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Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.

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 [

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

where

Similar to integer-order differentiation, Caputo fractional derivative operator is a linear operation

where

We use the ceiling function

For more details on fractional derivatives definitions and its properties see [

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 [

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

Under the zero boundary conditions

and the following initial condition

In the last few years appeared many papers to study this model (3)-(5) [

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 [

The organization of this paper is as follows. In the next section, we obtain the approximation of fractional derivative

The well known Chebyshev polynomials are defined on the interval

The analytic form of the Chebyshev polynomials

where

In order to use these polynomials on the interval

The analytic form of the shifted Chebyshev polynomials

The function

where the coefficients

In practice, only the first

Khader [

The main approximate formula of the fractional derivative of

Theorem 1 [

Let

where

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) [

The error in approximating

then

for all

Theorem 3 [

The Caputo fractional derivative of order

where

Theorem 4 [

The error

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

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

We now collocate Equation (18) at

For suitable collocation points we use roots of shifted chebyshev polynomial

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

Equation (19), together with

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 boundary conditions

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

We apply the proposed method with

Using Equation (19) we have

where

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:

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

We use the notation for the above system

where

The obtained numerical results by means of the proposed method are shown in

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

0.0 | ||

0.1 | ||

0.2 | ||

0.3 | ||

0.4 | ||

0.5 | ||

0.6 | ||

0.7 | ||

0.8 | ||

0.9 | ||

1.0 |

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.