Differential Equations and Nonlinear Mechanics

Volume 2008 (2008), Article ID 243459, 8 pages

http://dx.doi.org/10.1155/2008/243459

## Approximate Traveling Wave Solutions of Coupled Whitham-Broer-Kaup Shallow Water Equations by Homotopy Analysis Method

^{1}Department of Mechanical Engineering, Faculty of Engineering, Bu-Ali Sina University, Hamedan 65174-4161, Iran^{2}Department of Mechanical Engineering, Faculty of Engineering, Mazandaran University, P.O. Box 484, Babol 47415, Iran

Received 11 November 2007; Accepted 6 March 2008

Academic Editor: Shijun Liao

Copyright © 2008 M. M. Rashidi 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.

#### Abstract

The homotopy analysis method (HAM) is applied to obtain the approximate traveling wave solutions of the coupled Whitham-Broer-Kaup (WBK) equations in shallow water. Comparisons are made between the results of the proposed method and exact solutions. The results show that the homotopy analysis method is an attractive method in solving the systems of nonlinear partial differential equations.

#### 1. Introduction

In 1992, Liao [1] employed the basic ideas of the homotopy in topology to propose method for nonlinear problems, namely, homotopy analysis method (HAM), [2–6]. This method has many advantages over the classical methods; mainly, it is independent of any small or large quantities. So, the HAM can be applied no matter if governing equations and boundary/initial conditions contain small or large quantities or not. The HAM also avoids discretization and provides an efficient numerical solution with high accuracy, minimal calculation, and avoidance of physically unrealistic assumptions. Furthermore, the HAM always provides us with a family of solution expressions in the auxiliary parameter ; the convergence region and rate of each solution might be determined conveniently by the auxiliary parameter . This method has been successfully applied to solving many types of nonlinear problems [7–11].

A substantial amount of research work has been invested in the study of linear and nonlinear systems of partial differential equations (PDEs). Systems of nonlinear partial differential equations arise in many scientific models such as the propagation of shallow water waves and the Brusselator model of the chemical reaction-diffusion model.

Here, we consider the coupled Whitham-Broer-Kaup (WBK) equations which have been studied by Whitham [12], Broer [13], and Kaup [14]. The equations describe the propagation of shallow water waves, with different dispersion relations. The WBK equations are as follows: where is the horizontal velocity, is the height that deviates from equilibrium position of the liquid, and are constants which are represented in different diffusion powers [15]. The exact solutions of and are given by [16] where and are arbitrary constants. Above system is a very good model to describe dispersive waves. If and then the system represents the modified Boussinesq (MB) equations [16]. If and then the system represents the classical long-wave equations that describe shallow water wave with dispersion [15].

This Letter has been organized as follows. In Section 2, the basic concept of the HAM is introduced. In Section 3, we extend the application of the HAM to construct approximate solutions for the coupled WBK equations. Numerical experiments are presented in Section 4.

#### 2. Basic Concepts of HAM

Let us consider the following differential
equation:
where is a nonlinear operator; denotes independent variable; is an unknown function, respectively. For
simplicity, we ignore all boundary or initial conditions, which can be treated
in the similar way. By means of generalizing the traditional homotopy method,
Liao [5]
constructs the so-called *zero-order deformation equation*:
where is the embedding parameter; is a nonzero auxiliary parameter; is an auxiliary linear operator; is an initial guess of is an unknown function, respectively. It is
important that one has great freedom to choose auxiliary things in HAM.
Obviously, when and , it holds
respectively. Thus, as increases from 0 to 1, the solution varies from the initial guess to the solution . Expanding in Taylor series with respect to , we have
where
If the auxiliary linear operator, the
initial guess, and the auxiliary parameter are so properly chosen, the series (2.4)
converges at , then we have
which must be one of
solutions of original nonlinear equation, as proved by Liao [5].
As (2.2)
becomes
which is used mostly in the homotopy
perturbation method, whereas the solution is obtained directly, without using Taylor series [17, 18].

According to definition (2.5),
the governing equation can be deduced from the *zero-order deformation* (2.2).
Define the vector
Differentiating (2.2) times with respect to the embedding parameter and then
setting and finally dividing them by , we
have the so-called th-*order deformation equation*:
where

It should be emphasized that for is governed by the linear (2.9) with the linear boundary conditions that come from original problem, which can be easily solved by symbolic computation software such as Maple and Mathematica.

#### 3. Application

First, we consider the coupled Whitham-Broer-Kaup (WBK)
equations (1.1), with the initial conditions
For application of the
homotopy analysis method, we choose the initial approximations
and the linear operator
with the property
where is constant. From (1.1), we define a system
of nonlinear operators as
Using the above definition,
we construct the *zero-order deformation equations*:
Obviously, when and ,
Thus, as the embedding parameter increases from 0 to
1, and vary from the
initial approximations and to the solutions and respectively. Expanding and in Taylor
series with respect to , we
have
where
If the auxiliary linear operator, the initial approximations, and the auxiliary parameters and are so properly
chosen, the above series converge at , then we have
which must be one of
solutions of original system. Define the vectors
We gain the th-*order deformation equations*:
subject to initial conditions
where
Obviously, the solution of
the th-*order deformation equations* (3.12) for becomes
For simplicity, we suppose . From (3.2) and (3.15), we now successively obtain
We used 10 terms in evaluating the approximate solutions and .

