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\markboth{\footnotesize International Journal of Applied Mathematical Research}{\footnotesize International Journal of Applied Mathematical Research}

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\multirow{3}{*}{\includegraphics[width=1cm]{logo}}&{\scriptsize\emph{\textbf{International Journal of Applied Mathematical Research},  2 (xx) (2013) xxx-xxx}}\\
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\centerline {\Large{\bfThe Variational Homotopy Perturbation Method for Solving the K(2,2) 
Equations }}

\centerline{}

%\centerline{\Large{\bf Title second line}}

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%% My definition
\newcommand{\mvec}[1]{\mbox{\bfseries\itshape #1}}

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\centerline{\bf {Abdelkader Bouhassoun*, Mohamed Zellal}}

\centerline{}
{\small
\centerline{\emph{Laboratoire de  math\'{e}ematiques et ses applications (LAMAP) }}

\centerline{\emph{d\'{e}partement de math\'{e}matiques, facult\'{e} des sciences exactes}}

\centerline{\emph{University of Oran Senia, P.O. Box 1524 Oran, Algeria}}

\centerline{\emph{* a.bouhassoun@yahoo.fr}}}

\centerline{}

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\newtheorem{Theorem}{Theorem}[section]

\newtheorem{Definition}[Theorem]{Definition}

\newtheorem{Corollary}[Theorem]{Corollary}

\newtheorem{Lemma}[Theorem]{Lemma}

\newtheorem{Example}[Theorem]{Example}

\noindent\hrulefill
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\hspace{-20 pt}{\textbf{Abstract}\\}
\centerline{}
This paper deals with implementation of the variational homotopy pertubation method (VHPM) for solving the K(2,2) compacton equation. The numerical results show that the approach is easy to implement and accurate when it is compared with the exact solution. The suggested algorithm is quite efficient and is practically well suited for use in the nonlinear problems. The fact that the proposed technique solves nonlinear problems without using the Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.\\

\hspace{-20 pt}{{\footnotesize \emph{\textbf{Keywords}}:  \emph{variational homotopy, perturbation method, variational iteration method.}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\hrulefill
%=============================
\section{Introduction}
%=============================
Nonlinear differential equations are encountered in various fields in 
physics, chemistry, biology, mathematics and engineering. For example, 
Burgers' equation is used to describe various kinds of phenomena such as 
turbulence and the approximation theory of flow through a shock wave 
traveling in a viscous fluid \cite{2}. Numerical methods which are commonly used 
such as finite difference \cite{4}, finite element or characteristics method need 
large size of computational works and usually the effect of round-off error 
causes loss of accuracy in the results. Most nonlinear models of real-life 
problems are still very difficult to solve either numerically or 
theoretically.

In recent years, several methods have drawn particular attention, such as 
the Adomian decomposition method \cite{1}, the variational iteration method \cite{5}, 
the homotopy analysis method \cite{13}, and the homotopy perturbation method 
\cite{6,7,8,9}.

In this paper we consider the following nonlinear dispersive $K(m,n)$ 
equation:
\begin{equation}
\label{eq1}
u_t +a(u^m)_x +(u^n)_{xxx} =0
\end{equation}
developed in \cite{10} for describing the compacton $\left( {m>0,\mbox{ }1<n\le 
3} \right)$ which is a compact wave that preserves its shape after the 
interaction with another compact wave.

In the case $m=n=1$, this equation becomes 
\begin{equation}
\label{eq2}
u_t +(u^2)_x +(u^2)_{xxx} =0,
\end{equation}
has
\begin{equation}
\label{eq3}
u(x,t)=(4c/3)\cos ^2((x-ct)/4)
\end{equation}
as exact solution and it is developed in \cite{4} for describing the compacton 
solution. 
While taking $u(x,0)=(4/3)\cos ^2(x/4)$, and considering Eq. (\ref{eq2}), it derives the initial value problem
\begin{equation}
\label{eq4}
\left\{ {\begin{array}{l}
 u_t +(2u+6u_{xx} )u_x +2uu_{xxx} =0,\;\;\;t>0 \\ 
 u(x,0)=(4/3)\cos ^2(x/4) \\ 
 \end{array}} \right.\quad .
\end{equation}
In what follows, the variational iteration method is modified by introducing 
a transformation such that the solution is expressed by the series 
approximation. Precisely, we couple the classical variational iteration 
method with He's polynomials \cite{7,9} and construct a new homotopy to solve 
(\ref{eq4}). Our modification proposed in Section IV extends the variational 
iteration method with He's polynomials. This modification provides an 
accurate approximation for the $K(2,2)$ equation. This implies that our 
method provides a new idea of the variational iteration method with He's 
polynomials for finding an approximation of the nonlinear differential 
equations. Observing the numerical results, and comparing our approximation 
with the exact solution, the proposed method reveals to be very close to the 
exact solution. The details of the comparison results are displayed in Table 
1 and Table 2 in Section \ref{NUM}.
%==============================
\section{Variational Iteration Method} \label{VIM}
%==============================


