eu ON ADOMIAN ’ S DECOMPOSITION METHOD FOR SOLVING A FRACTIONAL ADVECTION-DISPERSION EQUATION

Abstract: In this paper, the Adomian’s decomposition method (ADM) is considered to solve a fractional advection-dispersion model. This model can be represented if the first order derivative in time is replaced by the Caputo fractional derivative of order α (0 < α ≤ 1). In addition, the space derivative orders are replaced by the alternative orders 0 < β ≤ 1 and 1 < γ ≤ 2. The obtained solutions are formulated in a convergent infinite series in terms of Mittage-Leffler functions. Finally, two illustrative examples are introduced to ensure the effectiveness of the used method.


Introduction
It is well known that, the advection-dispersion equation(ADE) is the mathemat-Received: March 24, 2015 c 2015 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author ical model that has commonly been applied to describe the Brownian motion of particles [1].(ADE) is used to describe the transport and spreading of substances (e.g.solute) or conserved property existing in a fluid due to the fluid's bulk motion.The classical (ADE) for the motion of a concentration field C(ξ, τ ) of one space variable ξ at time τ has the form (see [2]) where D and V are positive constants representing diffusion and drift coefficients respectively.The diffusion and drift coefficient may also be functions depending on the space variable ξ and the time τ to be in the form of Fokker-Planck equation(FPE) [3].For some applications of this equation, see [4,5,6].
Fractional derivatives play key role in modeling of particle transport in diffusion including the fractional Advection-dispersion equation(FADE) [3].Thus, (FADE) may be introduced to provide a good simulation for diffusion.Some special forms of the fractional (ADE) were solved analytically in [7].Rocca et al. [8] considered the fractional diffusion-advection equation for solar cosmicray transport and gave its general solution.The authors in [9] presented a level IV fugacity model coupled to a dispersion-advection equation to simulate the environmental concentration of a pesticide in rice fields.Zhuang and Liu [10] proposed an implicit difference approximation for the time fractional diffusion equation and gave error.Liu et al. [11] considered the space fractional Fokker-Planck equation with instantaneous source and presented a fractional method of lines.The authors in [12] developed practical numerical methods to solve one dimensional fractional advection-dispersion equations with variable coefficients on a finite domain.A. El-Sayed et al. [13] concerned with a model that describes the intermediate process between advection and dispersion via fractional derivative, an analytic approximate solution for that model is obtained by the powerful method called Adomian decomposition method(ADM).Adomian decomposition method (ADM) [14,15] is a mathematical technique for solving large classes of ordinary and partial differential equations that gives an analytic approximate solution.George Adomian(1923Adomian( -1996) ) is the first one who introduced this method.The basic idea of the (ADM) is based on representing the solution as infinite series.In fact, our actual problem that we can face is the convergence of that approximate solution to the exact solution.The convergence analysis of the method was studied by many researchers, for example see [16,17].Sufficient conditions for convergence are given in [18,19].Applying the (ADM) for solving Fractional differential equations (FDEs) comes from the method applied to the linear and nonlinear differential equations with integer derivatives.The application of the method to (FDEs) has been established in [20].For solving a system of fractional differential equations by (ADM) has been investigated in [21,22].
In this work, the fractional derivative in the Caputo sense is considered.Fractional advection-dispersion model is discussed for general case of the fractional derivative order.This paper is organized to have some important preliminaries in section 2. Section 3 deals with reviewing the basic idea of (ADM) briefly.Section 4 is concerned with applying the (ADM) to find the general solution of the (FADE) for different three cases of arbitrary fractional orders of derivatives.Finally, some illustrative examples are given in section 5.

Preliminaries
In this section we review some basic definitions in fractional calculus, and some of its properties [23].
Definition 2. The Riemann-Liouville fractional integral operator of order α ≥ 0 of a function f (x) ∈ L 1 is defined by where Γ denotes the gamma function.When a = 0, we can write I α a → I α .Some properties of the operator I α a can be found in [24].For f (x) ∈ L 1 , α, β ≥ 0 and γ > −1: Lemma 1.For α ∈ (m − 1, m), the Caputo fractional derivative of the power function x p is given by A two-parameter function of the Mittag-Leffler type is defined by the series expansion (see [24]): It follows from Definition 2.4 that the one parameter Mittag-Leffler function is .

