Ordinary and partial fractional differential equations (FDEs) have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics and engineering [1] [2] . Consequently, considerable attention has been given to the solutions of FDEs of physical interest. Most FDEs do not have exact solutions, so approximate and numerical techniques [3] [4] , must be used. Recently, several numerical methods to solve FDEs have been given such as variational iteration method [5] , homotopy perturbation method [3] [6] , Adomian decomposition method [7] [8] , homotopy analysis method [9] , collocation method [10] [11] and finite difference method [12] - [17] .

We introduce some necessary definitions and mathematical preliminaries of the fractional calculus theory that will be required in the present paper.

Definition 1

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 [2]

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] .

Anomalous, or non-Fickian, dispersion has been an active area of research in the physics community since the introduction of continuous time random walks (CTRW) by Montroll and Weiss [1965]. These random walks extended the predictive capability of models built on the stochastic process of Brownian motion, which is the basis for the classical advectiondispersion equation ( ADE ).

A fractional ADE . is a generalization of the classical ADE in which the second-order derivative is replaced with a fractional-order derivative. In contrast to the classical ADE , the fractional ADE has solutions that resemble the highly skewed and heavy-tailed breakthrough curves observed in field and laboratory studies.

When a fractional Fick’s law replaces the classical Fick’s law in an Eulerian evaluation of solute transport in a porous medium, the result is a fractional ADE.

It describes the spread of solute mass over large distances via a convolutional fractional derivative.

We consider the initial-boundary value problem of the fractional Advection-dispersion equation which is usually written in the following form

where, , is the source term, is a constant and is the Caputo fractional derivative with respect to and of order, where.

Under the zero boundary conditions

and the following initial condition

In the last few years appeared many papers to study this model (3)-(5) [1] [18] - [22] , the most of these papers study the ordinary case of such problem but in this paper we study the fractional case.

Our idea is to apply the Chebyshev collocation method to discretize (3) to get a linear system of ODEs thus greatly simplifying the problem, and use FDM [12] to solve the resulting system.

The organization of this paper is as follows. In the next section, we obtain the approximation of fractional derivative. In Section 3, we prove the error analysis of the proposed formula. In Section 4, we implement chebyshev collocation method to the solution of (3). As a result a system of ordinary differential equations is formed and the solution of the considered problem is introduced. In Section 5, we give some numerical results to clarify the proposed method. Also a conclusion is given in Section 6.

The well known Chebyshev polynomials are defined on the interval and can be determined with the aid of the following recurrence formula [23]

.

The analytic form of the Chebyshev polynomials of degree is given by

where denotes the integer part of. The orthogonality condition is

In order to use these polynomials on the interval we define the so called shifted Chebyshev polynomials by introducing the change of variable. So, the shifted Chebyshev polynomials are defined as.

The analytic form of the shifted Chebyshev polynomials of degree is given by

The function, which belongs to the space of square integrable in, may be expressed in terms of shifted Chebyshev polynomials as

where the coefficients are given by

In practice, only the first -terms of shifted Chebyshev polynomials are considered. Then we have

Khader [24] introduced a new approximate formula of the fractional derivative and used it to solve numerically the fractional diffusion equation.

The main approximate formula of the fractional derivative of is given in the following theorem.

Theorem 1 [24]

Let be approximated by Chebyshev polynomials as in (10) and also suppose then

where is given by

Also, in this section, special attention is given to study the convergence analysis and evaluate an upper bound of the error of the proposed approximate formula.

Theorem 2 (Chebyshev truncation theorem) [23]

The error in approximating by the sum of its first terms is bounded by the sum of the absolute values of all the neglected coefficients. If

then

for all, all, and all.

Theorem 3 [25]

The Caputo fractional derivative of order for the shifted Chebyshev polynomials can be expressed in terms of the shifted Chebyshev polynomials themselves in the following form

where

.

Theorem 4 [11]

The error in approximating by is bounded by

Consider the fractional Advection-dispersion equation of type given in Equation (3). In order to use Chebyshev collocation method, we first approximate as

From Equations (3), (17) and Theorem 1 we have

We now collocate Equation (18) at points, as

For suitable collocation points we use roots of shifted chebyshev polynomial.

Also, by substituting Equations (17) in the boundary conditions (4) we can obtain equations as follows

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.

In this section, we present a numerical example to illustrate the efficiency and the validation of the proposed numerical method when applied to solve numerically the fractional Advection-dispersion equation. Consider the ADE (3) with and the following source term

and the boundary conditions, with the initial condition.

The exact solution of Equation (3) in this case is

We apply the proposed method with, and approximate the solution as follows

Using Equation (19) we have

where are roots of shifted Chebyshev polynomial, i.e.

By using Equations (20) and (24) we can obtain the following system of ODEs

where

Now, to use FDM for solving the system (25)-(28), we will use the following notations: to be the integration time for. Define,. Then the system (25)-(28), is discretized in time and takes the following form

We can write the above system (29)-(32) in the following matrix form as follows

We use the notation for the above system

where and

The obtained numerical results by means of the proposed method are shown in Table 1 and Figures 1-4. In Table 1, the absolute error between the exact solution and the approximate solution at and with the final time are given. Also, in Figure 1 and Figure 2, comparison between the exact solution and the approximate solution at with time step, and, are presented, respectively. Also, in Figure 3 and Figure 4, the behavior of the approximate solution at and with different values of and are presented, respectively. From, these figures, we can see that the behavior of the approximate solution depends on the order of the fractional derivative.

The properties of the Chebyshev polynomials are used to reduce the fractional Advection-dispersion equation to the solution of system of ODEs which solved by using FDM. The fractional derivative is considered in the

Caputo sense. In this article, special attention is given to studying the convergence analysis and estimating an upper bound of the error for the proposed approximate formula of the fractional derivative. The solution obtained using the suggested method is in excellent agreement with the already existing ones and shows that this approach can be solved the problem effectively. From the resulted numerical solution, we can conclude that the used techniques in this work can be applied to many other problems. It is evident that the overall errors can be made smaller by adding new terms from the series (23). Comparisons are made between the approximate solution and the exact solution to illustrate the validity and the great potential of the technique. All computations in this paper are done using Matlab 8.