\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 66, pp. 1--12.\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-2014/66\hfil KdV equation with fractional time derivative]
{Solution of the KdV equation with fractional time derivative via
 variational method}

\author[Y. Zhang \hfil EJDE-2014/66\hfilneg]
{Youwei Zhang}  % in alphabetical order

\address{Youwei Zhang \newline
Department of Mathematics, Hexi University, Gansu
734000, China}
\email{ywzhang0288@163.com}

\thanks{Submitted November 5, 2013. Published March 7, 2014.}
\subjclass[2000]{35R11, 35G20}
\keywords{Riemann-Liouville fractional derivative;
 Euler-Lagrange equation;  \hfill\break\indent
 Riesz fractional derivative;  generalized KdV equation;
 He's variational iteration method; \hfill\break\indent solitary wave}

\begin{abstract}
 This article presents a formulation of the time-fractional generalized
 Korteweg-de Vries (KdV) equation using the Euler-Lagrange variational
 technique in the Riemann-Liouville derivative sense.
 It finds an approximate solitary wave solution, and shows that
 He's variational iteration method is an efficient technique in
 finding the solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction} \label{sec1}

During the past three decades or so, fractional calculus has
obtained considerable popularity and importance as generalizations
of integer-order evolution equation, and used to model some
meaningful things, such as fractional calculus can model price
volatility in finance \cite{a6,a7}, in hydrology to model fast
spreading of pollutants \cite{a8}, the most common hydrologic
application of fractional calculus is the generation of fractional
Brownian motion as a representation of aquifer material with
long-range correlation structure \cite{a9,a12}. Fractional
differential equation is used to to model the particle motions in a
heterogeneous environment and long particle jumps of the anomalous
diffusion in physics \cite{a13,a14}. Other exact description of the
applications of engineering, mechanics and mathematics et al., we
can refer to \cite{a16,a17,a18,a20,a21}. If the Lagrangian of
conservative system is constructed using fractional derivatives, the
resulting equation of motion can be nonconservative. Therefore, in
many cases, the real physical processes could be modeled in a
reliable manner using fractional-order differential equation rather
than integer-order equation \cite{a22}. Based on the stochastic
embedding theory, Cresson \cite{a23} defined the fractional
embedding of differential operators and provided a fractional
Euler-Lagrange equation for Lagrangian systems, then investigated a
fractional Noether-type theorem and a fractional Hamiltonian
formulation of fractional Lagrangian systems. The fractional
Noether-type theorem was proved by Frederico and Torres \cite{a231}.
For the discussion of fractional constants of motion see also
\cite{a232}, and a more general version of Noether-type's theorem,
valid for fractional problems of optimal control, in the
Riemann-Liouville sense, can be found in \cite{a233}.

The first necessary conditions of Euler-Lagrange were proved by Riewe in
references \cite{a26,a27}, and the first to obtain sufficient
optimality conditions for the Euler-Lagrange fractional equation
were Almeida and Torres in \cite{a234}. Herzallah and Baleanu
\cite{a24} presented the necessary and sufficient optimality
conditions for the Euler-Lagrange fractional equation of fractional
variational problems, hereof the first discussion about the space of
functions where fractional variational problems should be defined,
in order to guarantee existence of solutions, is given in
\cite{a241}. Euler-Lagrange equation for fractional variational
problems with multiple integrals were studied before in \cite{a251,
a252}. Malinowska \cite{a25} proves a fractional Noether-type
theorem for multidimensional Lagrangians and proved the fractional
Noether-type theorem for conservative and nonconservative
generalized physical systems. Wu and Baleanu \cite{a28} developed
some new variational iteration formulae to find approximate
solutions of fractional differential equation and determined the
Lagrange multiplier in a more accurate way. For generalized
fractional Euler-Lagrange equation and fractional order Van der
Pol-like oscillator, we can refer to the works by Odzijewicz
\cite{a29,a30}, Attari et al \cite{a31} respectively. Other the
known results we can see Baleanu et al \cite{a35} and Inokuti et al
\cite{a36}. In view of most of physical phenomena may be considered
as nonconservative, then they can be described using
fractional-order differential equation. Recently, several methods
have been used to solve nonlinear fractional evolution equation
using techniques of nonlinear analysis, such as Adomian
decomposition method \cite{a37}, homotopy analysis method
\cite{a38,a39} and homotopy perturbation method \cite{a40}. It was
mentioned that the variational iteration method has been used
successfully to solve different types of integer and fractional
nonlinear evolution equation.

