid flow, mechanics, viscoelasticity, biology, physics, science and engineering, and other applications [1] . Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. Half-order derivatives and integrals are proved to be more useful for the formulation of certain electrochemical problems than the classical models [2] . Thus, seeking solutions of nonlinear fractional differential equations (FDEs) is still a significant task. Except in a limited numbers of these equations, we have difficulty to find their analytical as well as approximate solutions. Therefore, there have been attempts to develop the new methods for obtaining analytical and approximate solutions of nonlinear FDEs. Recently, several methods have drawn special attention, such as homotopy perturbation method [3] , homotopy analysis method [4] , collocation method ([5] -[9] ) and finite difference method ([10] [11] ).
Our main goal in this paper is concerned with the application of Laguerre pseudo-spectral method to obtain the numerical solution of FWE of the following form
(1)
here the parameter 
refers to the fractional order of spatial derivatives with 
The function 
is a source term and 
is the coefficient function which is a given continues function satisfies Lipschitz condition. We also assume the following initial conditions
(2)
and the following Dirichlet boundary conditions
(3)
Note that at 
Equation (1) is the classical wave equation
Many authors studied the numerical solutions of the introduced problem (1) using different numerical methods such as, Adomian decomposition method [12] and finite difference methods ([13] [14] ) and others.
Our idea is to apply the Laguerre collocation method to discretize (1) to get a linear system of ordinary differential equations (ODEs) thus greatly simplifying the problem, and use the finite difference method (FDM) ([15] -[18] ) to solve the resulting system.
The structure of this paper is arranged in the following way: In Section 2, we introduce some basic definitions about Caputo fractional derivatives and properties of the generalized Laguerre polynomials. In Section 3, we introduce the fundamental theorems for the fractional derivatives of the generalized Laguerre polynomials and its convergence analysis. In Section 4, we give the procedure of solution for FWE. In Section 5, numerical example is given to solve FWE and show the accuracy of the presented method. Finally, in Section 6, the paper ends with a brief conclusion and some remarks.
2. Preliminaries and Notations
In this section, we present some necessary definitions and mathematical preliminaries of the fractional calculus theory required for our subsequent development.
2.1. The Caputo Fractional Derivative
Definition1.
The Caputo fractional derivative operator 
of order 
is defined in the following form
where
, 
is the gamma function.
Similar to integer-order differentiation, Caputo fractional derivative operator is a linear operation
where 
and ![]()
are constants. For the Caputo’s derivative we have ![]()
C is a constant and
![]()
(4)
We use the ceiling function ![]()
to denote the smallest integer greater than or equal to![]()
, and![]()
. Recall that for![]()
, the Caputo differential operator coincides with the usual differential operator of integer order. For more details on fractional derivatives definitions and its properties see ([2] [19] ).
2.2. The Definition and Properties of the Generalized Laguerre Polynomials
The generalized Laguerre polynomials![]()
, ![]()
are defined on the unbounded interval ![]()
and can be determined with the aid of the following recurrence formula
![]()
(5)
where, ![]()
and ![]()
The analytic form of these polynomials of degree n is given by [20]
![]()
(6)
![]()
These polynomials are orthogonal on the interval ![]()
with respect to the weight function![]()
. The orthogonality relation is
![]()
(7)
Also, they satisfy the differentiation formula
![]()
(8)
Any function ![]()
belongs to the space ![]()
of all square integrable functions on ![]()
with weight function![]()
, can be expanded in the following Laguerre series
![]()
(9)
where the coefficients ![]()
are given by
![]()
(10)
Consider only the first ![]()
terms of generalized Laguerre polynomials, so we can write
![]()
(11)
For more details on Laguerre polynomials, its definitions and properties see ([21] [22] ).
3. The Approximate Fractional Derivatives of ![]()
and Its Convergence Analysis
The main goal of this section is to introduce the following theorems to derive an approximate formula of the fractional derivatives of the generalized Laguerre polynomials and study the truncating error and its convergence analysis.
