IJDEInternational Journal of Differential Equations1687-96511687-9643Hindawi Publishing Corporation86402310.1155/2011/864023864023Research ArticleApplication of the Homotopy Perturbation Method for Solving the Foam Drainage Equation FereidoonAbdolhoseinYaghoobiHessameddinDavoudabadiMohammadrezaGanjiD. D.Faculty of Mechanical EngineeringSemnan UniversitySemnanIransemnan.ac.ir201138201120110805201104062011090620112011Copyright © 2011 Abdolhosein Fereidoon et al.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.

We use homotopy perturbation method (HPM) to handle the foam drainage equation. Foaming occurs in many distillation and absorption processes. The drainage of liquid foams involves the interplay of gravity, surface tension, and viscous forces. The concept of He's homotopy perturbation method is introduced briefly for applying this method for problem solving. The results of HPM as an analytical solution are then compared with those derived from Adomian's decomposition method (ADM) and the variational iteration method (VIM). The results reveal that the HPM is very effective and convenient in predicting the solution of such problems, and it is predicted that HPM can find a wide application in new engineering problems.

1. Introduction

Most scientific problems and physical phenomena occur nonlinearly. Except in a limited number of these problems, finding the exact analytical solutions of such problems are rather difficult. Therefore, there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions . In recent years, several such techniques have drawn special attention, such as Hirota’s bilinear method , the homogeneous balance method [3, 4], inverse scattering method , Adomian’s decomposition method (ADM) [6, 7], the variational iteration method (VIM) [8, 9], and homotopy analysis method (HAM)  as well as homotopy perturbation method (HPM).

The homotopy perturbation method was established by He . The method has been used by many authors to handle a wide variety of scientific and engineering applications to solve various functional equations. In this method, the solution is considered as the sum of an infinite series, which converges rapidly to accurate solutions. Recently, considerable research work has been conducted in applying this method to the linear and nonlinear equations .

Schematic illustration of the interdependence of drainage, coarsening, and rheology of foams .

The pursuit of analytical solutions for foam drainage equation is of intrinsic scientific interest. To the best of the authors’ knowledge, there is no paper that has solved the nonlinear foam drainage equation by HPM. In this paper, the basic idea of HPM is described, and then, it is applied to study the following nonlinear foam drainage equation . Finally, the results of HPM as an analytical solution are then compared with those derived from Adomian’s decomposition method  and the variational iteration method .

2. Basic Idea of Homotopy Perturbation Method

To explain this method, let us consider the following function:A(u)-f(r)=0,rΩ, with the boundary conditions ofB(u,un)=0,rΓ, where A is a general differential operator f(r) is a known analytic function, B is a boundary operator, and Γ is the boundary of the domain Ω. The operator A can be generally divided into two operators, L and N, where L is a linear and N a nonlinear operator. Equation (2.1) can be, therefore, written as follows:L(u)+N(u)-f(r)=0. Using the homotopy technique, we constructed a homotopy v(r,p):Ω×[0,1]R, which satisfiesH(v,p)=(1-p)[L(v)-L(u0)]+p[A(v)-f(r)]=0, orH(v,p)=L(v)-L(u0)+pL(u0)+p[N(v)-f(r)]=0, where p[0,1] is called homotopy parameter and u0 is an initial approximation for the solution of (2.1), which satisfies the boundary conditions. Obviously, from (2.4) or (2.5), we will haveH(v,0)=L(v)-L(u0)=0,H(v,1)=A(v)-f(r)=0. We can assume that the solution of (2.4) or (2.5) can be expressed as a series in p as follows:v=v0+pv1+p2v2+. Setting p=1 results in the approximate solution of  (2.1)u=limp1v=v0+v1+v2+.