The Basic Idea of the ADM
In reviewing the basic idea of the used method [14,15].Consider a nonlinear differential equation where L = D n , D = d dx , and n represents the highest order of the linear operator, R is the remaining linear operator grouping the lower derivatives which can be written as R = n−1 k=0 a k (x) d k dx k and N u represents the nonlinear term in equation (2).For the initial value problems, integrating (2) by the operator L −1 from 0 to x yields The method is based on setting the solution in an infinite series to be in the form and the nonlinear operator can be decomposed as where A n depends upon u 0 , u 1 , ..., u n , are called Adomian polynomials which are defined by The u n 's are determined recursively as Making use equation ( 5) and equation ( 6), the required solution (4) of the nonlinear differential equation ( 2) is obtained.
In this work we introduce the fractional advection-dispersion equation(FADE) in the general fractional form with 0 < α ≤ 1, β > 0, and γ > 0. For simplicity of the equation, we set t = τ V , x = ξ and C(ξ, τ ) is replaced by u(x, t) to become in the form where µ = D V , D t = ∂ ∂t , and Our main aim is to get solution for the (FADE) modeled in (8) with the initial condition given in (9) for different three cases.Case I is concerned with an exponential function as initial condition and for arbitrary fractional order 0 < α ≤ 1 and β = 1, γ = 2. Considering γ = 2β and any function f (x) for the initial condition (9), we get the solution for equation (8) in case II.Finally, case III is dealt with the space-time fractional (ADE) (8) for any arbitrary fractional order 0 < α, β ≤ 1 and 1 < γ ≤ 2 under the general initial condition u(x, 0) = f (x).
with the initial condition u(x, 0) = e −x (11) equation ( 10) can be written in the operator form as Applying the Adomian decomposition method, the solution u(x, t) can be given by the infinite series where each term of the series (13) will be determined recursively.So, the recursive relation can be written as So, the mth term takes the form Substituting ( 15) into (13), we have the required solution as where E α (x) denotes the Mittag-Leffler Function in one parameter, see [24]. Figure 1: The concentration u(x, t) of equation ( 16).
Remark 1.If α = 1, the solution ( 16) converges to the exact solution e (1+µ)t−x which is the same result obtained by El-Sayed in [13].

Case II: Space-time fractional ADE (γ = 2β)
Consider the space-time fractional ADE with the initial condition (9).So, the equivalent Adomian's recurrence relation of equation ( 17) is Applying this recurrence relation, we can get the first four terms of u n as follows Consequently, the m-th term can be calculated as Combining ( 19) with ( 13) we obtain the solution of equation ( 17) and (18a) as follows Remark 2. As a particular situation, let f (x) = e −x then we have where [x] denotes the ceiling function.
So, the first four terms of the Adomian's recursive relation can be written as In view of equation ( 13), we have which also can be obtained in the equivalent form Remark 4. Setting γ = 2β, the solution (24) reduces to equation (20).
Proof.γ = 2β, the solution ( 24) is equivalent to By Definition 2.4 and the binomial theorem, it easy to see

Test Examples
In this section, we give some examples for the different cases that we give their analytic solution in the previous sections.
Example 1.According to the relation (24), if α = 1, β = 1, and γ = 2 with the initial condition u(x, 0) = e −x .We get the solution to be in the form =e −x+(1+µ)t , which is the exact solution of the classical Advection-dispersion equation ( 1) under the initial condition u(x, 0) = e −x which is obtained in [13], see Figure 1a.

Conclusion
The main aim of this work is to use the Adomian's decomposition method (ADM) to solve the space-time fractional advection-dispersion equation (FADE).Different three cases of (FADE) based on the order of the derivatives were discussed.General formulas of the solutions for these three cases are analyzed under any initial conditions.These solutions are always given in terms of Mittag-Leffler functions.(ADM) is direct and powerful to get the exact solution.Finally, some examples are given to illustrate that our results generalize some important considerable works.Moreover, a graphical solution for an example worked out using Matlab program is given.