The KdV equation has been used to describe a wide range of physics
phenomena of the evolution and interaction to nonlinear waves. It
was derived from the propagation of dispersive shallow water waves
and is widely used in fluid dynamics, aerodynamics, continuum
mechanics, as a model for shock wave formation, solitons,
turbulence, boundary layer behavior, mass transport, and the
solution representing the water's free surface over a flat bed
\cite{a1,a2,a3}. Camassa and Holm \cite{a5} put forward the
derivation of solution as a model for dispersive shallow water waves
and discovered that it is formally integrable dimensional
Hamiltonian system, and that its solitary waves are solitons. Most
of classical mechanics techniques have studied conservative systems,
but almost of the processes observed in the physical real world are
nonconservative. In present paper, He's variational iteration method
\cite{a41,a42,a43,a44} is applied to solve time-fractional
generalized KdV equation
\[
{}_0^{R}D^{\alpha}_tu(x,t)+au^{p}(x,t)u_x(x,t)+bu_{xxx}
(x,t)=0,
\]
where $a,b$ are constants, $u(x,t)$ is a field variable, the
subscripts denote the partial differentiation of the function $u(x,
t)$ with respect to the parameter $x$ and $t$. $x\in\Omega(\Omega
\subseteq\mathbb{R})$ is a space coordinate in the propagation
direction of the field and $t\in T(=[0,t_0](t_0>0))$ is the time,
which occur in different contexts in mathematical physics. $a,b$ are
constant coefficients and not equal to zero. The dissipative
$u_{xxx}$ term provides damping at small scales, and the non-linear term
$u^{p}u_x$ $(p>0)$ (which has the same form as that in the KdV or
one-dimensional Navier-Stokes equation) stabilizes by transferring
energy between large and small scales. For $p=1$, we can refer to
the known results of time-fractional KdV equation: formulation and
solution using variational methods \cite{a45}. For $p>0$, $p\neq 1$,
there is a few of the formulation and solution to time-fractional
KdV equation. Thus the present paper considers that the formulation
and solution to time-fractional KdV equation as the index of the
nonlinear term satisfies $p>0$, $p\neq 1$. ${}_0^{R}D^{\alpha}_t$
denotes the Riesz fractional derivative. Making use of
the variational iteration method, this work motivation is devoted to
formulate a time-fractional generalized KdV equation and derives an
approximate solitary wave solution.

This paper is organized as follows: Section \ref{sec2} states some
background material from fractional calculus. Section \ref{sec3}
presents the principle of He's variational iteration method. Section
\ref{sec4} is devoted to describe the formulation of the
time-fractional generalized KdV equation using the Euler-Lagrange
variational technique and to derive an approximate solitary wave
solution. Section \ref{sec5} makes some analysis for the obtained
graphs and discusses the present work.

\section{Preliminaries} \label{sec2}

We recall the necessary definitions for the fractional calculus
(see \cite{a46,a47,a48}) which is used throughout the remaining
sections of this paper.

\begin{definition} \rm
A real multivariable function $f(x,t)$, $t>0$ is said to be in the
space $C_{\gamma}$, $\gamma\in\mathbb{R}$ with respect to $t$ if there
exists a real number $r(>\gamma)$, such that $f(x,t)=t^{r}f_{1}(x,t
)$, where $f_{1}(x,t)\in C(\Omega\times T)$. Obviously, $C_{\gamma}
\subset C_{\delta}$ if $\delta\leq\gamma$.
\end{definition}

\begin{definition}\rm
The left-hand side Riemann-Liouville fractional integral of a
function $f\in C_{\gamma}$, $(\gamma\geq-1)$ is defined by
\begin{gather*}
_0I^{\alpha}_tf(x,t)=\frac{1}{\Gamma(\alpha)}\int_0^{t}\frac{f(
x,\tau)}{(t-\tau)^{1-\alpha}}d\tau, \quad \alpha>0, \quad t\in T, \\
_0I^0_tf(x,t)=f(x,t).
\end{gather*}
\end{definition}

\begin{definition} \rm
The Riemann-Liouville fractional derivatives of the order $\alpha$, with
$n-1\leq \alpha<n$, of a function $f\in C_{\gamma}$, $(\gamma\geq-1)$ are
defined as
\begin{gather*}
_0D^{\alpha}_tf(x,t)=\frac{1}{\Gamma(n-\alpha)}\frac{\partial^
{n}}{\partial t^{n}}\int_0^{t}\frac{f(x,\tau)}{(t-\tau)^{\alpha+1
-n}}d\tau, \\
_tD^{\alpha}_{t_0}f(x,t)=\frac{1}{\Gamma(n-\alpha)}\frac{\partial^
{n}}{\partial t^{n}}\int_t^{t_0}\frac{f(x,\tau)}{(\tau-t)^{\alpha
+1-n}}d\tau, \quad t\in T.
\end{gather*}
\end{definition}