The series solutions contain the auxiliary parameter . The validity of the method is based on such an assumption that the series (2.4) converges at . It is the auxiliary parameter which ensures that this assumption can be satisfied. As pointed out by Liao [5], in general, by means of the so-called -curve, it is straightforward to choose a proper value of which ensures that the solution series is convergent. In this way, we choose in following computational works.

#### 4. Numerical Experiments

We now obtain numerical solutions of the coupled Whitham-Broer-Kaup (WBK) equations. In order to verify the efficiency of the proposed method in comparison with exact solutions, we report the absolute errors for and different values of and in the following examples.

*Example 4.1. *Consider the WBK equations (1.1), with the initial conditions
(3.1), and the exact solutions (1.2). In Table 1, we show the absolute error for and

*Example 4.2. *When and the WBK equations are reduced to the modified Boussinesq (MB) equations
[16]. We show the absolute error for MB equations in Table 2.

*Example 4.3. *When and the WBK equations are reduced to the
approximate long-wave (ALW) equations in shallow water [15]. Table 3 shows the
absolute error in this case.

#### 5. Conclusions

In this study, the homotopy analysis method (HAM) was used for finding the approximate traveling wave solutions of the Whitham-Broer-Kaup (WBK) equations in shallow water. A very good agreement between the results of the HAM and exact solutions was observed, which confirms the validity of the HAM. It should be emphasized that the HAM provides us with a convenient way to control the convergence of approximation series, which is a fundamental qualitative difference in analysis between the HAM and other methods. Furthermore, as the HAM does not require discretization, it is not affected by computation round off errors, and large computer memory as well as consumed time which are issues in the calculation procedure. The results show that the HAM is powerful mathematical tool for solving systems of nonlinear partial differential equations having wide applications in engineering.

#### References

- S.-J. Liao,
*The proposed homotopy analysis technique for the solutions of nonlinear problems*, Ph.D. thesis, Shanghai Jiao Tong University, Shanghai, China, 1992. - S.-J. Liao, “An approximate solution technique not depending on small parameters: a special example,”
*International Journal of Non-Linear Mechanics*, vol. 30, no. 3, pp. 371–380, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-J. Liao, “A kind of approximate solution technique which does not depend upon small parameters—II: an application in fluid mechanics,”
*International Journal of Non-Linear Mechanics*, vol. 32, no. 5, pp. 815–822, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - S.-J. Liao, “An explicit, totally analytic approximate solution for Blasius' viscous flow problems,”
*International Journal of Non-Linear Mechanics*, vol. 34, no. 4, pp. 759–778, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - S.-J. Liao,
*Beyond Perturbation: Introduction to the Homotopy Analysis Method*, vol. 2 of*CRC Series: Modern Mechanics and Mathematics*, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. View at Zentralblatt MATH · View at MathSciNet - S.-J. Liao, “On the homotopy analysis method for nonlinear problems,”
*Applied Mathematics and Computation*, vol. 147, no. 2, pp. 499–513, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Hayat, M. Khan, and M. Ayub, “On the explicit analytic solutions of an Oldroyd 6-constant fluid,”
*International Journal of Engineering Science*, vol. 42, no. 2, pp. 123–135, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - S.-J. Liao, “A new branch of solutions of boundary-layer flows over an impermeable stretched plate,”
*International Journal of Heat and Mass Transfer*, vol. 48, no. 12, pp. 2529–2539, 2005. View at Publisher · View at Google Scholar - S.-J. Liao and I. Pop, “Explicit analytic solution for similarity boundary layer equations,”
*International Journal of Heat and Mass Transfer*, vol. 47, no. 1, pp. 75–85, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Sajid, T. Hayat, and S. Asghar, “On the analytic solution of the steady flow of a fourth grade fluid,”
*Physics Letters A*, vol. 355, no. 1, pp. 18–26, 2006. View at Publisher · View at Google Scholar - Y. Tan, H. Xu, and S.-J. Liao, “Explicit series solution of travelling waves with a front of Fisher equation,”
*Chaos, Solitons & Fractals*, vol. 31, no. 2, pp. 462–472, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - G. B. Whitham, “Variational methods and applications to water waves,”
*Proceedings of the Royal Society of London A*, vol. 299, no. 1456, pp. 6–25, 1967. View at Google Scholar · View at Zentralblatt MATH - L. J. F. Broer, “Approximate equations for long water waves,”
*Applied Scientific Research*, vol. 31, no. 5, pp. 377–395, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. J. Kaup, “A higher-order water-wave equation and the method for solving it,”
*Progress of Theoretical Physics*, vol. 54, no. 2, pp. 396–408, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. A. Kupershmidt, “Mathematics of dispersive water waves,”
*Communications in Mathematical Physics*, vol. 99, no. 1, pp. 51–73, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Xie, Z. Yan, and H. Zhang, “Explicit and exact traveling wave solutions of Whitham-Broer-Kaup shallow water equations,”
*Physics Letters A*, vol. 285, no. 1-2, pp. 76–80, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “Homotopy perturbation method for solving boundary value problems,”
*Physics Letters A*, vol. 350, no. 1-2, pp. 87–88, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - J.-H. He, “Some asymptotic methods for strongly nonlinear equations,”
*International Journal of Modern Physics B*, vol. 20, no. 10, pp. 1141–1199, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH