MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 853127 10.1155/2013/853127 853127 Research Article Analytical Solutions of the Space-Time Fractional Derivative of Advection Dispersion Equation Atangana Abdon 1 0000-0002-1217-963X Kilicman Adem 2 Wu Guo-Cheng 1 Institute for Groundwater Studies University of the Free State P.O. Box 399 Bloemfontein South Africa ufs.ac.za 2 Department of Mathematics and Institute for Mathematical Research University Putra Malaysia P.O. Box 43400, Serdang, Selangor Malaysia upm.edu.my 2013 2 4 2013 2013 24 01 2013 01 03 2013 2013 Copyright © 2013 Abdon Atangana and Adem Kilicman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 0<β1, and the second-order space derivative is replaced with the Caputo or the Riemann-Liouville fractional derivative of order 1<α2. We derive the solution of the new equation in terms of Mittag-Leffler functions using Laplace transfrom. Some examples are given. The results from comparison let no doubt that the FADE is better in prediction than ADE.

1. Introduction

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 . The primary mechanism for the transport of improperly discarded hazardous waste through the environment is by the movement of water through the subsurface and surface waterways. Studying this movement requires that one must be able to measure the quantity of waste present at a particular point in space time. The measure universally for chemical pollution is the concentration. Analytical methods that handle solute transport in porous media are relatively easy to use . However, because of complexity of the equations involved, the analytical solutions are generally available restricted to either radial flow problems or to cases where velocity is uniform over the area of interest. Numerous analytical solutions are available for time-dependent solute transport within media having steady state and uniform flow. This work is devoted to the discussion underpinning the derivation of the analytical solution of space-time fractional derivative of advection-dispersion equation.

2. Governing Equations

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 . Here we shall review a case having a practical application.

Let us consider a one-dimensional model consisting of infinitely ling homogenous isotropic porous media with steady state uniform flow with seepage velocity v. We inject a particular chemical from one end of the model for a period of time t0 such that the input concentration varies as an exponential function of time . The value of that chemical concentration at any time t and at a distance x from the injection boundary, allowing for the decay and adsorption, may be obtained from the solution of the following set of equations : (1)D2c(x,t)x2-vc(x,t)x-λRc=Rc(x,t)t, where D is the dispersion coefficient and R the retardation factor, subject to the initial condition: (2)c(x,t)=0t=0,c(0,t)=c0exp(-γt)0<tt0, which means that the system is initially free of that chemical, γ and c0 are constants and boundaries conditions (3)c(x,t)x=0x. This indicates that the concentration of the gradient at the other end of the model remains unchanged. Note that the standard version of advection-dispersion equation does not allow for predicting the mass transform through the geological formation accurately; it is then important to investigate a possible analytical partial differential equation that can describe better this problem. In this work this possibility is further investigated for a rectangular symmetric form of (1), by replacing the classical first-order derivative of the concentration by a fractional derivative. Because the concepts of fractional (or noninteger) order derivatives may not be widely known, the concept is first briefly discussed below.

3. Fractional Calculus

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 . The concept of fractional order derivatives for a function, say f(x), is based on a generalization of the Abel integral: (4)D-nf(x)=f(x)dxn=1Γ(n)0x(x-t)n-1f(t)dt, where n is a nonzero positive integer and Γ(·) is the Gamma function .

This represents an integral of order n for the continuous function f(x), whenever f and all its derivatives vanish at the origin, x=0. This result can be extended to the concept of an integral of arbitrary order c, defined as (5)D-cf(x)=D-j-sf(x)=1Γ(c)0x(x-t)c-1f(t)dt, where c is a positive real number, j an integer such that 0<s1.

Let p now be the least positive integer larger than α such that α=m-ρ; 0<ρ1. Equation (4) can then be used to define the derivative of (positive) fractional order, say α, of a function f(x) as (6)Dcf(x)=Dp-ρf(x)=1Γ(ρ)0x(x-t)ρ-1dpf(t)dtpdt. Note that these results, like Abel’s integral, are only valid subject to the condition that f(k)(x)x=0 for k=0,1,2,,p.

