Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 30, 1-10; http://www.math.u-szeged.hu/ejqtde/

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

Adomian Method

Zoubir Dahmani Department of Mathematics, Mostaganem University, Mostaganem, Algeria Email : zzdahmani@yahoo.fr

Mohammed Mostefa Mesmoudi IUFM, University Paris 12, Créteil, France Email : mesmoudi@ac-creteil.fr

Rachid Bebbouchi Faculty of Mathematics, Houari Boumedienne University of Sciences and Technology, Algiers, Algeria Email : rbebbouchi@hotmail.com

Abstract 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.

Keywords Caputo Fractional derivative, Adomian method, Foam Drainage Equation, Evolution Equations. AMS Subject Classifications[2000]: 26A33, 35F25

1 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 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

D_{t}^{α}u − \frac{1}{2}uu_{xx} + 2u^{2}D_{x}^{β}u − (D_{x}^{β}u)^{2} = 0,  (1)

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

u_{t} − \frac{1}{2}uu_{xx} + 2u^{2}u_{x} − (u_{x})^{2} = 0.  (2)

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.

2 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.

\underline{Definition} 2.1 A real valued function f(x), x > 0 is said to be in the space C_{μ}, μ ∈ ℝ if there exists a real number p > μ such that f(x) = x^{p}f_{1}(x) where f_{1}(x) ∈ C([0, ∞)).

\underline{Definition} 2.2 A function f(x), x > 0 is said to be in the space C_{μ}^{n}, n ∈ ℕ, if f^{(n)} ∈ C_{μ}.

\underline{Definition} 2.3 The Riemann-Liouville fractional integral operator of order α ≥ 0, for a function f ∈ C_{μ}, (μ ≥ −1) is defined as

\begin{matrix}J^{α}f(x) = \frac{1}{Γ(α)} ∫_{0}^{x}(x − t)^{α−1}f(t)dt; α > 0, x > 0 \\ J^{0}f(x) = f(x). \end{matrix}  (3)

For the convenience of establishing the results for the fractional foam drainage equation, we give one basic property

J^{α}J^{β}f(x) = J^{α+β}f(x).  (4)

For the expression (3), when f(x) = x^{β} we get another expression that will be used later:

J^{α}x^{β} = \frac{Γ(β + 1)}{Γ(α + β + 1)}x^{α+β}.  (5)

\underline{Definition} 2.4 The fractional derivative of f ∈ C_{−1}^{n} in the Caputo’s sense is defined as

D^{α}f(t) = \{ \begin{matrix}\frac{1}{Γ(n−α)} ∫_{0}^{t}(t − τ)^{n−α−1}f^{(n)}(τ)dτ, & n − 1 < α < n, n ∈ ℕ^{∗}, \\ \frac{d^{n}}{dt^{n}} f(t), & α = n. \end{matrix} (6)

According to the Caputo’s derivative, we can easily obtain the following expressions:

D^{α}K = 0; K is a constant,  (7)
D^{α}t^{β} = \{ \begin{matrix}\frac{Γ(β+1)}{Γ(α+β−1)}t^{β−α}, & β > α − 1, \\ 0, & β ≤ α − 1. \end{matrix}  (8)

Details on Caputo’s derivative can be found in [21].

\underline{Remark} 2.1 In this paper, we consider equation (1) with time-and space-fractional derivative. When α ∈ ℝ^{+}, we have:

D_{t}^{α}u(x, t) = \frac{∂^{α}u(x, t)}{∂t^{α}} = \{ \begin{matrix}\frac{1}{Γ(n−α)} ∫_{0}^{t}(t − τ)^{n−α−1} \frac{∂^{n}u(x,τ)}{∂τ^{n}}dτ, & n − 1 < α < n \\ \frac{∂^{n}u(x,t)}{∂t^{n}}, & α = n. \end{matrix}

(9)

The form of the space fractional derivative is similar to the above and we just omit it here.