To illustrate the basic concepts of the VIM, we consider the following 
differential equation
\begin{equation}
\label{eq5}
L(u(x,t))+N(u(x,t)=g(x,t)\quad ,
\end{equation}
where $L$ is a linear operator, $N$ is a nonlinear operator and $g(x,t)$ is 
an inhomogeneous term. Then we can construct a correction functional as 
follows
\begin{equation}
\label{eq6}
u_{n+1} (x,t)=u_n (x,t)+\int\limits_0^t {\lambda \left\{ {L(u(x,\tau 
))+N(\tilde {u}(x,\tau ))-g(x,\tau )} \right\}} d\tau 
\end{equation}
where $\lambda $ is a general Lagrange multiplier, which can be identified 
optimally via variational theory. The second term on the right hand side is 
called the correction and is considered as a restricted variation, i.e., 
$\delta \tilde {u}_n =0$. By this method, it is required first to determine 
the Lagrangian multiplier $\lambda $ that will be identified optimally. The 
successive approximations $u_{n+1} (x,t)$, $n\ge 0$ of the solution $u(x,t)$ 
will be readily obtained upon using the determined Lagrangian multiplier and 
any selective function $u_0 (x,t)$. Consequently, the solution is given by
\begin{equation}
\label{eq7}
u(x,t)=\lim _{n\to \infty } u_n (x,t).
\end{equation}%
%=============================================
\section{Homotopy Perturbation Method} \label{HPM}
%=============================================

To illustrate the basic ideas of the HPM, we consider the following 
nonlinear differential equation
\begin{equation}
\label{eq8}
A(u)-f(r)=0,\;\;r\in \Omega ,
\end{equation}
with the boundary conditions 
\begin{equation}
\label{eq9}
B\left( {u,\frac{\partial u}{\partial n}} \right)=0,\;\;r\in \Gamma ,
\end{equation}
where $A$, is a general differential operator, $B$ is a boundary operator, 
$f(r)$is a known analytical function and $\Gamma $ is the boundary of the 
domain $\Omega $.

\noindent Generally speaking, the operator $A$ can be divided into two parts $L$ and 
$N$ where $L$ is the linear part, and $N$ the nonlinear part. Therefore Eq. 
(\ref{eq8}) can be rewritten as
\begin{equation}
\label{eq10}
L(u)+N(u)-f(r)=0.
\end{equation}
By the homotopy perturbation technique, we construct a homotopy 
$v(r,p):\Omega \times \left[ {0,1} \right]\to R$which satisfies
\begin{equation}
\label{eq11}
H(v,p)=(1-p)\left[ {L(v)-L(u_0 )} \right]+p\left[ {A(v)-f(r)} \right]=0,
\end{equation}
or
\begin{equation}
\label{eq12}
H(v,p)=L(v)-L(u_0 )+pL(u_0 )+p\left[ {N(v)-f(r)} \right]=0,
\end{equation}
where $p\in \left[ {0,1} \right]$ is an embedding parameter and $u_0 $ is an 
initial approximation of equation (\ref{eq8}) which satisfies the boundary 
conditions. Considering equation (\ref{eq12}) we will have
\begin{equation}
\label{eq13}
H(v,0)=L(v)-L(u_0 )=0
\end{equation}
and
\begin{equation}
\label{eq14}
H(v,1)=A(v)-f(r)=0.
\end{equation}
The changing process of $p$ from zero to unity is just that of $v(r,p)$ from 
$u_0 (r)$ to $u(r)$. In topology this is called deformation and $L(v)-L(u_0 
)$ and $A(v)-f(r)$ are called homotopy. According to the homotopy 
perturbation theory, we can first use the embedding parameter $p$ as a small 
parameter and assume that the solution of equation (\ref{eq11}) can be written as a 
power series in $p$
\begin{equation}
\label{eq15}
v=v_0 +pv_1 +p^2v_2 +.\;.\;.
\end{equation}
Setting $p=1$ one have the approximation solution of equation (n8) as the 
following
\begin{equation}
\label{eq16}
u=\mathop {\lim }\limits_{p\to 1} v=v_0 +v_1 +v_2 +.\;.\;.
\end{equation}
The convergence of series (\ref{eq16}) is discussed in \cite{3}. 
 

%===================================================
\section{Variational Homotopy Perturbation Method} \label{VHPM}
%===================================================

In the homotopy perturbation method \cite{11}, the basic assumption is that the 
solutions can be written as a power series in $p$
\begin{equation}
\label{eq17}
u=\sum\limits_{i=0}^{+\infty } {p^iu_i } =u_0 +pu_1 +p^2u_2 +.\;.\;.
\end{equation}
To illustrate the concept of the variational homotopy perturbation method 
[matinfar2] we consider the general differential equation (\ref{eq5}). We construct 
the correction functional (\ref{eq6}) and apply the homotopy perturbation method to 
obtain \cite{12,14}.
\begin{equation}
\label{eq18}
\sum\limits_{r=1}^{+\infty } {p^iu_i } (x,t)=u_0 (x,t)+p\int\limits_0^t 
{\lambda \left\{ {N(\sum\limits_{r=1}^{+\infty } {p^iu_i } (x,\tau 
))-g(x,\tau )} \right\}} d\tau .
\end{equation}
As we see, the procedure is formulated by the coupling of variational 
iteration method and homotopy perturbation method. A comparison of like 
powers of $p$ gives solutions of various orders.
%================================================
\section{Numerical Results} \label{NUM}
%==================================================
In this section we will examine the nonlinear dispersive equation $K(2,2)$ 
defined in Eq. (\ref{eq2}) and expressed in the form of the initial value problem 
(\ref{eq4}). We apply the VHPM developed in Section \ref{VHPM}, construct the correction 
functional and calculate the Lagrange multipliers optimally via variational 
theory.

\noindent The correction functional for (\ref{eq4}) reads
\begin{equation}
\label{eq19}
u_{n+1} =u_n +\int\limits_0^t {\lambda (\tau )\left\{ {(u_n )_\tau +((\tilde 
{u}_n )^2)_x +((\tilde {u}_n )^2)_{xxx} } \right\}} d\tau ,
\end{equation}
or
\begin{equation}
\label{eq20}
u_{n+1} =u_n +\int\limits_0^t {\lambda (\tau )\left\{ {\frac{\partial u_n 
}{\partial \tau }+\left( {2u_n +6\frac{\partial ^2u_n }{\partial x^2}} 
\right)\frac{\partial u_n }{\partial x}+2u_n \frac{\partial ^3u_n }{\partial 
x^3}} \right\}} d\tau 
\end{equation}
and which yields the stationary conditions
\begin{equation}
\label{eq21}
\quad
\quad
\left\{ {{\begin{array}{*{20}c}
 {\lambda ^'=0} \hfill \\
 {\lambda +1=0} \hfill \\
\end{array} }} \right..
\end{equation}
Therefore, the general Lagrange multiplier can be readily identified as 
$\lambda =-1$.