Theorem 1 [23]
Let ![]()
be approximated by the generalized Laguerre polynomials as (11) and also suppose ![]()
then, its Caputo fractional derivative can be written in the following form
![]()
(12)
where ![]()
is given by
![]()
(13)
Theorem 2
The Caputo fractional derivative of order ![]()
for the generalized Laguerre polynomials can be expressed in terms of the generalized Laguerre polynomials themselves in the following form
![]()
(14)
where ![]()
Proof. See [24] .
Theorem 3 [25] .
The error in approximating ![]()
by ![]()
is bounded by
![]()
(15)
![]()
(16)
where, ![]()
and ![]()
4. Solution of the Fractional Wave Equation
Consider the fractional wave equation of type given in Equation (1) in the interval![]()
. In order to use Laguerre collocation method, we first approximate ![]()
as
![]()
(17)
From Equations (1), (17) and Theorem 1, we have
![]()
(18)
we now collocate Equation (18) at ![]()
points![]()
, ![]()
as
![]()
(19)
For suitable collocation points we use roots of shifted Laguerre polynomial![]()
.
Also, by substituting Equations (17) and (11) in the boundary conditions (3) we can obtain ![]()
equations as follows
![]()
(20)
where ![]()
Equation (19), together with ![]()
equations of the boundary conditions (20), give ![]()
of ordinary differential equations which can be solved, for the unknowns![]()
, ![]()
, using the finite difference method, as described in the following section.
5. Numerical Results
In this section, we implement the proposed method to solve FWE (1) with![]()
, of the form
![]()
where the coefficient and the source functions are ![]()
and ![]()
the initial and Dirichlet conditions are![]()
, ![]()
, ![]()
The exact solution to this problem is ![]()
We apply the method with![]()
, and approximate the solution as follows
![]()
(21)
Using Equation (19) we have
![]()
(22)
where ![]()
are roots of Laguerre polynomial![]()
, i.e. ![]()
and ![]()
By using Equations (20) and (22) we obtain the following system of ODEs
![]()
(23)
![]()
(24)
![]()
(25)
![]()
(26)
where ![]()
and ![]()
are defined in (20) and
![]()
![]()
![]()
Now, in order to use FDM to solve the system (23)-(26), we will use the notations ![]()
to be the integration time![]()
, ![]()
, for ![]()
Define![]()
,![]()
. Then, the system (23)-(26), will discretize in time and take the following form
![]()
(27)
![]()
(28)
![]()
(29)
![]()
(30)
we can write the above system (27)-(30) in the following matrix form as follows
![]()
(31)
The above system can be rewritten in a matrix form as follows
![]()
(32)
where ![]()
and ![]()
The obtained numerical results by means of the proposed method are shown in Table 1 and (Figure 1 and Figure 2). In Table 1, the absolute errors between the exact solution uex and approximate solution uapprox at m = 3 and m = 5 with the final time T = 2 are given. Also, in Figure 1 and Figure 2, comparison between the exact solution and the approximate solution at T = 1 with time step τ = 0.0025, and m = 3, m = 5 respectively are presented.
6. Conclusion and Remarks
This article is devoted to introducing an accurate numerical technique for solving the fractional wave equation. The prosed method depends on the approximate formula for the Caputo fractional derivative of the generalized Laguerre polynomials derived above. In the proposed method, the properties of the Laguerre polynomials are used to reduce FWE to solve a system of ODEs which solved by using FDM. The results show that the introduced algorithm converges as the number of m terms is increased. The solution is expressed as a truncated Laguerre series and so it can be easily evaluated for arbitrary values of time using any computer program without any computational effort. Although we only considered a model problem in this paper, the main idea and the
![]()
Table 1. The absolute error between the exact solution and the approximate solution at m = 3, m = 5 and T = 2.
![]()
Figure 1. Comparison between the exact solution and the approximate solution at T = 1 with τ = 0.0025, m = 3.
![]()
Figure 2. Comparison between the exact solution and the approximate solution at T = 1 with τ = 0.0025, m = 5.
used techniques are also applicable to many other problems. It is evident that the overall errors can be made smaller by adding new terms from the series (21). In the end, from our numerical results using the proposed method, we can see that, the solutions are in excellent agreement with the exact solution. All computations are made by Matlab.