3 Applications of the ADM Method

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

D_{t}^{α}u = \frac{1}{2}uu_{xx} − 2u^{2}D_{x}^{β}u + (D_{x}^{β}u)^{2}; 0 < α ≤ 1, 0 < β ≤ 1,  (10)

where the operators D_{t}^{α} and D_{x}^{β} stand for the fractional derivative and are defined as in (6). Take the initial condition as

u(x, 0) = f(x).  (11)

Applying the operator J^{α}, the inverse of D^{α} on corresponding sub-equation of Eq.(10), using the initial condition (11), yields:

u(x, t) = f(x) − 2J^{α}Φ_{1}(u(x, t)) + \frac{1}{2}J^{α}Φ_{2}(u(x, t)) + J^{α}Φ_{3}(u(x, t)), (12)

where Φ_{1}(u) = u^{2}D_{x}^{β}u , Φ_{2}(u) = uu_{xx} and Φ_{3}(u) = (D_{x}^{β}u)^{2}. Following Adomian decomposition method [4, 5], the solution is represented as infinite series like

u(x, t) = ∑_{n=0}^{∞} u_{n}(x, t).  (13)

The nonlinear operators Φ_{1}(u), Φ_{2}(u) and Φ_{3}(u) are decomposed in these forms

Φ_{1}(u) = ∑_{n=0}^{∞} A_{n}, Φ_{2}(u) = ∑_{n=0}^{∞} B_{n}, Φ_{3}(u) = ∑_{n=0}^{∞} C_{n},  (14)

where A_{n}, B_{n} and C_{n} are the so-called Adomian polynomials and have the form

A_{n} = \frac{1}{n!} \frac{d^{n}}{dλ^{n}} [Φ_{1}( ∑_{k=0}^{∞} λ^{k}u_{k})]_{λ=0} = \frac{1}{n!} \frac{d^{n}}{dλ^{n}} [( ∑_{k=0}^{∞} λ^{k}u_{k})^{2}D_{x}^{β}( ∑_{k=0}^{∞} λ^{k}u_{k})]_{λ=0},

B_{n} = \frac{1}{n!} \frac{d^{n}}{dλ^{n}} [Φ_{2}( ∑_{k=0}^{∞} λ^{k}u_{k})]_{λ=0} = \frac{1}{n!} \frac{d^{n}}{dλ^{n}} [( ∑_{k=0}^{∞} λ^{k}u_{k})( ∑_{k=0}^{∞} λ^{k}u_{kxx})]_{λ=0},

C_{n} = \frac{1}{n!} \frac{d^{n}}{dλ^{n}} [Φ_{3}( ∑_{k=0}^{∞} λ^{k}u_{k})]_{λ=0} = \frac{1}{n!} \frac{d^{n}}{dλ^{n}} [(D_{x}^{β}(∑_{k=0}^{∞} λ^{k}u_{k}))^{2}]_{λ=0}.

(15)

In fact, these Adomian polynomials can be easily calculated. Here we give the first three components of these polynomials:

\begin{matrix}A_{0} = u_{0}^{2}D_{x}^{β}u_{0}, \\ A_{1} = 2u_{0}u_{1}D_{x}^{β}u_{0} + u_{0}^{2}D_{x}^{β}u_{1}, \\ A_{2} = u_{1}^{2}D_{x}^{β}u_{0} + 2u_{0}u_{2}D_{x}^{β}u_{0} + 2u_{0}u_{1}D_{x}^{β}u_{1} + u_{0}^{2}D_{x}^{β}u_{2}, \\ A_{3} = u_{1}^{2}D_{x}^{β}u_{1} + 2u_{1}u_{2}D_{x}^{β}u_{0} + 2u_{0}u_{2}D_{x}^{β}u_{1} + 2u_{0}u_{1}D_{x}^{β}u_{2} \\ +2u_{0}u_{3}D_{x}^{β}u_{0} + u_{0}^{2}D_{x}^{β}u_{0}^{2}D_{x}^{β}u_{3}. \end{matrix}  (16)