3. Implementation of HPM

In this section, we obtain an analytical solution of (1.1). We first use the transformationA(x,t)=u2(x,t), to convert (1.1) tout+2u2ux-(ux)2-12uxxu=0, with initial conditionu(x,0)=-ctanh(cx). To solve (3.2) with the initial condition (3.3), according to the homotopy perturbation method, we construct the following He’s polynomials corresponding to (2.5):L(v)=vt,N(v)=2v2vx-(vx)2-12vxxv,H(v,p)=vt-u0t+pu0t+p[2v2vx-(vx)2-12vxxv]. Substituting v=v0+pv1+p2v2+ into (3.2) and rearranging the resultant equation based on powers of p-terms, one hasp0:v0t-u0t=0,p1:v1t+u0t+[2v02v0x-(v0x)2-12v0xxv0]=0,p2:v2t+4v0v1v0x+2v02v1x-2v0xv1x-12v0xxv1-12v1xxv0=0, with the following conditions:v0(x,0)=-ctanh(cx),vi(x,0)=0for  i=1,2,. With the effective initial approximation for v0 from the condition (3.6), the solutions of (3.5) are obtained as follows:v0(x,t)=-ctanh(cx),v1(x,t)=2c2tcosh(2cx)2,v2(x,t)=c7/2t2sinh(cx)cosh(cx)3. In the same manner, the rest of components were obtained using the Maple package. According to the HPM, we can conclude thatu(x,t)=v0(x,t)+v1(x,t)+v2(x,t). Therefore, we will haveu(x,t)=-ctanh(cx)+2c2tcosh(2cx)2+c7/2t2sinh(cx)cosh(cx)3. By substituting (3.10) into (3.1), we can find the solution of (1.1).

4. Results and Discussion

In this section, we present the results with HPM to show the efficiency of the method, described in the previous section for solving (1.1). By Figures 2, 3, and 4, we may simply compare the HPM solution and exact solution of (1.1) for c=1, respectively. It is easy to verify the accuracy of the obtained results if we graphically compare HPM solutions with the exact ones. The absolute error of HPM is drawn in Figure 5 at t=0.01. It can be seen from this figure that the absolute error is very small and the mentioned method is very accurate. The effect of velocity of the wave, c, is demonstrated in Figure 6.

The solution of (1.1) at c=1 (a) HPM (b) Exact.

The solution of (1.1) at c=2 (a) HPM (b) Exact.

The solution of (1.1) at c=3 (a) HPM (b) Exact.

The absolute error of HPM at t=0.01 (a) c=1 (b) c=2.

The effect of c at t=0.01.

It is important to see the difference between the results obtained by ADM  and VIM  and the results which are obtained by the analytic solution using by the HPM. These can be seen by comparing Tables 1, 2, and 3. The results which are obtained by the HPM and the exact solutions are quite similar. It is shown the accuracy of the HPM. The solution of the HPM is more accurate than the ADM. While using HPM, the difficulty of calculating Adomian's polynomials does not occur. Another benefit of using the HPM is that the time consumption of the numerical calculation is less than the time consumption of ADM.

Comparison between absolute errors of ADM, VIM, and HPM for t=0.001, c=3.

−10 4.44089E-160 0
−8 2.24265E-1302.4E-18
−6 2.29754E-1002.1059E-15
−4 2.34498E-702.14918E-12
−2 0.000236656 02.17229E-9
−1 0.00523834 05.10302E-8
0 5.2479E-801.45797E-9

Comparison between absolute errors of ADM, VIM, and HPM for t=0.01, c=3.

−10 1.77636E-15 0 2.00000E-18
−8 1.86962E-12 0 2.01600E-15
−6 1.9087E-9 0 2.05744E-12
−4 1.94811E-66E-082.09993E-9
−2 0.00197296 6.0881E-05 0.000002123
−1 0.0485679 0.000050433
0 0.00051592 8.08544E-3 0.000014557

Comparison between absolute errors of ADM, VIM, and HPM for t=0.1, c=3.

−10 1.42109E-14 0 1.59300E-15
−8 1.40941E-11 0 1.62593E-12
−6 1.43842E-85E-091.65952E-9
−4 0.0000146727 4.527E-061.69379E-6
−2 0.0064218 4.59239E-03 0.001716375
−1 0.094494 0.043767689
0 3.71367 6.8374E-01 0.126259125
5. Conclusion

In this work, He’s homotopy perturbation method has been successfully utilized to derive approximate explicit analytical solution for the nonlinear foam drainage equation. The results show that this perturbation scheme provides excellent approximations to the solution of this nonlinear equation with high accuracy and avoids linearization and physically unrealistic assumptions. This new method is extremely simple, easy to apply, needs less computation, and accelerates the convergence to the solutions. The results obtained here are compared with results of exact solution. The current work illustrates that the HPM is indeed a powerful analytical technique for most types of nonlinear problems and several such problems in scientific studies and engineering may be solved by this method.

Acknowledgments

The authors gratefully acknowledge the support of the Department of Mechanical Engineering and the talented office of Semnan University for funding the current research grant.