3.1. Properties

Properties of the operator can be found in [14, 15]; we mention only the following.

For fCμ,  μ-1,α,β0 and γ>-1: (7)D-αD-βf(x)=D-α-βf(x),D-αD-βf(x)=D-βD-αf(x),D-αxγ=Γ(γ+1)Γ(α+γ+1)xα+γ.

3.2. Formulation of Space-Time Fractional Derivative of Hydrodynamic Advection-Dispersion Equation

In order to include explicitly the possible effect of flow geometry into the mathematical model, the Cartesian component of the gradient of concentration, xc(x,t) is replaced by the Riemann-Liouville fractional derivatives of order β,  xβc(x,t) and x2c(x,t) is replaced by xαc(x,t) with 0<β1<α<l, as follows: (8)Dαc(x,t)xα-vβc(x,t)xβ-λRc=Rc(x,t)t. This provides a generalized form of the classical equation governing the transport of the solute (1): this integrodifferential equation does contain the additional parameter α and β, which can be viewed as new physical parameters that characterize the transport through the geological formations. The same transformation generates also a more general form for the boundary condition at the other end of the model: (9)βc(x,t)xβ=0x. Relations (8) and (9), together with the initial condition described in (2), represent a complete set of equations for which a solution exists. The integrodifferential character of the relations makes the search for analytical solutions for the problem very difficult however. In this work an analytical solution in series form will be discussed in the next section.

4. Analytical Solution 4.1. The Riemann-Liouville Derivative

The method here consists of applying the Laplace transform on both sides of (8) to have (10)Dαc(x,s)xα-vβc(x,s)xβ-R(λ+s)c(x,s)=Rc(x,0) with the initial condition (2) and further transformation; the above equation can then be in the following form: (11)αc(x,s)xα-μβc(x,s)xβ-τc(x,s), where s is the variable of Laplace for the time-component, μ=v/D, and τ=R(λ+s)/D. Let c(x,s)=y(x); then (11) becomes (12)αy(x)xα-μβy(x)xβ-τy(x)=0. Applying the Laplace operator on both sides of (11), on the space component and replacing, we have the following equation : (13)(y)(p)=i=1lhipi-1pα-μpβ-τ, where p is the Laplace variable for the space component and hi=xα-ic(0+).

For p and |τp-β/(pα-β-μ)|<1, we have the following expression 1/(pα-μpβ-τ) which can be written in the form of series as follows : (14)pi-1pα-μpβ-τ=pi-1j=0τnp-β-βn(pα-β-μ)n+1. And hence replacing the above expression in (13) yields the following representation: (15)(y)(p)=i=12hij=0τnp-β-βn+i-1(pα-β-μ)n+1. The above expression is then simplified further, for p and |μpβ-α|<1, we have first (16)τnp-β-βn+i-1(pα-β-μ)n+1=τnp(α-β)-(α+βn-i+1)(pα-β-μ)n+1. And secondly the above equation can now be expressed as follows: (17)=1n!{xαn+α-i(x)nEα-β,α+nβ+1-i(μxα-β)}, where (18)(x)nEα,β(x)=j=0Γ(n+j+1)Γ(nα+β+αj)xjj!. Hence the solutions of (12) can be given as follows: (19)yi(x)=n=0τnn!xαn+α-i(x)nEα-β,α+βn+1-i(μxα-β). Thus it follows that the solution of (12) is given as (20)y(x)=i=12hiyi(x), so that (21)c(x,s)=i=12hici(x,s). Thus the series solution of (8) can be now given by applying the inverse Laplace operator on c(x,s) to have (22)ci(x,t)=-1(n=0τnn!xαn+α-i(x)nEα-β,α+βn+1-i(μxα-β)). Since the inverse Laplace operator is a linear operator, it follows that (23)ci(x,t)=n=0-1(τn)n!xαn+α-i(x)nEα-β,α+βn+1-i(μxα-β). Replacing τn=(R(λ+s)/D)n=(R/D)n(λ+s)n so that (24)-1{τn}=(RD)n-1{(λ+s)n}=(RD)nexp[-λt]t-1-nΓ(-n),ci(x,t)=n=0(R/D)nexp[-λt]t-1-n/Γ(-n)n!xαn+α-i×(x)nEα-β,α+βn+1-i(μxα-β),c(x,t)=i=12hici(x,t),c1(x,t)=n=0(R/D)nexp[-λt]t-1-n/Γ(-n)n!xαn+α-1×(x)nEα-β,α+βn(μxα-β),c2(x,t)=n=0(R/D)nexp[-λt]t-1-n/Γ(-n)n!xαn+α-2×(x)nEα-β,α+αn-1(μxα-β). To find the coefficient hi,  i=1,2, we need to apply the boundaries and initial condition on c(x,t) which yields to (25)hi=c02.

