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Fractional advection-dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in porous medium. A space-time fractional advection-dispersion equation (FADE) is a generalization of the classical ADE in which the first-order space derivative is replaced with Caputo or Riemann-Liouville derivative of order

The description of transport is closely related to the terms convection, diffusion, dispersion, and retardation as well as decomposition. First, it is assumed that there are no interactions between the species dissolved in water and the surrounding solid phase [

A relatively complete set of one-dimensional analytical solutions for convective-dispersive solute equations has been recently published by Van Genuchten and Alves in 1982 [

Let us consider a one-dimensional model consisting of infinitely ling homogenous isotropic porous media with steady state uniform flow with seepage velocity

Fractional calculus has been used to model physical and engineering processes, which are found to be best described by fractional differential equations. It is worth noting that the standard mathematical models of integer-order derivatives, including nonlinear models, do not work adequately in many cases. In the recent years, fractional calculus has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, notably control theory, and signal and image processing. In the past several decades, the investigation of travelling-wave solutions for nonlinear equations has played an important role in the study of nonlinear physical phenomena [

This represents an integral of order

Let

Properties of the operator can be found in [

For

In order to include explicitly the possible effect of flow geometry into the mathematical model, the Cartesian component of the gradient of concentration,

The method here consists of applying the Laplace transform on both sides of (

For

Our concern here is to consider (

The Riemann-Liouville derivative has certain disadvantages when trying to model realworld phenomena with fractional differential equations [

For the Caputo derivative, the Laplace transform is based on the formula

Our concern here is to consider (

Up to this section we expressed the solution of the fractional advection-dispersion equation in terms of Mittag-Leffler function. This function is cumbersome to be used in real world problem, especially when the users of this solution are from the field of geohydrology. Since the solution is in series form, one will need first to know how many terms of the series expansion can be used to simulate real world problem. Therefore to accommodate the users of this solution, we propose the approximate solution of the fractional advection-dispersion equation to be in the form of

Figures

Comparison of FADE and ADE for

Comparison of FADE and ADE for

Figures

Simulation of the FADE (

Simulation of the ADE (

Simulation of the FADE (

Simulation of the FADE (

Simulation of the FADE (

From Figures

To test the accuracy and efficiency of FADE, we compare the solution of FADE, ADE, and the experimental data from field observation. Figures

Comparison of FADE, ADE, and experimental data from real world;

Comparison of FADE, ADE, and experimental data from real world;

The numerical simulation in Figures

An excellent literature review revealed that the fractional advection-dispersion equation has proven to be useful in modeling contaminant flow in heterogeneous porous media [

The classical hydrodynamic advection-dispersion equation has been generalized using the concept of fractional order derivatives. This leads to the formulation of a new (generalized) form of the hydrodynamic advection-dispersion equation. A general solution of the new equation was given in terms of Mittag-Leffler functions for two general cases including Riemann-Liouville fractional derivative and the Caputo fractional derivative. The solutions of FADE are not only function of time and space but also a function of the order of the derivative. If these orders are integer, we recover the standard ADE. Figures