The first three components of B_{n} are

\begin{matrix}B_{0} = u_{0}u_{0xx}, \\ B_{1} = u_{0}u_{1xx} + u_{1}u_{0xx}, \\ B_{2} = u_{2}u_{0xx} + u_{0}u_{2xx} + u_{1}u_{1xx}, \\ B_{3} = u_{3}u_{0xx} + u_{0}u_{3xx} + u_{2}u_{1xx} + u_{1}u_{2xx}, \end{matrix}  (17)

and those of C_{n} are given by

\begin{matrix}C_{0} = (D_{x}^{β}u_{0})^{2}, \\ C_{1} = 2D_{x}^{β}u_{0}D_{x}^{β}u_{1}, \\ C_{2} = (D_{x}^{β}u_{1})^{2} + 2D_{x}^{β}u_{0}D_{x}^{β}u_{2}, \\ C_{3} = 2D_{x}^{β}u_{1}D_{x}^{β}u_{2} + 2D_{x}^{β}u_{0}D_{x}^{β}u_{3}. \end{matrix}  (18)

Other polynomials can be generated in a like manner. Substituting the decomposition series (13) and (14) into Eq.(12), yields the following recursive formula:

\begin{matrix}u_{0}(x, t) = f(x), \\ u_{n+1}(x, t) = −2J^{α}(A_{n}) + \frac{1}{2}J^{α}(B_{n}) + J^{α}(C_{n}); n ≥ 0. \end{matrix}  (19)

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.

3.1 Numerical Solutions of Time-Fractional Foam Drainage Equation

Consider the following form of the time-fractional equation

D_{t}^{α}u = \frac{1}{2}uu_{xx} − 2u^{2}u_{x} + u_{x}^{2}; 0 < α ≤ 1,  (20)

with the initial condition

u(x, 0) = f(x) = −\sqrt{c} tanh \sqrt{c}(x).  (21)

where c is the velocity of wavefront [23]. The exact solution of (20) for the special case α = β = 1 is

u(x, t) = \{ \begin{matrix}−\sqrt{c} tanh(\sqrt{c}(x − ct)); & x ≤ ct \\ 0; & x > ct. \end{matrix}  (22)

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:

\begin{matrix}u_{0} = f(x), \\ u_{1} = −2J^{α}(A_{0}) + \frac{1}{2}J^{α}(B_{0}) + J^{α}(C_{0}) \\ = −2J^{α}(u_{0}^{2}u_{0x}) + \frac{1}{2}J^{α}(u_{0}u_{0xx}) + J^{α}(u_{0x}^{2}) = f_{1} \frac{t^{α}}{Γ(α+1)}, \\ u_{2} = −2J^{α}(A_{1}) + \frac{1}{2}J^{α}(B_{1}) + J^{α}(C_{1}) \\ = −2J^{α}(2u_{0}u_{1}u_{0x} + u_{0}^{2}u_{1x}) + \frac{1}{2}J^{α}(u_{0}u_{1xx} + u_{1}u_{0xx}) + J^{α}(2u_{0x}u_{1x}) \\ = f_{2} \frac{t^{2α}}{Γ(2α+1)}, \end{matrix} (23)

where

f(x) = −\sqrt{c} tanh(\sqrt{c}x), f_{1}(x) = −2f^{2}f_{x} + \frac{1}{2}ff_{xx} + f_{x}^{2},

f_{2}(x) = −2(2ff_{x}f_{1} + f^{2}f_{1x}) + \frac{1}{2}ff_{1xx} + \frac{1}{2}f_{1}f_{xx} + 2f_{x}f_{1x},

f_{3}(x) = −2f_{x}f_{1}^{2} \frac{Γ(2α+1)}{Γ^{2}(α+1)} − 4ff_{x}f_{2} − 4ff_{1}f_{1x} \frac{Γ(2α+1)}{Γ^{2}(α+1)} − 2f^{2}f_{2x} + \frac{1}{2}f_{xx}f_{2}

+\frac{1}{2}ff_{2xx} + \frac{1}{2}f_{1}f_{1xx} \frac{Γ(2α+1)}{Γ^{2}(α+1)} + f_{1x}^{2} \frac{Γ(2α+1)}{Γ^{2}(α+1)} + 2f_{x}f_{2x}.