Example 1.

Our concern here is to consider (8) when α=2 and 0<β1. Following the discursion presented earlier, the analytical solution of space-time fractional derivative of hydrodynamic advection-dispersion equation has its two solutions given by (26)c1(x,t)=n=0(RD)nexp[-λt]t-1-nΓ(-n)n!x2n+1×(x)nE2-β,nβ+1(RDx2-β),c2(x,t)=n=0(RD)nexp[-λt]t-1-nΓ(-n)n!x2n×(x)nE2-β,nβ+1(RDx2-β). The above solutions form the fundamental system of solution when β<1.

4.2. The Caputo Derivative

The Riemann-Liouville derivative has certain disadvantages when trying to model realworld phenomena with fractional differential equations . Therefore, we investigate the solution of space-time Caputo fractional derivative of hydrodynamic advection-dispersion equation.

For the Caputo derivative, the Laplace transform is based on the formula (27)(cDαy)(s)=sα(y)(s)-i=01hisα-i-1 with (28)hi=yi(0)(i=0,1). Thus applying the Laplace transform in both sides of (8) on the component of time and applying again the Laplace transform on the component of space yield (29)(y)(p)=i=02-1hipα-i-1pα-μpβ-τ-μi=01-1hipβ-i-1pα-μpβ-τ. For p and |τp-β/(pα-β-μ)|<1, in analogy of the discursion presented earlier for the case of Riemann-Liouville, we have the following: (30)(y)(p)=i=02-1hin=0τnp(α-β)-(βn+i+1)(pα-β-μ)n+1-μi=01-1τnp(α-β)-(βn+i+1+α-β)(pα-β-μ)n+1. Hence for p and |μpβ-α|<1, we have that (31)p(α-β)-(βn+j+1)(pα-β-μ)n+1=1n!([xnα+i(x)nEα-β,βn+i+1(μxα-β)]),p(α-β)-(βn+j+1+α-β)(pα-β-μ)n+1=1n![xnα+i+α-β(x)nEα-β,βn+i+1+α-β(μxα-β)]. Thus from the above expression we derive the following solution to the space-time Caputo fractional derivative of hydrodynamic advection-dispersion equation (8): (32)c(x,t)=i=02-1hici(x,t)-μi=01-1hici(x,t), where for i=0(33)ci(x,t)=n=0(RD)nexp[-λt]t-1-nΓ(-n)n!xnα+i×(x)nEα-β,βn+i+1(μxα-β)-μn=0(RD)nexp[-λt]t-1-nΓ(-n)n!xnα+i+α-β×(x)nEα-β,βn+i+1+α-β(μxα-β) and for i=1(34)ci(x,t)=n=0(RD)nexp[-λt]t-1-nΓ(-n)n!xnα+i×(x)nEα-β,βn+i+1(μxα-β). And the coefficients hi are found by applying the initial and boundary conditions on c(x,t).

Example 2.

