Homotopy Analysis Method for Solving Foam Drainage Equation with Space- and Time-Fractional Derivatives

Abstract

The analytical solution of the foam drainage equation with time- and space-fractional derivatives was derived by means of the homotopy analysis method (HAM). The fractional derivatives are described in the Caputo sense. Some examples are given and comparisons are made; the comparisons show that the homotopy analysis method is very effective and convenient. By choosing different values of the parameters 𝛼,𝛽 in general formal numerical solutions, as a result, a very rapidly convergent series solution is obtained.

1. Introduction

Many phenomena in engineering, physics, chemistry, and other science can be described very successfully by models using the theory of derivatives and integrals of fractional order. Interest in the concept of differentiation and integration to noninteger order has existed since the development of the classical calculus [1–3]. By implication, mathematical modeling of many physical systems are governed by linear and nonlinear fractional differential equations in various applications in fluid mechanics, viscoelasticity, chemistry, physics, biology, and engineering.

Since many fractional differential equations are nonlinear and do not have exact analytical solutions, various numerical and analytic methods have been used to solve these equations. The Adomian decomposition method (ADM) [4], the homotopy perturbation method (HPM) [5], the variational iteration method (VIM) [6], and other methods have been used to provide analytical approximation to linear and nonlinear problems [7, 8]. However, the convergence region of the corresponding results is rather small.

In 1992, Liao [9–13] employed the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely, Homotopy Analysis Method (HAM). This method has been successfully applied to solve many types of nonlinear problems in science and engineering, such as the viscous flows of non-Newtonian fluids [14], the KdV-type equations [15], higher-dimensional initial boundary value problems of variable coefficients [16], and finance problems [17]. The HAM contains a certain auxiliary parameter ℎ which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution.

The HAM offers certain advantages over routine numerical methods. Numerical methods use discretization which gives rise to rounding off errors causing loss of accuracy and requires large computer memory and time. This computational method yields analytical solutions and has certain advantages over standard numerical methods. The HAM method is better since it does not involve discretization of the variables and hence is free from rounding off errors and does not require large computer memory or time.

The study of foam drainage equation is very significant for that the equation is a simple model of the flow of liquid through channels (Plateau borders [18]) and nodes (intersection of four channels) between the bubbles, driven by gravity and capillarity [19]. It has been studied by many authors [20–22]. The study for the foam drainage equation with time and space-fractional derivatives of this form𝐷^𝛼_𝑡1𝑢=2𝑢𝑢_𝑥𝑥−2𝑢²𝐷^𝛽_𝑥𝑢+𝐷^𝛽_𝑥𝑢²,0<𝛼,𝛽≤1,𝑥>0, (1.1)

has been investigated by the ADM and VIM method in [23, 24]. The fractional derivatives are considered in the Caputo sense. When 𝛼=𝛽=1, the fractional equation reduces to the foam drainage equation of the form𝑢_𝑡=12𝑢𝑢_𝑥𝑥−2𝑢²𝑢_𝑥+𝑢_𝑥². (1.2)

In this paper, we extend the application of HAM to obtain analytic solutions to the space- and time-fractional foam drainage equation. Two cases of special interest such as the time-fractional foam drainage equation and the space-fractional foam drainage equation are discussed in details. Further, we give comparative remarks with the results obtained using ADM and VIM method (see [23, 24]).

The paper has been organized as follows. Notations and basic definitions are given in Section 2. In Section 3 the homotopy analysis method is described. In Section 4 we extend the method to solve the space- and time-fractional foam drainage equation. Discussion and conclusions are presented in Section 5.

2. Description on the Fractional Calculus

Definition 2.1. A real function 𝑓(𝑡),𝑡>0 is said to be in the space 𝐶_𝜇,𝜇∈𝑅 if there exists a real number 𝑝>𝜇, such that 𝑓(𝑡)=𝑡^𝑝𝑓₁(𝑡) where 𝑓₁∈(0,∞), and it is said to be in the space 𝐶^𝜇_𝑛 l if and only if ℎ(𝑛)∈𝐶_𝜇,𝑛∈𝑁. Clearly 𝐶_𝜇⊂𝐶_𝜈 if 𝜈≤𝜇.

