\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 195, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/195\hfil Approximate solutions]
{Approximate solutions of general perturbed KdV-Burgers equations}

\author[B. Hong, D. Lu \hfil EJDE-2014/195\hfilneg]
{Baojian Hong, Dianchen Lu}  % in alphabetical order

\address{Baojian Hong \newline
Faculty of Science, Jiangsu University,
Zhenjiang, Jiangsu 212013, China. \newline
Department of Basic Courses,
Nanjing Institute of Technology, Nanjing 211167, China}
\email{hbj@njit.edu.cn}

\address{Dianchen Lu \newline
Faculty of Science, Jiangsu University,
Zhenjiang, Jiangsu 212013, China}
\email{dclu@ujs.edu.cn, Tel +86 13815158840, fax +86 (511) 88791128}

\thanks{Submitted December 20, 2013. Published September 18, 2014.}
\subjclass[2000]{35A24, 35Q53}
\keywords{Homotopy analysis method; perturbed KdV-Burgers equation;
\hfill\break\indent $\hbar$-curve; approximate solution}

\begin{abstract}
 In this article, we present some approximate analytical solutions to the
 general perturbed KdV-Burgers equation with nonlinear terms of any
 order by applying the homotopy analysis method (HAM).
 While compared with the Adomain decomposition method (ADM) and the
 homotopy perturbation method (HPM), the HAM contains the auxiliary
 convergence-control parameter $\hbar$ and the control function
 $H(x,t)$, which provides a useful way to adjust and control the
 convergence region of solution series. The numerical results reveal
 that HAM is accurate and effective when it is applied to the perturbed PDEs.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

With the development of soliton theory in nonlinear science,
searching for analytical solitary wave solutions or approximate
solutions of nonlinear partial differential equations (NLPDEs)
plays an important and significant role in the study of dynamics
of those nonlinear phenomena \cite{g2}. Many authors presented various
powerful method to deal with this problem, such as inverse
scattering transformation \cite{a4}, Hirota bilinear method \cite{s1},
homogeneous balance method \cite{w3}, B\"{a}cklund transformation \cite{w2},
Darboux transformation \cite{m1}, the elliptic integral method \cite{f1},
the first integral method \cite{f2,f3} and so on.
Because of the complexity of NLPDEs,
It is difficult for us to find exact solutions in a straightforward way.
One has to propose and develop some approximate methods for nonlinear
theory, such as the multiple-scale method \cite{t1}, the variational iteration
method \cite{g1}, the indirect matching method \cite{w4}, the renormalization
method \cite{m2}, and the homotopy perturbation method \cite{h2} etc.
The common essential point of these methods is to study nonlinear systems
by using the approximation method.

The homotopy analysis method (HAM) was introduced in
1992 \cite{l2,l3}, which yields a fast convergence for most of the
selected problems. It also shows a high accuracy and a rapid
convergence to solutions of the nonlinear partial evolution
equations. After this, many types of nonlinear problems were solved
with the aid of HAM, such as the nonlinear  Vakhnenko equation \cite{w5},
the Glauert-jet problem \cite{b1}, a generalized Hirota-Satsuma coupled KdV
equation \cite{a2}, and a smoking habit model \cite{g3,z2} etc.

The rest of this article is organized as follows.
In Section 2, we obtain some exact solutions of the general perturbed KdV-Burgers
equation by using the mapping deformation method.
In Section 3, we apply HAM to construct approximate solutions for the general
perturbed KdV-Burgers equation.
In Section 4, we discuss the accuracy of these solutions
with the small perturbation term as illustrations.
Also we present a short conclusion.

\section{Exact solutions}

Consider the general perturbed KdV-Burgers equation
\begin{equation}\label{e1}
u_t + \alpha u^pu_x + \beta u^{2p}u_x + \gamma u_{xx} +
\delta u_{xxx} = f( {u} ),
\end{equation}
where $\alpha ,\beta ,\gamma ,\delta ,p$ are arbitrary constants,
and $f = f( {u} )$ is a perturbed term, which is a
sufficiently smooth function in a corresponding domain. If we let $f
= 0$, we can get the well-known KdV-Burgers equation with
nonlinear terms of any order \cite{h1,f1,k1,l1,w1,z1}:
\begin{equation} \label{e2}
u_t + \alpha u^pu_x + \beta u^{2p}u_x + \gamma u_{xx} +
\delta u_{xxx} = 0.
\end{equation}

This equation with $p \ge 1$ arises in modeling waves generated by a
wavemaker in a channel and waves incoming from deep water into
nearshore zones and some profound results have been  described in \cite{p1}.
In fact, if one takes different values for $\alpha ,\beta
,\gamma ,\delta ,p$ and $f$, equation \eqref{e1} includes quite a few
equations as particular cases such as KdV equation, MKdV equation, CKdV equation,
Burgers equation, and KdV-Burgers equation as follows:
Fitzhugh-Nagumo equation \cite{a3}:
\begin{equation}
 u_t - u_{xx} = f = u(u - \alpha )(1 - u);\label{e3}
\end{equation}
Burgers-Huxley equation \cite{m3}:
\begin{equation}
 u_t + \alpha u^\delta u_x - \lambda u_{xx} = f = \beta
u(1 - u^\delta )(\eta u^\delta - \gamma );\label{e4}
\end{equation}
Burgers-Fisher equation \cite{m3}:
\begin{equation}
u_t + \alpha u^\delta u_x - u_{xx} = f = \beta u(1 -u^\delta ).\label{e5}
\end{equation}

By using the general mapping deformation method \cite{f3}, we know
that  \eqref{e2} admits the following solutions:
\begin{gather}
u_1 = \{A_1 (K - \sqrt {K^2} \tanh [(\frac{p\gamma }{2K(2 + p)\delta
}\pm \frac{p\alpha }{2K^{2}(2 + p)}\sqrt {\frac{ - K^2(1 + 2p)}{(1 +
p)\beta \delta }} )\sqrt {K^2} \xi _1 ]\}^{1/p};\label{e6}
\\
u_2 = \{ - \frac{c(1 + p)}{2\alpha } - \frac{c(1 + p)}{2\alpha }
\tanh{\frac{cp}{\gamma}(x+ct+\xi _0
)\}^{1/p}},\beta=\delta=0; \label{e7}
\\
\xi _1 = x + [\frac{(1 + p)\gamma ^2}{(2 + p)^2\delta } + \frac{(1 +
2p)\alpha ^2}{(1 + p)(2 + p)^2\beta }\pm \frac{p\alpha \gamma }{K(2
+ p)^2\beta }\sqrt {\frac{ - K^2(1 + 2p)}{(1 + p)\beta \delta }} ]t
+ \xi _0;\label{e8}
\end{gather}
where
\[
A_1 = - \frac{(1 + 2p)\alpha }{2K(2 + p)\beta }\pm
\frac{\gamma }{2K^2(2 + p)}\sqrt {\frac{ - K^2(1 + p)(1 + 2p)}{\beta
\delta }},\, K,\,\xi_0
\]
 and $c$ are arbitrary constants.

Note that $i\tanh(i\xi)=-\tan\xi$,
$\tanh(\xi+\frac{\pi}{2}i)=\coth(\xi)$,
$i\coth(i\xi)=\cot\xi$, $i=\sqrt{-1}$. Also note that
the solution $u_{1,2} $ contains all results presented in \cite{l1}.

\section{Homotopy analysis method (HAM)}

To describe the basic idea of the HAM, let us consider the
nonlinear equation, in a standard form,
\begin{equation}
N[u(x,t)] = 0,\label{e9}
\end{equation}
where $N$ is a nonlinear operator, $u(x,t)$ is an unknown function,
$x$ and $t$ denote the spatial and temporal independent variables,
respectively.

By using the basic idea of the traditional homotopy method \cite{l2},
we construct the zero-order deformation equation
\begin{equation}
 ( {1 - q} )L[\phi (x,t;q) - u_0 (x,t)] =
q\hbar H(x,t)N[\phi (x,t;q)],\label{e10}
\end{equation}
where $q \in [0,1]$ is the embedding parameter, $\hbar $ is a
nonzero auxiliary parameter, $H(x,t)$ is an auxiliary function, $L$
is an auxiliary linear operator, $ {u}_0 (x,t)$ is an initial estimate
of $u(x,t)$ and $\phi (x,t;q)$ is an unknown function. It is
important that we have much freedom to choose auxiliary things in
HAM. Obviously, when $q = 0$ and $q = 1$, it holds
\begin{equation}
 \phi (x,t;0) = u_0 (x,t),\quad \phi (x,t;1) = u(x,t).\label{e11}
\end{equation}
Thus, as $q$ increases from 0 to 1, the function $\phi (x,t;q)$
varies from the initial value $u_0 (x,t)$ to the exact solution
$u(x,t)$. Expanding $\phi (x,t;q)$ in the Taylor series with respect to
$q$, we have
\begin{equation}
 \phi (x,t;q) = u_0 + \sum_{m = 1}^\infty {u_m }
q^m;\quad u_0 = u_0 (x,t),\quad u_m = u_m ( {x,t} ),\label{e12}
\end{equation}
where
\begin{equation}
 u_m ( {x,t} ) = \frac{1}{m!}\frac{\partial
^m}{\partial q^m}\phi (x,t;q)\big|_{q=0}\,. \label{e13}
\end{equation}