(24)

Then we can have the numerical solution of time-fractional equation (20) under the series form

u(x, t) = f(x) + f_{1}(x)\frac{t^{α}}{Γ(α + 1)} + f_{2}(x)\frac{t^{2α}}{Γ(2α + 1)} + f_{3}(x)\frac{t^{3α}}{Γ(3α + 1)} + ...

(25)

In order to check the efficiency of the proposed ADM for the equation (20), we draw figures for the numerical solutions with α = \frac{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.

3.2 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

u_{t} = \frac{1}{2}uu_{xx} − 2u^{2}D_{x}^{β}u + (D_{x}^{β}u)^{2}; 0 < β ≤ 1.  (26)

Assuming the condition as

u(x, 0) = f(x) = x^{2}.  (27)

(a)  (b)

Figure 1: Representing time-fractional solutions of Eq.(20). In (a), solution obtained by the Adomian method. In (b), the exact solution (22).

Initial condition has been taken as the above polynomial to avoid heavy calculation of fractional differentiation.

In order to estimate the numerical solution of equation (26), substituting (14), (15) and the initial condition (27) into (19), we get the Adomian solution. Here, we give the first few terms of the series solution:

u_{0} = x^{2},
u_{1} = −2J(A_{0}) + \frac{1}{2}J(B_{0}) + J(C_{0})
= −2J(u_{0}^{2}D_{x}^{β}u_{0}) + \frac{1}{2}J(u_{0}u_{0xx}) + J(D_{x}^{β}u_{0})^{2}
= (f_{1}x^{6−β} + x^{2} + f_{2}x^{4−2β})t,
u_{2} = −2J(A_{1}) + \frac{1}{2}J(B_{1}) + J(C_{1})

= −2J(2u_{0}u_{1}D_{x}^{β}u_{0} + u_{0}^{2}D_{x}^{β}u_{1}) + \frac{1}{2}J(u_{0}u_{1xx} + u_{1}u_{0xx}) + J(2D_{x}^{β}u_{0}D_{x}^{β}u_{1})

= (f_{3}x^{10−2β} + f_{4}x^{8−3β} + f_{5}x^{6−β} + f_{6}x^{6−4β} + f_{7}x^{4−2β} + x^{2})\frac{t^{2}}{2} ,

(28)

where

f(x) = x^{2}, f_{1} = −2\frac{Γ(3)}{Γ(3 − β)}, f_{2} = [\frac{Γ(3)}{Γ(3 − β)}]^{2},

f_{3} = [\frac{Γ(7−β)}{Γ(7−2β)} − 4\frac{Γ(3)}{Γ(3−β)}]f_{1},

f_{4} = (\frac{Γ(5−2β)}{Γ(5−3β)} − 4\frac{Γ(3)}{Γ(3−β)})f_{2} + 2(\frac{Γ(3)}{Γ(3−β)} \frac{Γ(7−β)}{Γ(7−2β)})f_{1}

f_{5} = ((6 − β)(5 − β) + 2)\frac{f_{1}}{2} − 3\frac{Γ(3)}{Γ(3−β)},
f_{6} = 2\frac{Γ(3)}{Γ(3−β)} \frac{Γ(5−2β)}{Γ(5−3β)}f_{2},
f_{7} = ((4 − 2β)(3 − 2β) + 6)\frac{f_{2}}{2} .

(29)

Then we obtain a numerical solution of space-fractional Eq.(26) in series form

\begin{matrix}u(x, t) = x^{2} + (f_{1}x^{6−β} + f_{2}x^{4−2β} + x^{2})t+ \\ +(f_{3}x^{10−2β} + f_{4}x^{8−3β} + f_{5}x^{6−β} + f_{6}x^{6−4β} + f_{7}x^{4−2β} + x^{2})\frac{t^{2}}{2} + ... \end{matrix} (30)

Figures 2(a, b) show, respectively, the numerical solutions given by expression (30) for space-fractional Eq.(26) with β = \frac{1}{2} and β = 1. From these figures, we can appreciate the convergence rapidity of Adomian solutions.

(a)  (b)

Figure 2: Representing space-fractional solutions of Eq.(26). In (a), solution obtained by the Adomian method for β = 1/2. In (b), solution obtained by the Adomian method for β = 1.

4 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.

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(Received May 30, 2008)