Definition 2.2. The Riemann-Liouville fractional integral operator (𝐽^𝛼) of order 𝛼≥0, of a function 𝑓∈𝐶_𝜇,𝜇≥−1, is defined as 𝐽^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝐽𝑑𝑡,𝛼>0,𝑥>0,₀𝑓(𝑥)=𝑓(𝑥). (2.1)Γ(𝛼) is the well-known Gamma function. Some of the properties of the operator 𝐽^𝛼, which we will need here, are as follows:
for 𝑓∈𝐶_𝜇,𝜇≥−1,𝛼,𝛽≥0 and 𝛾≥−1,𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛼+𝛽𝑓(𝑥),𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛽𝐽^𝛼𝑓(𝑥),𝐽^𝛼𝑡^𝛾=Γ𝛾+1Γ𝑡𝛼+𝛾+1^𝛼+𝛾. (2.2)

Definition 2.3. For the concept of fractional derivative, there exist many mathematical definitions [2, 25–28]. In this paper, the two most commonly used definitions: the Caputo derivative and its reverse operator Riemann-Liouville integral are adopted. That is because Caputo fractional derivative [2] allows the traditional assumption of initial and boundary conditions. The Caputo fractional derivative is defined as 𝐷^𝛼_𝑡𝑢()=𝜕𝑥,𝑡^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=1Γ(𝑛−𝛼)^𝑡₀(𝑡−𝜏)^{𝑛−𝛼−1}𝜕^𝑛𝑢(𝑥,𝑡)𝜕𝑡^𝑛𝜕𝑑𝜏,𝑛−1<𝛼<𝑛,_𝑛𝑢(𝑥,𝑡)𝜕𝑡^𝑛,𝛼=𝑛∈𝑁. (2.3) Here, we also need two basic properties about them: 𝐷^𝛼𝐽^𝛼𝑓(𝑥)(𝑥),𝐽=𝑓_𝛼𝐷^𝛼𝑓(𝑥)=𝑓(𝑥)−^∞_𝑘=0𝑓^(𝑘)0⁺𝑥^𝑘𝑘!,𝑥>0. (2.4)

Definition 2.4. The MittagLeffler function 𝐸_𝛼(𝑧) with 𝑎>0 is defined by the following series representation, valid in the whole complex plane: 𝐸_𝛼(𝑧)=^∞_𝑛=0𝑧^𝑛Γ(𝛼𝑛+1),𝛼>0,𝑧∈𝐶. (2.5)

3. Basic Idea of HAM

To describe the basic ideas of the HAM, we consider the following differential equation:𝑁𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=0,𝑡>0, (3.1)

where 𝑁 is nonlinear operator, 𝐷^𝛼_𝑡 stand for the fractional derivative and is defined as in (2.3), 𝑥,𝑡 denotes independent variables, and 𝑢(𝑥,𝑡) is an unknown function, respectively.

By means of generalizing the traditional homotopy method, Liao [9] constructs the so-called zero-order deformation equation1−𝑞𝐿𝜙𝑥,𝑡,𝑞−𝑢₀(𝑥,𝑡)=𝑞ℎ𝐻(𝑡)𝑁𝐷^𝛼_𝑡𝜙𝑥,𝑡,𝑞, (3.2)

where 𝑞∈[0,1] is the embedding parameter, ℎ≠0 is a nonzero auxiliary parameter, 𝐻(𝑡)≠0 is an auxiliary function, 𝐿 is an auxiliary linear operator, 𝑢₀(𝑥,𝑡) is initial guesse of 𝑢(𝑥,𝑡), and 𝜙(𝑥,𝑡,𝑞) is unknown function. It is important that one has great freedom to choose auxiliary things in HAM. Obviously, when 𝑞=0 and 𝑞=1, it holds that 𝜙(𝑥,𝑡,0)=𝑢₀(𝑥,𝑡),𝜙(𝑥,𝑡,1)=𝑢(𝑥,𝑡), (3.3)

respectively. Thus, as 𝑞 increases from 0 to 1, the solution 𝜙(𝑥,𝑡,𝑞) varies from the initial guess 𝑢₀(𝑥,𝑡) to the solution 𝑢(𝑥,𝑡). Expanding 𝜙(𝑥,𝑡,𝑞) in Taylor series with respect to 𝑞, we have𝜙𝑥,𝑡,𝑞=𝑢₀(𝑥,𝑡)+^+∞_𝑚=1𝑢_𝑚(𝑥,𝑡)𝑞^𝑚, (3.4)

where𝑢_𝑚()=1𝑥,𝑡𝜕𝑚!^𝑚𝜙𝑥,𝑡,𝑞𝜕𝑞^𝑚_𝑞=0. (3.5)

