- Research
- Open access
- Published:

# Approximate analytic solutions of the modified Kawahara equation with homotopy analysis method

*Advances in Difference Equations*
**volume 2012**, Article number: 178 (2012)

## Abstract

In this paper, we applied the homotopy analysis method (HAM) to solve the modified Kawahara equation. Numerical results demonstrate that the methods provide efficient approaches to solving the modified Kawahara equation. It is shown that the method, with the help of symbolic computation, is very effective and powerful for discrete nonlinear evolution equations in mathematical physics.

## 1 Introduction

In the past several decades, the investigation of traveling-wave solutions for nonlinear equations has played an important role in the study of nonlinear physical phenomena. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction, and convection are very important in nonlinear wave equations. In recent years, many authors paid attention to study solitonic solutions of nonlinear equations by using a variety of powerful methods such as the variational iteration method (VIM) [1, 2] and homotopy perturbation method (HPM) [3, 4]. Exp-function method [5–8], sine-cosine method [9], and homogeneous balance method [10–12] have been proposed for obtaining exact and approximate analytic solutions.

The aim of this paper is to directly apply the optimal HAM [13, 14] to reconsider the traveling-wave solution of the Kawahara equation. The method used here contains three convergence-control parameters to accelerate the convergence of homotopy series solution. The optimal convergence-control parameters can be determined by minimizing the averaged residual error. The results obtained in this paper show that the solutions given by the optimal HAM give much better approximations and converge much faster than those given by the usual HAM. The homotopy analysis method (HAM) [15–18] is a general analytic approach to get series solutions of various types of nonlinear equations, including algebraic equations, ordinary differential equations, partial differential equations, differential-integral equations, differential-difference equation, and coupled equations of them.

## 2 The homotopy analysis method

In this paper, we use the homotopy analysis method to solve the problem.

This method was proposed by the Chinese mathematician Liao [15]. We apply Liao’s basic ideas to the nonlinear partial differential equations. Let us consider the nonlinear partial differential equation

Based on the constructed zero-order deformation equation, we give the following zero-order deformation equation in the similar way:

*L* is an auxiliary linear integer-order operator and it possesses the property L(C)=0. *U* is an unknown function. Expanding *U* in Taylor series with respect to *q*, one has

where

As h=-1, Eq. (2.2) becomes

which is used mostly in the homotopy perturbation method (HPM) [19–22]. Thus, HPM is a special case of HAM.

Differentiating the equation *m* times with respect to the embedding parameter *q* and then setting q=0 and finally dividing them by *m*!, we have the *m* th-order deformation equation

where

and

These equations can be easily solved with software such as Maple, Matlab and so on.

The *m* th-order deformation Eq. (2.6) is linear, and thus can be easily solved, especially by means of a symbolic computation software such as Maple, Matlab and so on.

## 3 Test problem

We first consider the modified Kawahara equation [23]

where *p*, *q* are nonzero real constants. We solve the nonlinear partial differential equation with the HAM method. We consider Eq. (3.1) with initial condition

where K=\frac{1}{2}\sqrt{\frac{-p}{5q}} is constant. The exact solution is given for modified Kawahara equation by [23]

with c=\frac{25q-4{p}^{2}}{25q}.

Furthermore, Eq. (3.1) suggests defining the nonlinear fractional partial differential operator

Applying the above definition, we construct the zeroth-order deformation equation

For q=0 and q=1 respectively, we can write

According to Eqs. (2.6)-(2.7), we gain the *m* th-order deformation equation

where

Now, the solution of Eq. (3.7) for m\ge 1 becomes

From Eqs. (3.1), (3.6), and (3.9), for h=-1, we now successively get

As shown in Table 1, we note through the results of the preceding table that the solutions we have obtained are very precise and that we have compared our solution (HAM) to HPM and exact solution. HAM is easily more than the other method. It is obvious that two components only were sufficient to determine the exact solution of Eq. (3.1). Figures 1 and 2 show the evolution results. From Figures 1 and 2, it is easy to conclude that the solution continuously depends on the derivative. Where, Figures 1 and 2 are approximation and exact solution respectly. The exact solution of this test problem is as follows [23]:

## 4 Conclusions

In this paper, we applied the homotopy analysis method to the Kawahara equation. The homotopy analysis method was successfully used to obtain the exact solutions of Kawahara equation. As a result, some new generalized solitary solutions with parameters are obtained. It may be important to explain some physical phenomena by setting the parameters as special values. Finally, the method is straightforward, concise, and is a powerful mathematical method for solving nonlinear problems.