Substituting this value of the Lagrangian multiplier into functional (\ref{eq19}) or 
its equivalent equation (\ref{eq20}) gives the iteration formula
\begin{equation}
\label{eq22}
u_{n+1} (x,t)=u_n (x,t)-\int\limits_0^t {\left\{ {\begin{array}{l}
 \frac{\partial u_n (x,\tau )}{\partial \tau }+\left( {2u_n (x,\tau 
)+6\frac{\partial ^2u_n (x,\tau )}{\partial x^2}} \right)\frac{\partial u_n 
(x,\tau )}{\partial x} \\ 
 +2u_n (x,\tau )\frac{\partial ^3u_n (x,\tau )}{\partial x^3} \\ 
 \end{array}} \right\}} d\tau \quad .
\end{equation}
Applying the variational homotopy perturbation method, one obtains 
\begin{equation}
\label{eq23}
\sum\limits_{i=0}^{+\infty } {p^iu_i } (x,t)=u_0 (x,t)-p\int\limits_0^t 
{\left\{ {\begin{array}{l}
 \frac{\partial \sum\limits_{i=0}^{+\infty } {p^iu_i } (x,\tau )}{\partial 
\tau }+\left( {\begin{array}{l}
 2\sum\limits_{i=0}^{+\infty } {p^iu_i } (x,\tau )+ \\ 
 6\frac{\partial ^2\sum\limits_{i=0}^{+\infty } {p^iu_i } (x,\tau 
)}{\partial x^2} \\ 
 \end{array}} \right)\frac{\partial \sum\limits_{i=0}^{+\infty } {p^iu_i } 
(x,\tau )}{\partial x} \\ 
 +2\sum\limits_{i=0}^{+\infty } {p^iu_i } (x,\tau )\frac{\partial 
^3\sum\limits_{i=0}^{+\infty } {p^iu_i } (x,\tau )}{\partial x^3} \\ 
 \end{array}} \right\}} d\tau .
\end{equation}
Comparing the coefficient of like powers of $p$ we obtain the following set of linear partial differential equations
\begin{equation}
\label{eq24}
u_1 (x,t)=-\int\limits_0^t {\left\{ {2u_0 (x,\tau )\frac{\partial ^3u_0 
(x,\tau )}{\partial x^3}+\left( {2u_0 (x,\tau )+6\frac{\partial ^2u_0 
(x,\tau )}{\partial x^2}} \right)\frac{\partial u_0 (x,\tau )}{\partial x}} 
\right\}} d\tau 
\end{equation}
\begin{equation}
\label{eq25}
u_2 (x,t)=-\int\limits_0^t {\left\{ {\begin{array}{l}
 6\frac{\partial ^2u_0 (x,\tau )}{\partial x^2}\frac{\partial u_1 (x,\tau 
)}{\partial x}+2\frac{\partial ^3u_0 (x,\tau )}{\partial x^3}u_1 (x,\tau ) 
\\ 
 +6\frac{\partial u_0 (x,\tau )}{\partial x}\frac{\partial ^2u_1 (x,\tau 
)}{\partial x^2}+2u_0 (x,\tau )\frac{\partial ^3u_1 (x,\tau )}{\partial x^3} 
\\ 
 +2u_0 (x,\tau )\frac{\partial u_1 (x,\tau )}{\partial x}+2\frac{\partial 
u_0 (x,\tau )}{\partial x}u_1 (x,\tau ) \\ 
 \end{array}} \right\}} d\tau 
\end{equation}
\begin{equation}
\label{eq26}
u_3 (x,t)=-\int\limits_0^t {\left\{ {\begin{array}{l}
 6\frac{\partial ^2u_0 (x,\tau )}{\partial x^2}\frac{\partial u_2 (x,\tau 
)}{\partial x}+2u_1 (x,\tau )\frac{\partial ^3u_1 (x,\tau )}{\partial x^3} 
\\ 
 +2u_0 (x,\tau )\frac{\partial u_2 (x,\tau )}{\partial x}+2u_1 (x,\tau 
)\frac{\partial u_1 (x,\tau )}{\partial x} \\ 
 +2u_2 (x,\tau )\frac{\partial ^3u_0 (x,\tau )}{\partial x^3}+2u_2 (x,\tau 
)\frac{\partial u_0 (x,\tau )}{\partial x} \\ 
 +6\frac{\partial u_0 (x,\tau )}{\partial x}\frac{\partial ^2u_2 (x,\tau 
)}{\partial x^2}+2u_0 (x,\tau )\frac{\partial ^3u_2 (x,\tau )}{\partial x^3} 
\\ 
 +6\frac{\partial u_1 (x,\tau )}{\partial x}\frac{\partial ^2u_1 (x,\tau 
)}{\partial x^2} \\ 
 \end{array}} \right\}} d\tau 
\end{equation}
\begin{center}
.
\end{center}

\begin{center}
.
\end{center}

\begin{center}
.
\end{center}

\noindent and so on, in the same manner the rest of components can be obtained using 
the Maple package.