If the auxiliary linear operator, the initial estimate, the auxiliary
parameter and the auxiliary function are  properly chosen such
that they are smooth enough, the Taylor's series \eqref{e12} with respect
to $q$ converges at $q = 1$, and we have
\begin{equation}
 u = \phi (x,t;1) = \sum_{m = 0}^\infty {u_m}.\label{e14}
\end{equation}
The above series solutions generally converge very rapidly. A
classical approach of convergence of this type of series has already
presented by Abbaoui and Cherruault \cite{a1}. Liao  proved that it
must be one of the exact solutions of the original nonlinear
equation \cite{l3}. As $\hbar = - 1$ and $H(x,t) = 1$, equation \eqref{e10} becomes
\begin{equation}
 ( {1 - q} )L[\phi (x,t;q) - u_0 (x,t)] +qN[\phi (x,t;q)] = 0,\label{e15}
\end{equation}
which is frequently used in the homotopy perturbation method (HPM). The
comparison between HAM and HPM can be found in \cite{l4}. As $H(x,t) =
1$, equation  \eqref{e10} becomes
\begin{equation}
 ( {1 - q} )L[\phi (x,t;q) - u_0 (x,t)] =
q\hbar N[\phi (x,t;q)].\label{e16}
\end{equation}

According to definition \eqref{e13}, the governing equation can be deduced
from the zero-order deformation equation \eqref{e10}. Define the vector
\begin{equation}
 \vec {u}_m (x,t) = \{u_0 ,u_1 ,u_2 , \dots ,u_m\}.\label{e17}
\end{equation}
Differentiating equation  \eqref{e10} $m$ times with respect to the embedding
parameter $q$, then setting $q = 0$ and dividing them by $m!$, we
get the $m$th-order deformation equation
\begin{equation}
 L[u_m (x,t) - \chi _m u_{m - 1} (x,t)] = \hbar
H(x,t)R_{m - 1} (\vec {u}_{m - 1} ,x,t),\label{e18}
\end{equation}
where
\begin{gather}
 R_{m - 1} (\vec {u}_{m - 1} ,x,t) = \frac{1}{(m -1)!}
\frac{\partial ^{m - 1}}{\partial q^{m - 1}}N[\phi (x,t;q)]\big|_{q = 0}\,,
 \label{e19} \\
\chi _m = \begin{cases}
 0, &m \le 1, \\
 1, &m \ge 2.
\end{cases} \label{e20}
\end{gather}

It is notable that the $m$-th order deformation
equation \eqref{e18} is linear, and $u_m ( {x,t} )$ for $m \ge 1$
can be easily solved by the boundary conditions and the symbolic
computation software such as Mathematica and
Matlab.

To solve  \eqref{e1} by means of HAM, we choose the initial approximation
\begin{equation}
 u_0 (x,t) = \tilde {u}_0 (x,t)\big|_{t = 0}
= g(x),\label{e21}
\end{equation}
where $\tilde {u}_0 (x,t)$ is an arbitrary exact solution of  \eqref{e2}.
According to  \eqref{e1}, we define the nonlinear operator
\begin{equation}
 N[\phi ] = \phi _t + \alpha \phi ^p\phi _x + \beta \phi
^{2p}\phi _x + \gamma \phi _{xx} + \delta \phi _{xxx} - f(\phi
),\quad \phi = \phi (x,t;q).\label{e22}
\end{equation}
By following the process above, it is straightforward to
choose $H(x,t) = 1$, the base functions $g_n (x)t^n$, $n \ge 0$, and the
linear operator
\begin{equation}
 L[\phi (x,t;q)] = \frac{\partial \phi (x,t;q)}{\partial t},\label{e23}
\end{equation}
with the condition
\begin{equation}
 L[c(x)] = 0.\label{e24}
\end{equation}

