Homotopy analysis method for the Kawahara equation

doi:10.1016/j.nonrwa.2008.11.005

Nonlinear Analysis: Real World Applications

Volume 11, Issue 1, February 2010, Pages 307-312

https://doi.org/10.1016/j.nonrwa.2008.11.005 Get rights and content

Abstract

The homotopy analysis method (HAM) is used to find a family of travelling-wave solutions of the Kawahara equation. This approximate solution, which is obtained as a series of exponentials, has a reasonable residual error. The homotopy analysis method contains the auxiliary parameter $ħ$ , which provides us with a simple way to adjust and control the convergence region of series solution. This method is reliable and manageable.

Introduction

Nonlinear equations are widely used as models to describe complex physical phenomena in various fields of sciences. The world around us is inherently nonlinear. Nonlinear problems are more difficult to solve than linear ones. Recently, a powerful analytic method for nonlinear problems, the so-called homotopy analysis method (HAM), has been developed by Liao [1]. The homotopy analysis method (HAM) is a powerful analytical tool for nonlinear problems. This technique provides us with a simple way to ensure the convergence of the solution series, so that we can always get accurate enough approximations. Furthermore, the homotopy analysis method logically contains the nonperturbation methods such as Lyapunov’s artificial small parameter method, the $δ$ -expansion method and Adomian’s decomposition method and homotopy perturbation method [1], [2], [3], [4], [5]. HAM has been applied successfully to many nonlinear problems in engineering and science, such as application in heat radiation [6], solitary-wave solutions for the fifth-order KdV equation [7], generalized Benjamin–Bona–Mahony equation [8], boundary-layer flows over an impermeable stretched plate [9], unsteady boundary-layer flows over a stretching flat plate [10], exponentially decaying boundary layers [11], and many other problems (see [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], for example).

In this paper we apply the homotopy analysis method for finding the travelling-wave solutions of the Kawahara equation given by $u_{t} + α u u_{x} + β u_{3 x} + γ u_{5 x} = 0,$ where $α$ , $β$ and $γ$ are some arbitrary constants, see [23] and the references therein. The Kawahara equation (1) occurs in the theory of magneto-acoustic waves in a plasma and in the theory of shallow water waves with surface tension. This equation was first proposed by Kawahara in 1972, as a model equation describing solitary-wave propagation in media [24]. The Kawahara equation (1) has been the subject of extensive research work in recent publications [25], [26], [27].

Section snippets

Mathematical formulation

Consider the travelling-wave solutions of Eq. (1). It is convenient to introduce a new dependent variable $w (ξ)$ defined by $u (x, t) = a w (ξ),$ where $ξ = x - c t$ , $a$ is the amplitude, and $c$ is the wave speed. Substitution of $u$ given by (2) into Eq. (1) gives $γ w^{(5)} + β w^{‴} + α a w w^{'} - c w^{'} = 0,$ and integrating once gives $γ w^{(4)} + β w^{″} + \frac{α}{2} a w^{2} - c w = 0,$ where the prime denotes the differentiation with respect to $ξ$ .

Write $w (ξ) \approx B exp (- μ ξ), as ξ \to \infty,$ where $μ > 0$ and $B$ are constants. Substituting (4) into (3) and balancing the main term yields

Result analysis

For simplicity, in the rest of this section and in the next section, we set $c = 1$ or $c = - 1$ , for finding a real value for $μ$ from (5). In this section we consider two cases.

Conclusions

In this paper, the homotopy analysis method (HAM) [1] is applied to obtain the travelling-wave solution of the Kawahara equation. HAM provides us with a convenient way to control the convergence of approximation series, which is a fundamental qualitative difference in analysis between HAM and other methods. Our examples show that, the best value for $ħ$ is not −1. So, examples show the flexibility and potential of the homotopy analysis method for complicated nonlinear problems in engineering.

References (27)

S. Abbasbandy
Phys. Lett. A
(2006)
S. Abbasbandy
Phys. Lett. A
(2007)
F.M. Allan
Appl. Math. Comput.
(2007)
S. Abbasbandy
Int. Commun. Heat Mass
(2007)
S.J. Liao
Int. J. Heat Mass Transfer
(2005)
S.J. Liao et al.
Int. J. Heat Mass Transfer
(2006)
Y. Tan et al.
Chaos, Solitons Fractals
(2007)
W. Wu et al.
Chaos, Solitons Fractals
(2005)
T. Hayat et al.
Phys. Lett. A
(2007)
M. Sajid et al.
Phys. Lett. A
(2006)

Y. Tan et al.

Commun. Nonlinear Sci. Numer. Simul.

(2008)

S. Abbasbandy

Appl. Math. Model.

(2008)

T. Hayat et al.

Three-dimensional flow over a stretching surface in a viscoelastic fluid

Nonlinear Anal. RWA

(2008)

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Homotopy analysis method for the Kawahara equation

Abstract

Introduction

Section snippets

Mathematical formulation

Result analysis

Conclusions

Phys. Lett. A

Phys. Lett. A

Appl. Math. Comput.

Int. Commun. Heat Mass

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

Chaos, Solitons Fractals

Chaos, Solitons Fractals

Phys. Lett. A

Phys. Lett. A

Commun. Nonlinear Sci. Numer. Simul.

Appl. Math. Model.

Nonlinear Anal. RWA