\noindent Consequently, while taking the initial value $u(x,0)=\textstyle{4 \over 
3}\cos ^2(\textstyle{x \over 4})$, and according to Eqs. (\ref{eq24})-(\ref{eq26}), the 
first few components of the variational homotopy perturbation solution for 
Eq. (\ref{eq19}) are derived as follows 
\[
u_0 (x,t)=u(x,0)=\textstyle{4 \over 3}\cos ^2(\textstyle{x \over 4})
\]
\[
u_1 (x,t)=\textstyle{2 \over 3}\cos (\textstyle{x \over 4})\sin 
(\textstyle{x \over 4})t
\]
\[
u_2 (x,t)=\textstyle{{-1} \over {12}}\left( {-1+2\cos ^2(\textstyle{x \over 
4})} \right)t^2
\]
\[
u_3 (x,t)=\textstyle{{-1} \over {36}}\left( {\cos (\textstyle{x \over 
4})\sin (\textstyle{x \over 4})} \right)t^3
\]
.

.

.

\noindent The other components of the VHPM can be determined in a similar way. 
Finally, the approximate solution of Eq. (\ref{eq19}) in a series form is
\begin{equation}
\label{eq27}
u(x,t)\simeq u_0 (x,t)+u_1 (x,t)+u_2 (x,t)+.\;.\;.
\end{equation}
Consequently, the third-order term approximate solution for Eq. (\ref{eq19}) is 
given by
\begin{equation}
\label{eq28}
\begin{array}{l}
 u(x,t)=\textstyle{4 \over 3}\cos ^2(\textstyle{x \over 4})+\textstyle{2 
\over 3}\cos (\textstyle{x \over 4})\sin (\textstyle{x \over 4})t \\ 
 -\textstyle{1 \over {12}}\left( {-1+2\cos ^2(\textstyle{x \over 4})} 
\right)t^2-\textstyle{1 \over {36}}\left( {\cos (\textstyle{x \over 4})\sin 
(\textstyle{x \over 4})} \right)t^3 \\ 
 \end{array}
\end{equation}
and this will, in the limit of infinitely many terms, yield the closed form 
solution
\begin{equation}
\label{eq29}
u(x,t)=(\frac{4}{3})\cos ^2(\frac{x-t}{4}).
\end{equation}
represented in figure 1.

\noindent On the other hand, a development of the exact solution (\ref{eq29}) in Taylor series 
over t=0 to order 3 gives:
\begin{equation}
\label{eq30}
\begin{array}{l}
 u(x,t)=\textstyle{4 \over 3}\cos ^2(\textstyle{x \over 4})+\textstyle{2 
\over 3}\cos (\textstyle{x \over 4})\sin (\textstyle{x \over 4})t \\ 
 -\textstyle{1 \over {12}}\left( {-1+2\cos ^2(\textstyle{x \over 4})} 
\right)t^2-\textstyle{1 \over {36}}\left( {\cos (\textstyle{x \over 4})\sin 
(\textstyle{x \over 4})} \right)t^3+{\rm O}(t^4) \\ 
 \end{array}
\end{equation}
which confirms our result.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%% figue 1%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[htbp]
\centerline{\includegraphics[width=2.50in,height=2.00in]{The1.eps}}
\caption{Graphic representation of the exact solution (\ref{eq29}) of the initial 
value problem (\ref{eq4}).}
\label{fig1}
\end{figure}


\begin{figure}[htbp]
\centerline{\includegraphics[width=2.50in,height=2.00in]{The2.eps}}
\caption{Approximate solution (\ref{eq27}) of the Eq. (\ref{eq19}) given by the VHPM method 
with third order.}
\label{fig2}
\end{figure}