From equations \eqref{e18}, \eqref{e19} and \eqref{e22}, we have
\begin{equation}
\begin{aligned}
R_{m - 1} (\vec {u}_{m - 1},x,t)
& = u_{m - 1,t} + \gamma u_{m - 1,xx}
+ \delta u_{m - 1,xxx}+ \alpha D_{m - 1} (\phi ^p\phi _x )\\
&\quad + \beta D_{m - 1} (\phi ^{2p}\phi _x ) - F(u_0 ,u_1 ,
\dots ,u_{m - 1} ),
\end{aligned}\label{e25}
\end{equation}
where
\begin{align*}
D_{m - 1} (\phi ^n\phi _x )
&= \sum_{k_1 = 0}^n \sum_{k_2 = 0}^{k_1 } \sum_{k_3 = 0}^{k_2 }  \dots
\sum_{k_{m - 1} = 0}^{k_{m - 2} } \sum_{i = 0}^{m -
1} C_n^{k_1 } C_{k_1 }^{k_2 } C_{k_2 }^{k_3 } \dots \\
&\quad\times C_{k_{m - 2}}^{k_{m - 1} } u_0^{n - k_1 }
u_1^{k_1 - k_2 } u_2^{k_2 -k_3 } \dots
u_{m - 2}^{k_{m - 2} - k_{n - 1} }
u_{m - 1}^{k_{m -1} } u_{i\xi },
\end{align*}
and $n \ge k_1 \ge k_2 \ge \dots \ge k_{m - 1} \ge 0 \in N$, with
$\sum_{j = 1}^{m - 1} {k_j + i = m - 1}$,
$i = 0, \dots ,m -1$. Furthermore, we have
\[
F(u_0 ,u_1 , \dots ,u_{m - 1} )
= \frac{1}{(n - 1)!}\frac{\partial ^{(m - 1)}}{\partial q^{m - 1}}f(\phi)
\big|_{q=0}\,.
\]
The solution of the $m$-th order deformation equation \eqref{e18} with the
initial condition $u_m ( {x,0} ) = 0$ for $m \ge 1$
becomes
\begin{equation}
u_m = \chi _m u_{m - 1} + L^{ - 1}[\hbar R_{m - 1} (\vec
{u}_{m - 1} ,x,t)].\label{e26}
\end{equation}
Thus, from equations \eqref{e21}, \eqref{e25} and \eqref{e26}, we can
successively obtain
\begin{gather}
u_0 = \tilde {u}_0 (x,0) = g(x),\label{e27} \\
u_1 = - \hbar t[\tilde {c}_0(x) + f(u_0 )], \tilde
{c}_0(x) = \frac{\partial }{\partial t}\tilde {u}_0 (x,t)\big|_{t = 0}\,,
\label{e28} \\
u_2 = (1 + \hbar )u_1 + \hbar t[\alpha u_0^p u_{1,x} +
\beta u_0^{2p} u_{1,x} + \gamma u_{1,xx} + \delta u_{1,xxx} - f_u
(u_0 )u_1 ],\label{e29} \\
\dots \notag \\
\begin{aligned}
u_m &= (1 + \hbar )u_{m - 1} + \hbar t[\gamma u_{1,xx} + \delta
u_{1,xxx} + \alpha D_{m - 1} (\phi ^p\phi _x ) \\
&\quad + \beta D_{m - 1} (\phi ^{2p}\phi _x ) - F(u_0 ,u_1 , \dots ,u_{m - 1} )].
\end{aligned} \label{e30}
\end{gather}
Consequently, we obtain the following $m$-th order approximate solution, and
exact solution of \eqref{e1}:
\begin{equation}
u_{m,{\rm appr}} = \sum_{k = 0}^m {u_k }, \quad u_{\rm exact} =
\phi (x,t;1) = \mathop {\lim }_{m \to \infty } \sum_{k
= 0}^m {u_k }. \label{e31}
\end{equation}

\section{Examples and discussion}

In this section, three specific examples about equation \eqref{e1} are presented
to illustrate the effectiveness of the HAM. We plot the
$\hbar$-curves of $u_{\rm appr}'' (0,0)$ and $u_{\rm appr}''' (0,0)$ to
discover the valid region of $\hbar $, which corresponds to the line
segment nearly parallel to the horizontal axis. A comparison
among the initial exact solution for the traditional unperturbed
equation when $f=0$, the exact solution for the perturbed equation
when $f\neq0$ and the fourth order of approximate solution for the perturbed equation
is given through numerical simulations.

\begin{example} \label{examp1} \rm
Consider the CKdV equation with a small perturbed term
\begin{equation}
 u_t + 6uu_x - 6u^2u_x + u_{xxx} = \varepsilon u^2, \quad
0 < \varepsilon \ll 1,\label{e32}
\end{equation}
with the initial exact solution
\begin{equation}
\tilde {u}_0 (x,t) = \frac{1}{2} - \frac{1}{2}\tanh
[\frac{1}{2}(x - t)].\label{e33}
\end{equation}
From the preceding section, we have
\begin{gather*}
u_0 = \frac{1}{2} - \frac{1}{2}\tanh (\frac{1}{2}x),  \tilde
{c}_0(x) = \frac{1}{4}\sec h^2(\frac{1}{2}x),\\
u_1 = - \hbar t \{\frac{1}{4}\sec h^2(\frac{1}{2}x)
+ \varepsilon [\frac{1}{2} - \frac{1}{2}\tanh (\frac{1}{2}x)]^2\},
\\
\begin{aligned}
 u_2
&= - (1 + \hbar )\hbar t\{\frac{1}{4}\sec h^2(\frac{1}{2}x)
+ \varepsilon [\frac{1}{2} - \frac{1}{2}\tanh (\frac{1}{2}x)]^2\} \\
&\quad -\hbar ^2t^2\Big\{6[\frac{1}{2} - \frac{1}{2}\tanh (\frac{1}{2}x)]
 \{\frac{1}{4}\sec h^2(\frac{1}{2}x) + \varepsilon [\frac{1}{2} -
\frac{1}{2}\tanh (\frac{1}{2}x)]^2\}_x \\
&\quad + 6\hbar ^2t^2[\frac{1}{2} -\frac{1}{2}\tanh (\frac{1}{2}x)]^2
\{\frac{1}{4}\sec h^2(\frac{1}{2}x) \\
&\quad + \varepsilon [\frac{1}{2} - \frac{1}{2}\tanh (\frac{1}{2}x)]^2\}_x
- \hbar ^2t^2\{\frac{1}{4}\sec h^2(\frac{1}{2}x) \\
&\quad + \varepsilon [\frac{1}{2}- \frac{1}{2}\tanh (\frac{1}{2}x)]^2\}_{xxx} \\
&\quad + 2\varepsilon \hbar ^2t^2[\frac{1}{2} - \frac{1}{2}\tanh
(\frac{1}{2}x)]\{\frac{1}{4}\sec h^2(\frac{1}{2}x) + \varepsilon
[\frac{1}{2} - \frac{1}{2}\tanh (\frac{1}{2}x)]^2\}
\\
&= \frac{\hbar t}{32}[\cosh (\frac{x}{2}) - \sinh (\frac{x}{2})]\sec
h^5(\frac{x}{2})\{\hbar (5t - 3 - 3\varepsilon ) - 3 - 3\varepsilon \\
&\quad + 2\hbar t\varepsilon (1 + \varepsilon ) + 2\cosh (x)[2\varepsilon - 2 -
2\hbar (1 + \varepsilon ) + \hbar t(2\varepsilon ^2 + 7\varepsilon - 3)] \\
&\quad + [\hbar (t - \varepsilon - 1 + 2t\varepsilon ^2) - \varepsilon - 1]\cosh
(2x) - 2\sinh (\frac{x}{2})[1 - \varepsilon + \hbar - \varepsilon \hbar \\
&\quad + \hbar t(2 - 3\varepsilon + 2\varepsilon ^2) + (1 - \varepsilon )\cosh x +
\hbar (1 - t - \varepsilon + 2t\varepsilon ^2)\cosh x)]\Big\},
\end{aligned} \\
 \dots \\
\begin{aligned}
 u_{\rm appr}
&= \frac{1}{2} - \frac{1}{2}\tanh (\frac{1}{2}x)
- \hbar \big\{\frac{1}{4}\sec h^2(\frac{1}{2}x)
+ \varepsilon [\frac{1}{2} - \frac{1}{2}\tanh (\frac{1}{2}x)]^2\big\}t \\
&\quad + \frac{\hbar t}{32}[\cosh (\frac{x}{2}) - \sinh (\frac{x}{2})]
 \sec h^5(\frac{x}{2})
\Big\{\hbar (5t - 3 - 3\varepsilon ) - 3 - 3\varepsilon +
2\hbar t\varepsilon (1 + \varepsilon ) \\
&\quad + 2\cosh (x)[2\varepsilon - 2-2\hbar (1 + \varepsilon )
 + \hbar t(2\varepsilon ^2 + 7\varepsilon - 3)] \\
&\quad + [\hbar (t - \varepsilon -1 + 2t\varepsilon ^2) - \varepsilon - 1]\cosh (2x)
- 2\sinh  (\frac{x}{2})[1- \varepsilon + \hbar - \varepsilon \hbar \\
&\quad + \hbar t(2 - 3\varepsilon + 2\varepsilon ^2)
+ (1 - \varepsilon )\cosh x + \hbar (1 - t - \varepsilon
+ 2t\varepsilon ^2)\cosh x]\} + \dots .
\end{aligned}
\end{gather*}