## References

He JH:

**Variational iteration method for delay differential equations.***Commun. Nonlinear Sci. Numer. Simul.*1997, 2(4):235–236. 10.1016/S1007-5704(97)90008-3He JH:

**Approximate solution of nonlinear differential equations with convolution product nonlinearities.***Comput. Methods Appl. Mech. Eng.*1998, 167(1–2):69–73. 10.1016/S0045-7825(98)00109-1He JH:

**Homotopy perturbation method: a new nonlinear analytical technique.***Appl. Math. Comput.*2003, 135: 73–79. 10.1016/S0096-3003(01)00312-5Jin L:

**Application of variational iteration method and homotopy perturbation method to the modified Kawahara equation.***Math. Comput. Model.*2009, 49: 573–578. 10.1016/j.mcm.2008.06.017He J-H, Wu X-H:

**Exp-function method for nonlinear wave equations.***Chaos Solitons Fractals*2006, 30: 700–708. 10.1016/j.chaos.2006.03.020He JH, Abdou MA:

**New periodic solutions for nonlinear evolution equations using Exp-function method.***Chaos Solitons Fractals*2007, 34(5):1421–1429. 10.1016/j.chaos.2006.05.072Abbasbandy S:

**Homotopy analysis method for the Kawahara equation.***Nonlinear Anal., Real World Appl.*2010, 11: 307–312. 10.1016/j.nonrwa.2008.11.005Yusufoglu E, Bekir A:

**Symbolic computation and new families of exact travelling solutions for the Kawahara and modified Kawahara equations.***Comput. Math. Appl.*2008, 55: 1113–1121. 10.1016/j.camwa.2007.06.018Wazwaz AM:

**The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants.***Commun. Nonlinear Sci. Numer. Simul.*2006, 11(2):148–160. 10.1016/j.cnsns.2004.07.001Fan E, Zhang H:

**A note on the homogeneous balance method.***Phys. Lett. A*1998, 246: 403–406. 10.1016/S0375-9601(98)00547-7Senthilvelan M:

**On the extended applications of homogeneous balance method.***Appl. Math. Comput.*2001, 123: 381–388. 10.1016/S0096-3003(00)00076-XWang ML:

**Exact solutions for a compound KdV-Burgers equation.***Phys. Lett. A*1996, 213: 279–287. 10.1016/0375-9601(96)00103-XDehghan M, Manafian J, Saadatmandi A:

**Solving nonlinear fractional partial differential equations using the homotopy analysis method.***Numer. Methods Partial Differ. Equ.*2009. doi:10.1002/num.20460Assas LMB:

**New exact solutions for the Kawahara equation using Exp-function method.***J. Comput. Appl. Math.*2009, 233: 97–102. 10.1016/j.cam.2009.07.016Liao, SJ: The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University, Shanghai, China (1992)

Liao SJ:

**An approximate solution technique not depending on small parameters: a special example.***Int. J. Non-Linear Mech.*1995, 30(3):371–380. 10.1016/0020-7462(94)00054-ELiao SJ:

**A kind of approximate solution technique which does not depend upon small parameters - II. An application in fluid mechanics.***Int. J. Non-Linear Mech.*1997, 32(5):815–822. 10.1016/S0020-7462(96)00101-1Liao SJ

**CRC Series: Modern Mechanics and Mathematics 2.**In*Beyond Perturbation: Introduction to the Homotopy Analysis Method*. Chapman & Hall/CRC, Boca Raton; 2004.Dehghan M, Shakeri F:

**Use of He’s homotopy perturbation method for solving a partial differential equation arising in modeling of flow in porous media.***J. Porous Media*2008, 11: 765–778. 10.1615/JPorMedia.v11.i8.50Shakeri F, Dehghan M:

**Solution of the delay differential equations via homotopy perturbation method.***Math. Comput. Model.*2008, 48: 486–498. 10.1016/j.mcm.2007.09.016Dehghan M, Shakeri F:

**Solution of an integro-differential equation arising in oscillating magnetic fields using He’s homotopy perturbation method.***Prog. Electromagn. Res.*2008, 78: 361–376.Saadatmandi A, Dehghan M, Eftekhari A:

**Application of He’s homotopy perturbation method for non-linear system of second-order boundary value problems.***Nonlinear Anal., Real World Appl.*2009, 10: 1912–1922. 10.1016/j.nonrwa.2008.02.032Daoreji S:

**New exact travelling wave solutions for the Kawahara and modified Kawahara equations.***Chaos Solitons Fractals*2004, 19: 147–150. 10.1016/S0960-0779(03)00102-4

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Competing interests

The author declares that they have no competing interests.

## Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Kurulay, M. Approximate analytic solutions of the modified Kawahara equation with homotopy analysis method.
*Adv Differ Equ* **2012**, 178 (2012). https://doi.org/10.1186/1687-1847-2012-178

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/1687-1847-2012-178