If the auxiliary linear operator, the initial guess, the auxiliary parameter ℎ, and the auxiliary function are so properly chosen, the series (3.4) converges at 𝑞=1, then we have𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+^+∞_𝑚=1𝑢_𝑚(𝑥,𝑡), (3.6)

which must be one of solutions of original nonlinear equation, as proved by Liao [11]. As ℎ=−1 and 𝐻(𝑡)=1, (3.2) becomes1−𝑞𝐿𝜙₁𝑥,𝑡,𝑞−𝑢₀(𝑥,𝑡)+𝑞𝑁𝐷^𝛼_𝑡𝜙𝑥,𝑡,𝑞=0, (3.7)

which is used mostly in the homotopy perturbation method [29], whereas the solution obtained directly, without using Taylor series. According to definition (3.5), the governing equation can be deduced from the zero-order deformation equation (3.2). Define the vector 𝑢_𝑛=𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),…,𝑢_𝑛(𝑥,𝑡). (3.8)

Differentiating (3.2) 𝑚 times with respect to the embedding parameter 𝑞 and then setting 𝑞=0 and finally dividing them by 𝑚!, we have the so-called 𝑚th-order deformation equation𝐿𝑢_𝑚(𝑥,𝑡)−𝜒_𝑚𝑢_𝑚−1(𝑥,𝑡)=ℎ𝐻(𝑡)𝑅_𝑚𝑢_𝑚−1, (3.9)

where𝑅_𝑚𝑢_𝑚−1=1()!𝜕𝑚−1^𝑚−1𝑁𝐷^𝛼_𝑡𝜙𝑥,𝑡,𝑞𝜕𝑞^𝑚−1_𝑞=0,𝜒_𝑚=0,𝑚⩽1,1,𝑚>1. (3.10)

Applying the Riemann-Liouville integral operator 𝐽^𝛼 on both side of (3.9), we have𝑢_𝑚(𝑥,𝑡)=𝜒_𝑚𝑢_𝑚−1(𝑥,𝑡)−𝜒_𝑚^𝑛−1_𝑖=0𝑢^𝑖_𝑚−10⁺𝑡^𝑖(𝑡)𝐽𝑖!+ℎ𝐻^𝛼𝑅_𝑚𝑢_𝑚−1. (3.11)

It should be emphasized that 𝑢_𝑚(𝑥,𝑡) for 𝑚⩾1 is governed by the linear equation (3.9) under the linear boundary conditions that come from original problem, which can be easily solved by symbolic computation software such as Matlab. For the convergence of the above method we refer the reader to Liao's work.

Liao [10] proved that, as long as a series solution given by the homotopy analysis method converges, it must be one of exact solutions. So, it is important to ensure that the solution series is convergent. Note that the solution series contain the auxiliary parameter ℎ, which we can choose properly by plotting the so-called ℎ-curves to ensure solution series converge.

Remark 3.1. The parameters 𝛼 and 𝛽 can be arbitrarily chosen as, integer or fraction, bigger or smaller than 1. When the parameter is bigger than 1, we will need more initial and boundary conditions such as 𝑢^′₀(𝑥,0),𝑢₀^′′(𝑥,0),… and the calculations will become more complicated correspondingly. In order to illustrate the problem and make it convenient for the readers, we only confine the parameter to [0,1] to discuss.

4. Application

In this section we apply this method for solving foam drainage equation with time- and space-fractional derivatives.

Example 4.1. Consider the following form of the time-fractional equation: 𝐷^𝛼_𝑡1𝑢=2𝑢𝑢_𝑥𝑥−2𝑢²𝑢_𝑥+𝑢²_𝑥,0<𝛼≤1,𝑥>0, (4.1) with the initial condition 𝑢(𝑥,0)=−√𝑐tanh√𝑐𝑥, (4.2) where 𝑐 is the velocity of wavefront [15].
The exact solution of (4.1) for the special case 𝛼=𝛽=1 is𝑢()=−√𝑥,𝑡𝑐tanh√𝑐(𝑥−𝑐𝑡),𝑥⩽𝑐𝑡,0,𝑥>𝑐𝑡. (4.3) For application of homotopy analysis method, in view of (4.1) and the initial condition given in (4.2), it in convenient to choose 𝑢₀(𝑥,𝑡)=−√𝑐tanh√𝑐𝑥, (4.4) as the initial approximate. We choose the linear operator 𝐿𝜙𝑥,𝑡;𝑞=𝐷^𝛼_𝑡, (4.5) with the property 𝐿(𝑐)=0, where 𝑐 is constant of integration. Furthermore, we define a nonlinear operator as 𝑁𝜙𝑥,𝑡,𝑞=𝐷^𝛼_𝑡𝜙−1𝑥,𝑡,𝑞2𝜙𝑥,𝑡,𝑞𝜙_𝑥𝑥𝑥,𝑡,𝑞+2𝜙𝑥,𝑡,𝑞²𝜙_𝑥−𝜙𝑥,𝑡,𝑞_𝑥𝑥,𝑡,𝑞². (4.6) We construct the zeroth-order and the 𝑚th-order deformation equations where 𝑅_𝑚𝑢_𝑚−1=𝐷^𝛼_𝑡𝑢_𝑚−1−12^𝑚−1_𝑘=0𝑢_𝑘𝑢_{𝑚−1−𝑘𝑥𝑥}+2^𝑚−1𝑘_𝑘=0𝑗=0𝑢_𝑗𝑢_𝑘−𝑗𝑢_{𝑚−1−𝑘𝑥}−^𝑚−1_𝑘=0𝑢_𝑘𝑥𝑢_{𝑚−1−𝑘𝑥}. (4.7) We now successively obtain 𝑢₁()=1𝑥,𝑡Γ()ℎ√𝛼+1−1+tanh𝑐𝑥²𝑡^𝛼𝑐²,𝑢₂()=1𝑥,𝑡Γ(𝛼+1)²√𝑐tanh√𝑐𝑥Γ(𝛼+1)²+ℎ√𝑐tanh√𝑐𝑥Γ(𝛼+1)²−ℎ𝑡^𝛼𝑐²Γ(𝛼+1)+ℎ𝑡^𝛼𝑐²Γ(𝛼+1)tanh√𝑐𝑥²−ℎ²𝑡^𝛼𝑐²Γ(𝛼+1)+ℎ²𝑡^𝛼𝑐²Γ(𝛼+1)tanh√𝑐𝑥²+ℎ²𝑡^2𝑎𝑐^7/2tanh√𝑐𝑥−2ℎ²𝑡^2𝑎𝑐^7/2tanh√𝑐𝑥³,⋮ (4.8) By taking 𝛼=1,ℎ=−1, we reproduce the solution of problem as follows: 𝑢()=1𝑥,𝑡Γ(𝛼+1)³−√𝑐tanh√𝑐𝑥Γ(𝛼+1)³+𝑡^𝛼𝑐²Γ(𝛼+1)²−𝑡^𝛼𝑐²Γ(𝛼+1)²tanh√𝑐𝑥²+2𝑡^2𝛼𝑐^7/2tanh√𝑐𝑥Γ(𝛼+1)−2𝑡^2𝛼𝑐^7/2tanh√𝑐𝑥³Γ(𝛼+1)+13𝑡^3𝛼𝑐⁵tanh√𝑐𝑥²−13𝑡^3𝛼𝑐⁵tanh√𝑐𝑥⁴+3𝑡^3𝛼𝑐⁵tanh√𝑐𝑥⁶−3𝑡^3𝛼𝑐⁵. (4.9) Figures 1 and 2 show the HAM and exact solutions of time-fractional foam drainage equation with ℎ=−1,𝑛=3,𝛼=1. It is obvious that, when 𝛼=1, the solution is nearly identical with the exact solution. Figures 3 and 4 show the approximate solutions of time-fractional foam drainage equation with ℎ=−1,𝑛=3,𝛼=0.5 and 𝛼=0.75, respectively.

Figure 1 

HAM solution with 𝛼=1.

Figure 2 

Exact solution.

Figure 3 

HAM solution with 1𝛼=2.

Figure 4 

HAM solution with 3𝛼=4.

Remark 4.2. This example has been solved using ADM and VIM in [23, 24]. The graphs drawn and Tables by ℎ=−1 are in excellent agreement with that graphs drawn with ADM and VIM.

Example 4.3. Considering the operator form of the space-fractional equation 𝑢_𝑡=12𝑢𝑢_𝑥𝑥−2𝑢²𝐷^𝛽_𝑥𝑢+𝐷^𝛽_𝑥𝑢²,0<𝛽≤1,𝑥>0, (4.10) with the initial condition 𝑢(𝑥,0)=𝑥². (4.11) For application of homotopy analysis method, in view of (4.10) and the initial condition given in (4.2), it is inconvenient to choose 𝑢₀(𝑥,𝑡)=𝑥². (4.12) Initial condition has been taken as the above polynomial to avoid heavy calculation of fractional differentiation.We choose the linear operator 𝐿𝜙𝑥,𝑡,𝑞=𝜕𝜙𝑥,𝑡,𝑞,𝜕𝑡 (4.13) with the property 𝐿(𝑐)=0, where 𝑐 is constant of integration. Furthermore, we define a nonlinear operator as 𝑁𝜙𝑥,𝑡,𝑞=𝜙_𝑡−1𝑥,𝑡,𝑞2𝜙𝑥,𝑡,𝑞𝜙_𝑥𝑥𝑥,𝑡,𝑞+2𝜙𝑥,𝑡,𝑞²𝐷^𝛽_𝑥𝜙−𝐷𝑥,𝑡,𝑞_𝛽𝑥𝜙𝑥,𝑡,𝑞². (4.14) We construct the zeroth-order and the 𝑚th-order deformation equations where 𝑅_𝑚𝑢_𝑚−1=𝑢_𝑡𝑚−1−12^𝑚−1_𝑘=0𝑢_𝑘𝑢_{𝑚−1−𝑘𝑥𝑥}+2^𝑚−1𝑘_𝑘=0𝑗=0𝑢_𝑗𝑢_𝑘−𝑗𝐷^𝛽_𝑥𝑢_{𝑚−1−𝑘}−^𝑚−1_𝑘=0𝐷^𝛽_𝑥𝑢_𝑘𝐷^𝛽_𝑥𝑢_{𝑚−1−𝑘}. (4.15) We now successively obtain 𝑢₁(𝑥,𝑡)=−ℎ𝑥²𝑡+4ℎ𝑥^6−𝛽𝑡Γ3−𝛽−4ℎ𝑥^4−2𝛽𝑡Γ3−𝛽²,𝑢₂()=1𝑥,𝑡Γ3−𝛽²−14ℎ²𝑡²𝑥^4−2𝛽𝛽+4ℎ²𝑡²𝑥^4−2𝛽𝛽²+18ℎ𝑥^4−2𝛽ℎ𝑡²−4ℎ²𝑡𝑥^4−2𝛽−4ℎ𝑡𝑥^4−2𝛽+ℎ²𝑥^8−3𝛽𝜋^1/2−4+𝛽−5+𝛽−6+𝛽𝑡²4^𝛽()+8Γ7/2−𝛽128ℎ²𝑥^6−4𝛽4^−𝛽𝑡²Γ(5/2)−𝛽𝜋^1/2Γ+15−3𝛽Γ3−𝛽−ℎ²𝑥^10−2𝛽𝜋^1/2−4+𝛽−5+𝛽−6+𝛽𝑡²4^𝛽)−16Γ(7/2−𝛽64ℎ²𝑥^8−3𝛽4^−𝛽𝑡²Γ(5/2)−𝛽𝜋^1/2Γ5−3𝛽+4ℎ𝑡𝑥^6−𝛽+4ℎ²𝑡𝑥^6−𝛽−38ℎ²𝑡62𝑥^6−𝛽+11ℎ²𝑡²𝛽²𝑥^6−𝛽−16ℎ²𝑡²𝑥^8−3𝛽Γ3−𝛽³−ℎ𝑥²𝑡−ℎ²𝑥²𝑡+𝑡²𝑥²ℎ².⋮ (4.16) Figures 5 and 6 show the HAM solutions of space-fractional foam drainage equation with ℎ=−1,𝑛=3,𝛽=0.5 and 𝛽=1, respectively.

Figure 5 

HAM solution with 𝛽=1/2.

Figure 6 

HAM solution with 𝛽=1.

Remark 4.4. This example has been solved using ADM and VIM in [23, 24]. The graphs drawn and Tables by ℎ=−1 are in excellent agreement with that graphs drawn with ADM and VIM.

5. Conclusion

In this paper, we have successfully developed HAM for solving space- and time-fractional foam drainage equation. HAM provides us with a convenient way to control the convergence of approximation series by adapting ℎ, which is a fundamental qualitative difference in analysis between HAM and other methods. The obtained results demonstrate the reliability of the HAM and its wider applicability to fractional differential equation. It, therefore, provides more realistic series solutions that generally converge very rapidly in real physical problems.

Matlab has been used for computations in this paper.