The Foam Drainage Equation with Time- and Space-Fractional Derivatives Solved by The Adomian Method

In this paper, by introducing the fractional derivative in the sense of Caputo, we apply the Adomian decomposition method for the foam drainage equation with time- and space-fractional derivative. As a result, numerical solutions are obtained in a form of rapidly convergent series with easily computable components.


Introduction
Since the introduction by Adomian of the decomposition method [4,5] at the begin of 1980s, the algorithm has been widely used for obtaining analytic solutions of physically significant equations [4,5,6,16,26,27,28,29,30,31].With this method, we can easily obtain approximate solutions in the form of a rapidly convergent infinite series with each term computed conveniently.As EJQTDE, 2008 No. 30, p. 1 it is known, for the nonlinear equations with derivatives of integer order, many methods are used to derive approximation solutions [3,4,5,7,10].However, for the fractional differential equations, there are only limited approaches, such as Laplace transform method [21], the Fourier transform method [17], the iteration method [22] and the operational method [20,23].In recent years, the fractional differential equations have attracted great attention; they are used in many areas of physics and engineering [9,15,32], like phenomena in electromagnetic theory, acoustics, electrochemistry and material science [11,21,22,32].The study to foam drainage equation is very significant for that the equation is a simple model of the flow of liquid through channels ( Plateau borders [34] ) and nodes ( intersection of four channels) between the bubbles, driven by gravity and capillarity [18,33].It has been studied by many authors [12,13,24].However, as we know, the study for the foam drainage equation with time-and space-fractional derivatives of this form by the Adomian method (ADM) has not been investigated.Here α and β are the parameters standing for the order of the fractional time and space derivatives, respectively and they satisfy 0 < α, β ≤ 1 and x > 0. In fact, different response equations can be obtained when at lest one of the parameters varies.When α = β = 1, the fractional equation reduces to the foam drainage equation of the form We introduce Caputo fractional derivative and apply the ADM to derive numerical solutions of the foam drainage equation with time-and space-fractional derivatives.
The paper is organized as follows.In Sec.II, some necessary details on the fractional calculus are provided.In Sec.III, the foam drainage equation with time and space-fractional derivative is studied with the ADM.Finally, conclusions follow.

Description of Fractional Calculus
There are several mathematical definitions about fractional derivative [21,22].
Here, we adopt the two usually used definitions: the Caputo and its reverse operator Riemann-Liouville.That is because Caputo fractional derivative allows traditional initial condition assumption and boundary conditions.More details one can consults [21].In the following, we will give the necessary notation and basic definitions.
Definition 2.1 A real valued function f (x), x > 0 is said to be in the space C µ , µ ∈ if there exists a real number p > µ such that f (x) = x p f 1 (x) where EJQTDE, 2008 No. 30, p. 2 Definition 2.3 The Riemann-Liouville fractional integral operator of order α ≥ 0, for a function f ∈ C µ , (µ ≥ −1) is defined as For the convenience of establishing the results for the fractional foam drainage equation, we give one basic property For the expression (3), when f (x) = x β we get another expression that will be used later: Definition 2.4 The fractional derivative of f ∈ C n −1 in the Caputo's sense is defined as According to the Caputo's derivative, we can easily obtain the following expressions: Details on Caputo's derivative can be found in [21].
Remark 2.1 In this paper, we consider equation ( 1) with time-and space-fractional derivative.When α ∈ + , we have: The form of the space fractional derivative is similar to the above and we just omit it here.
Consider the foam drainage equation with time and space-fractional derivatives Eq.( 1).In order to solve numerical solutions for this equation by using ADM method, we rewrite it in the operator form where the operators D α t and D β x stand for the fractional derivative and are defined as in (6).Take the initial condition as Applying the operator J α , the inverse of D α on corresponding sub-equation of Eq.( 10), using the initial condition (11), yields: where Φ Following Adomian decomposition method [4,5], the solution is represented as infinite series like The nonlinear operators Φ 1 (u), Φ 2 (u) and Φ 3 (u) are decomposed in these forms where A n , B n and C n are the so-called Adomian polynomials and have the form .
(15) In fact, these Adomian polynomials can be easily calculated.Here we give the first three components of these polynomials: EJQTDE, 2008 No. 30, p. 4 The first three components of B n are and those of C n are given by Other polynomials can be generated in a like manner.Substituting the decomposition series ( 13) and ( 14) into Eq.(12), yields the following recursive formula: The Adomian decomposition method converges generally very quickly.Details about its convergence and convergence speed can be found in [1,2,8,14].Here, according to the above steps, we will derive the numerical solution for the equation with time and space-fractional derivative in details.

Numerical Solutions of Time-Fractional Foam Drainage Equation
Consider the following form of the time-fractional equation with the initial condition where c is the velocity of wavefront [23].
The exact solution of (20) for the special case EJQTDE, 2008 No. 30, p. 5 In order to obtain numerical solution of equation ( 20), substituting the initial condition ( 21) and using the Adomian polynomials (16,17,18) into the expression (19), we can compute the results.For simplicity, we only give the first few terms of series: where Then we can have the numerical solution of time-fractional equation ( 20) under the series form In order to check the efficiency of the proposed ADM for the equation (20), we draw figures for the numerical solutions with α = 1 2 as well as the exact solution (22) when α = β = 1. Figure 1(a) stands for the numerical solution of (25). Figure 1(b) shows the exact solution of equation (22).From these figures, we can appreciate how closely are the two solutions.This is to say that good approximations are achieved using the ADM method.

Numerical Solutions of Space-Fractional Foam Drainage Equation
In this section, we will take the space-fractional equation as another example to illustrate the efficiency of the method.As the main computation method is the same as the above, we will omit the heavy calculation and only give some necessary expressions.
Considering the operator form of the space-fractional equation Assuming the condition as Initial condition has been taken as the above polynomial to avoid heavy calculation of fractional differentiation.

Conclusion
In this paper, the ADM has been successfully applied to derive explicit numerical solutions for the time-and space-fractional foam drainage equation.The above procedure shows that the ADM method is efficient and powerful in solving wide classes of equations in particular evolution fractional order equations.