Our concern here is to consider (8) when α=2 and 0<β1. Following the discursion presented earlier, the analytical solution of space-time fractional derivative of hydrodynamic advection-dispersion equation has its two solutions given by (35)c1(x,t)=n=0(RD)nexp[-λt]t-1-nΓ(-n)n!x2n×(x)nE2-β,βn+1(μx2-β)-μn=0(RD)nexp[-λt]t-1-nΓ(-n)n!x2n+2-β×(x)nE2-β,βn+3-β(μx2-β),c2(x,t)=n=0(RD)nexp[-λt]t-1-nΓ(-n)n!x2n+1×(x)nE2-β,βn+2(μx2-β). These solutions are linearly independent and they provide the fundamental system of solutions to space-time Caputo fractional derivative of hydrodynamic advection-dispersion equation. An approximation of this series is given below for possible simulation. Some other analytical methods and their recent development for solving nonlinear fractional partial differential equation can found in the work done by  and the excellent book for analytical and numerical methods.

5. Numerical Simulation

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 (36)c(x,t)=c0exp(-γt)2×[exp(xα(qr-ur)2Dr)erfc(xα-urt2(Drt)1/β)+exp(xα(qr+ur)2Dr)erfc(xα+urt2(Drt)1/β)]. The above solutions take into account the effect of the fractional derivative order. Now notice that if one set α=1 and β=2, we recover the solution of the advection-dispersion equation (37)c(x,t)=c0exp(-γt)2×[expx(qr-ur)2Drerfcx-urt2Drt+expx(qr+ur)2Drerfcx+urt2Drt]. To access the effect of the fractional order derivative into the solution of the advection-dispersion equation, we compare both solutions (36) and (37) with the theoretical values firstly and secondly we compare both solutions with experimental data obtained from one of the experimental sites of the Institute for Groundwater Studies (IGS). We shall start with the theoretical values.

Figures 1 and 2 show the numerical simulation of the plume first as a function of time and second as a function of time and space with the FADE and ADE for theoretical values. Figures 1 and 2 show the comparison of the approximate and exact solutions of FADE and ADE, respectively, as function of time for a fixed distance x. These figures are plotted via Mathematica.

Comparison of FADE and ADE for α=0.95 and β=1.9.

Comparison of FADE and ADE for α=1 and β=1.98.

Figures 37 show the density plots of the theoretical simulation of the plume by the FADE as function of time and fractional order derivative. The figures are simulated via MATLAB. Here, Figure 3 is the simulation of the concentration for FADE through the geological formation, for c0=100,  α=0.45,  β=2,  D=2;  q=1,  γ=0.25, and λ=1. Figure 4 is the simulation of FADE for c0=100,  α=1,  β=2,  D=2;  q=1,  γ=0.25, and λ=1. Figure 5 is the simulation of FADE for c0=100,  α=0.55,  β=1.55,  D=2;  q=1,  γ=0.25, and λ=1. And finally Figure 6 is the simulation of FADE for c0=100,  α=0.25,  β=1.55,  D=2;  q=1,  γ=0.25, and λ=1. Figure 7 is the simulation of FADE for c0=100,  α=0.25,  β=1.95,  D=2;  q=1,  γ=0.25, and λ=1.

Simulation of the FADE (c0=100,  α=0.45,  β=2,  D=2;  q=1,  γ=0.25, and λ=1).

Simulation of the ADE (c0=100,  α=1,  β=2,  D=2;  q=1,  γ=0.25, and λ=1).

Simulation of the FADE  (c0=100,  α=0.55,  β=1.55,  D=2;  q=1,  γ=0.25, and λ=1).

Simulation of the FADE (c0=100,  α=0.25,  β=1.55,  D=2;  q=1,  γ=0.25, and λ=1).

Simulation of the FADE (c0=100,  α=0.55,  β=1.95,  D=2;  q=1,  γ=0.25, and λ=1).

From Figures 37 one can see that the solutions of FADE are not only a 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 3 and 5 show that the order of the derivative can be used to simulate the real-world problem and this makes the fractional version of ADE better than the ADE.

To test the accuracy and efficiency of FADE, we compare the solution of FADE, ADE, and the experimental data from field observation. Figures 8 and 9 show the comparison between FADE, ADE, and measured data for different values of α and μ.

Comparison of FADE, ADE, and experimental data from real world; Dr=4.5,  β=1.95,  α=0.99, and qr=0.51.

Comparison of FADE, ADE, and experimental data from real world; Dr=2.5, β=1.36, α=0.3,  qr=0.4, and c0=150.