The $\hbar$-curves of $u_{\rm appr}'' (0,0)$ and
$u_{\rm appr}''' (0,0)$ to equation \eqref{e32} are shown in Figure 1.
A comparison between the initial exact solution and the approximate solution
of the fourth order is provided in Figure 2 (a)-(b),
which indicates that the solution series
\eqref{e31} is convergent when $ - 1.2 \le \hbar < 0$, and the approximate
solution for $\hbar = - 0.1$ and $\hbar = - 1$ (HPM) is compared.
We can see that the best value of $\hbar $ in this case is not $ -1$.
\end{example}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1} % 1.eps
\end{center}
\caption{$\hbar $-curves of $u_{\rm appr}'' (0,0)$ and $u_{\rm appr}''' (0,0)$
at the fourth order approximation} \label{fig1}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig2a}  % 2.eps
\includegraphics[width=0.48\textwidth]{fig2b}  \\ % 3.eps
(a)\hfil (b)
\end{center}
\caption{Comparison between the curves of
initial exact solution and the fourth order approximate solution
with $\hbar =-0.1,-1$.} \label{fig2}
\end{figure}

\begin{example} \label{examp2} \rm
Consider the KdV-Burgers equation with a small perturbed term
\begin{equation}
 u_t + 6uu_x + u_{xx} - u_{xxx} = \varepsilon \sin u,\quad
0 < \varepsilon \ll 1,\label{e34}
\end{equation}
with the initial exact solution
\begin{equation}
 \tilde {u}_0 (x,t) = \frac{1}{50} \big\{1 - \coth [ -
\frac{1}{10}(x - \frac{6}{25}t)] \big\}^2.\label{e35}
\end{equation}
From the preceding section, we have
\begin{gather*}
u_0 = \frac{1}{50}[1 - \coth ( - \frac{1}{10}x)]^2, \quad
\tilde {c}_0(x) = \frac{3}{3125}\csc h^2(\frac{1}{10}x)[1 + \coth
(\frac{1}{10}x)],
\\
u_1 = - \hbar \varepsilon \sin \{\frac{1}{50}[1 - \coth ( -
\frac{1}{10}x)]^2\}t - \frac{3}{3125}\hbar
t\csc h^2(\frac{1}{10}x)[1 + \coth (\frac{1}{10}x)],\\
 \dots \\
\begin{aligned}
u_{\rm appr} &= \frac{1}{50}[1 - \coth ( - \frac{1}{10}x)]^2 - \hbar
\varepsilon \sin \{\frac{1}{50}[1 - \coth ( - \frac{1}{10}x)]^2\}t
\\
&\quad - \frac{3}{3125}\hbar t\csc h^2(\frac{1}{10}x)[1 + \coth
(\frac{1}{10}x)] + u_2 + \dots .
\end{aligned}
\end{gather*}
The $\hbar $-curves of $u_{\rm appr}'' (0,0)$ and
$u_{\rm appr}'''(0,0)$ to equation \eqref{e34} are shown in
Figure \ref{fig3}(a). A comparison between the
initial exact solution and the approximate solution of the fourth order
are shown in Figure \ref{fig3}(b).
\end{example}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig3a}   % 4.eps
\includegraphics[width=0.48\textwidth]{fig3b}  \\% 5.eps
(a)\hfil (b)
\end{center}
\caption{(a) The $\hbar $-curves of
$u_{\rm appr}'' ($10ln2$,0)$ and $u_{\rm appr}''' ($10ln2$,0)$ at the
4th order of approximation.
(b) Comparison between the curves of initial exact solution and the fourth order
 of approximate solution.} \label{fig3}
\end{figure}