In figure \ref{fig1}, we have represented the graph of the exact solution of Eq. (\ref{eq20}). 
As we see, there is practically no difference between the graph of the 
approximate series solution in Fig. \ref{fig2} and the exact solution in Fig. \ref{fig1}. 
Additionally, we see in table 1, that the calculation of error between the 
exact solution and that obtained by the VHPM method shows that the resulting 
value is very close to the exact solution.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Table 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{center}
\caption{\small The VHPM results for 3 iterations in comparison with the exact 
solution of the K(2,2) equation with initial conditions of Eq. (\ref{eq4}).
}
\begin{tabular}{ccccccc}
\hline
%
\multicolumn{1}{c}{$t/x$} & \multicolumn{1}{c}{$0.1$} & $0.2$ & \multicolumn{1}{c}{$0.3$} & \multicolumn{1}{c}{$0.4$} & $0.5$\\
%\cline{5-6}
%& & & & \multicolumn{1}{c}{\cite{Ismail-etal04}} &
%\multicolumn{1}{c}{Present work}  \\
\hline
0.1 & 1.73466*10^{-7}     & 	1.72903*10^{-7}    & 1.71907*10^{-7}  & 1.70481*10^{-7}$   &  $1.68629*10^{-7}$  \\
$0.2$ & $2.77616*10^{-6}$  &  $	2.76852*10^{-6}$ & $	2.75397*10^{-6}$ & $	2.73253*10^{-6}$ &  $2.70427*10^{-6}$ \\
$0.3$ &  $0.0000140555$  &  $0.0000140239$  &$	0.0000139572$  &  $0.0000138556$    &  $	0.0000137194$ \\
$0.4$ &  $0.0000444185$  &  $0.0000443408$  & $0.0000441522$   &  $	0.0000438533$    &  $0.0000434448$   \\
$0.5$ &  $0.000108417$  &  $		0.000108281$  & $0.000107875$   &  $	0.000107199$    &  $	0.000106255$  \\
%
\hline
\end{tabular}
\end{center}
\end{table}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Table 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{center}
\caption{\small The VHPM results for 6 iterations in comparison with the exact 
solution of the K(2,2) equation with initial conditions of Eq. \ref{eq4}.
}
\begin{tabular}{ccccccc}
\hline
%
\multicolumn{1}{c}{$t/x$} & \multicolumn{1}{c}{$0.1$} & $0.2$ & \multicolumn{1}{c}{$0.3$} & \multicolumn{1}{c}{$0.4$} & $0.5$\\
%\cline{5-6}
%& & & & \multicolumn{1}{c}{\cite{Ismail-etal04}} &
%\multicolumn{1}{c}{Present work}  \\
\hline
$0.1$ & $1.44542*10^{-11}$  &  $1.44051*10^{-11}$    & $1.43199*10^{-11}$  & $1.41993*10^{-11}$   &  $1.40423*10^{-11}$  \\
$0.2$ & $9.25265*10^{-10}$  &  $9.22456*10^{-10}$    & $9.17342*10^{-10}$ & $9.09935*10^{-10}$ &  $9.00253*10^{-10}$ \\
$0.3$ &  $1.05408*10^{-8}$  &  $1.05125*10^{-8}$  &$1.0458*10^{-8}$  &  $1.03774*10^{-8}$    &  $	1.02708*10^{-8}$ \\
$0.4$ &  $5.92275*10^{-8}$  &  $5.909*10^{-8}$  & $5.88049*10^{-8}$   &  $	5.83727*10^{-8}$    &  $5.77947*10^{-8}$   \\
$0.5$ &  $2.25925*10^{-7}$  &  $	2.25481*10^{-7}$  & $2.24474*10^{-7}$   &  $	2.22906*10^{-7}$    &  $2.2078*10^{-7}$  \\
%
\hline
\end{tabular}
\end{center}
\end{table}
%
%==========================
\section{Conclusion}
%==========================
In this paper, we have studied the one-dimensional K(2,2) equation by using 
the variational homotopy perturbation method. The results show that the 
proposed method is powerful for finding the numerical solutions and can be 
used to obtain the series solution for the general case K(m,n) equations, 
where m and n can be different from 2. We have seen that the VHPM method 
requires the evaluation of the Lagrangian multiplier $\lambda $, while the 
ADM method requires the evaluation of the Adomian polynomials. As the 
evaluation of the Adomian polynomials for every nonlinear terms requires 
more and more algebraic calculations, it should be better to used the VHPM 
method to overcome this difficulty. We can integrate the equation directly 
without use calculation of Adomian polynomials. Moreover, an observation of 
the error analysis in Table 1 and 2 shows that more accuracy can be obtained 
by adding terms in the series.



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