\documentclass[12pt,twoside,fleqn]{article}
\usepackage[margin=2cm]{geometry}
\usepackage{graphicx}
\usepackage{multirow}
\renewcommand{\baselinestretch}{1}
\setcounter{page}{1}
\pagestyle{myheadings}
\thispagestyle{empty}
\markboth{\footnotesize International Journal of Applied Mathematical Research}{\footnotesize International Journal of Applied Mathematical Research}
\date{}
\begin{document}
{\renewcommand{\arraystretch}{0.65}
\begin{table}[ht]
\begin{tabular}{ll}
\multirow{3}{*}{\includegraphics[width=1cm]{logo}}&{\scriptsize\emph{\textbf{International Journal of Applied Mathematical Research}, 2 (xx) (2013) xxx-xxx}}\\
&{\scriptsize\emph{\copyright Science Publishing Corporation}}\\
&{\scriptsize\emph{www.sciencepubco.com/index.php/IJAMR}}
\end{tabular}
\end{table}}
%\renewcommand{\arraystretch}{1}
\centerline{}
\centerline{}
\centerline{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline {\Large{\bfThe Variational Homotopy Perturbation Method for Solving the K(2,2)
Equations }}
\centerline{}
%\centerline{\Large{\bf Title second line}}
\centerline{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% My definition
\newcommand{\mvec}[1]{\mbox{\bfseries\itshape #1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Definition}[Theorem]{Definition}
\newtheorem{Corollary}[Theorem]{Corollary}
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Example}[Theorem]{Example}
\noindent\hrulefill
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{ }10 \\
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.
\begin{thebibliography}{99}
{\small
\bibitem{1}{ G. Adomian, "Solving frontier problems of physics: The decomposition method", Kluwer Academic Press, Boston, 1994.}
\bibitem{2}{A. Refik Bahadir, "A fully implicit finite-difference scheme for two dimensional Burgers' equations", Applied Mathematics and Computation, 137, (2003), pp. 131-137.}
\bibitem{3}{J. Biazar, H. Ghazvini, "Convergence of the homotopy perturbation method for partial differential equations", Nonlinear Analysis: Real World Applications, 10, (2009), pp. 2633-2640.}
\bibitem{4}{J. de Frutos, M.A Lopez-Marcus and L.M Sanz-Serna, "A finite difference sheme for thr K(2,2) compacton equation", J. Comp. Phys., 120, (1995), pp. 248-254.}
\bibitem{5}{J. H. He, "Approximate solution of nonlinear differential equations with convolution product nonlinearities", Comput. Methods Appl. Mech. Eng.,167, (1998), pp. 57--68.}
\bibitem{6}{J. H. He, "New Interpretation of homotopy-perturbation method", Int. J. Mod. Phys. B, 20, (2006), pp. 2561-2568.}
\bibitem{7}{S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, "Some relatively new techniques for nonlinear problems", Math. Probl. En., Article ID 234849, (2009), 25 pp.}
\bibitem{8}{S. Momani, Z. Odibat, "Homotopy perturbation method for nonlinear partial differential equations of fractional order", Physics Letters A, 365, (2007), pp. 345--350.}
\bibitem{9}{S. T. Mohyud-Din, "Solving heat and wave-like equations using He's polynomials", Math. Probl. En., vol. 2009, Article ID 427516, (2009) 12 pages.}
\bibitem{10}{P. Rosenau, J.M. Hyman, "Compactons: solitons with finite wavelengths", Phys. Rev. Lett., 70, 1993, pp. 564--567.}
\bibitem{11}{Yanqin Liu, "Variational Homotopy Perturbation Method for Solving Fractional Initial Boundary Value Problems", Abstract and Applied Analysis, Art. ID 727031, (2012), 10 pp.}
\bibitem{12}{S. T. Mohyud-Din, A. Yildirim, S. A. Sezer, and M. Usman, "Modified Variational Iteration Method for Free-Convective Boundary-Layer Equation Using Pad'e Approximation", Mathematical Problems in Engineering, Article ID 318298, 11 pp., 2010.}
\bibitem{13}{S. J. Liao, Beyond perturbation: Introduction to the homotopy analysis method, Champan {\&} Hall/CRC Press, Boca Raton, 2004}.
\bibitem{14}{M. Matinfar, M. Saeidy, Z. Raeisi, "Modified variational iteration method for heat equation using He's polynomials", Bull. Math. Anal. Appl. 3 no. 2, (2011), pp. 238--24}
}
\end{thebibliography}
\end{document}
?
?