\begin{example} \label{examp3} \rm
Consider the Burgers-Fisher equation
\begin{equation}
 u_t + u^2u_x - u_{xx} = \varepsilon u(1 - u^2),\quad
0 < \varepsilon \le 1,\label{e36}
\end{equation}
with the initial exact solution and the exact solution
\begin{gather}
\tilde {u}_0 (x,t) = \sqrt {\frac{1}{2} -
\frac{1}{2}\tanh [\frac{1}{3}x - \frac{1}{9}t + \xi _0 ]},\label{e37}
\\
u_{\rm exact} = \sqrt {\frac{1}{2} - \frac{1}{2}\tanh [\frac{1}{3}x -
\frac{1 + 9\varepsilon }{9}t + \xi _0 ]} .\label{e38}
\end{gather}
Following the process above, we have
\begin{gather*}
u_0 = \sqrt {\frac{1}{2} - \frac{1}{2}\tanh (\frac{1}{3}x)} , \quad
\tilde{c}_0(x) = \sec h^2(\frac{1}{3}x) / 18\sqrt {2 - 2\tanh
(\frac{1}{3}x)} , \\
u_1 = - \frac{\hbar t\sec h^2(\frac{1}{3}x)}{18\sqrt {2 - 2\tanh
(\frac{1}{3}x)} } - \hbar t\varepsilon \sqrt {\frac{1}{2} -
\frac{1}{2}\tanh (\frac{1}{3}x)} (\frac{1}{2} + \frac{1}{2}\tanh
(\frac{1}{3}x)), \\
 \dots \\
\begin{aligned}
 u_{\rm appr}
&= \sqrt {\frac{1}{2} - \frac{1}{2}\tanh (\frac{1}{3}x)} -
\frac{\hbar t\sec h^2(\frac{1}{3}x)}{18\sqrt {2 - 2\tanh
(\frac{1}{3}x)} }\\
&\quad - \hbar t\varepsilon \sqrt {\frac{1}{2}
 - \frac{1}{2}\tanh (\frac{1}{3}x)}
(\frac{1}{2} + \frac{1}{2}\tanh (\frac{1}{3}x)) + u_2 + \dots
\end{aligned}
\end{gather*}

The $\hbar$-curves of $u_{\rm appr}'' (0,0)$ and $u_{\rm appr}'''(0,0)$
to equation \eqref{e36} are shown in Figure \ref{fig4}(a). A comparison between the
initial exact solution and the approximate solution of the fourth order
is shown in Figure \ref{fig4}(b).
\end{example}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig4a}  %6.eps
\includegraphics[width=0.48\textwidth]{fig4b} \\ %7.eps
(a)\hfil (b)
\end{center}
\caption{(a) The $\hbar $-curves of $u_{\rm appr}'' (0,0)$ and
$u_{\rm appr}''' (0,0)$ at the 4th order of approximation.
(b) Comparison between the curves of initial exact
solution, exact solution and the fourth order of approximate
solution.} \label{fig4}
\end{figure}

\subsection*{Conclusion}
In this work, the HAM has been applied to find the
approximate solutions of the general perturbed KdV-Burgers equation.
Numerical simulations show that, compared to HPM, this method provides us
more accuracy and reductions in the size of calculations. In
addition, the results of the HPM can be obtained as a special case
of the HAM when $\hbar = - 1$. The parameter $\hbar $ provides us
with a simpler way to adjust and control the convergence region of
solution series for large values of $t$. It was shown that
the HAM is a very powerful and efficient technique for solving
various kinds of nonlinear systems in science and engineering
without any assumptions and restrictions, and the auxiliary
parameter $\hbar $ plays a critical role within the frame of the
HAM which can be determined by the $\hbar $-curves.

\subsection*{Acknowledgements}
This work was supported by the National Natural Science Foundation of
China (Grant No. 61070231), the Graduate Student Innovation Project
of Jiangsu Province (Grant No. CXLX13\_673) and the General
Program of Innovation Foundation of Nanjing Institute of
Technology (Grant No. CKJB201218).

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\end{document}