\begin{lemma} \label{lem01}
The integration formula of Riemann-Liouville fractional derivative,
for order $0<\alpha<1$,
\[
\int_{T}f(x,t){}_0D^{\alpha}_tg(x,t)dt=\int_{T}g(x,t)
{}_tD^{\alpha}_{t_0}f(x,t)dt
\]
is valid under the assumption that $f, g\in C(\Omega\times T)$ and
that for arbitrary $x\in \Omega$, ${}_tD^{\alpha}_{t_0}f$,
${}_0D^{\alpha}_tg$ exist at every point $t\in T$ and are
continuous in $t$.
\end{lemma}

\begin{definition}\rm
The Riesz fractional integral of  order $\alpha$, $n-1\leq\alpha<n$, of a
function $f\in C_{\gamma}$, $(\gamma\geq-1)$ is defined as
\[
{}_0^{R}I_t^{\alpha}f(x,t)
=\frac{1}{2}\big({}_0I_t
^{\alpha}f(x,t)+{}_tI_{t_0}^{\alpha}f(x,t)\big)=\frac{1}{2\Gamma
(\alpha)}\int_0^{t_0}|t-\tau|^{\alpha-1}f(x,\tau)d\tau,
\]
where ${}_0I_t^{\alpha}$ and ${}_tI_{t_0}^{\alpha}$
are respectively the left- and right-hand side Riemann-Liouville
fractional integral operators.
\end{definition}

\begin{definition} \label{def} \rm
The Riesz fractional derivative of the order $\alpha$, $n-1\leq\alpha<n$ of a
function $f\in C_{\gamma}$, $(\gamma\geq-1)$ is defined by
\begin{align*}
{}_0^{R}D_t^{\alpha}f(x,t)
&=\frac{1}{2}\big({}_0D_t ^{\alpha}f(x,t)+(-1)^{n}{}_tD_{t_0}^{\alpha}f(x,t)\big) \\
&=\frac{1}{2\Gamma(n-\alpha)}\frac{d^{n}}{dt^{n}}\int_0^{t_0}|t-
\tau|^{n-\alpha-1}f(x,\tau)d\tau,
\end{align*}
\end{definition}
where ${}_0D_t^{\alpha}$ and ${}_tD_{t_0}^{\alpha}$
are respectively the left- and right-hand side Riemann-Liouville
fractional differential operators.

\begin{lemma} \label{lem02}
Let $\alpha>0$ and $\beta>0$ be such that $n-1<\alpha<n$, $m-1<
\beta<m$ and $\alpha+\beta<n$, and let $f\in L_{1}(\Omega\times T)$
and ${}_0I_t^{m-\alpha}f\in AC^{m}(\Omega\times T)$. Then
we have the following index rule:
\[
{}_0^{R}D_t^{\alpha}\big({}_0^{R}D_t^{\beta}f(x,t)\big)
={}_0^{R}D_t^{\alpha+\beta}f(x,t)-\sum_{i=1}^{m}{}_0^{R}
D_t^{\beta-i}f(x,t)|_{t=0}\frac{t^{-\alpha-i}}{\Gamma(1-\alpha-i)}.
\]
\end{lemma}

\begin{remark}\label{rem01} \rm
One can express the Riesz fractional differential operator
${}_0^{R}D_t^{\alpha-1}$ of the order $0<\alpha<1$ as the Riesz
fractional integral operator ${}_0^{R}I_{\tau}^{1-\alpha}$,
i.e.
\[
{}_0^{R}D_t^{\alpha-1}f(x,t)={}_0^{R}I_t^{1-\alpha}
f(x,t), \quad t\in T.
\]
\end{remark}

\section{Variational iteration method} \label{sec3}

The variational iteration method provides an effective procedure for
explicit and solitary wave solutions of a wide and general class of
differential systems representing real physical problems. Moreover,
the variational iteration method can overcome the foregoing
restrictions and limitations of approximate techniques so that it
provides us with a possibility to analyze strongly nonlinear
evolution equation. Therefore, we extend this method to solve the
time-fractional KdV equation. The basic features of the variational
iteration method outlined as follows.

Considering a nonlinear evolution equation that consists of a linear part
$\mathcal {L}u(x,t)$, nonlinear part $\mathcal {N}u(x,t)$, and a
free term $g(x,t)$ represented as
\begin{equation} \label{3.1}
\mathcal {L}u(x,t)+\mathcal {N}u(x,t)=g(x,t).
\end{equation}
According to the variational iteration method, the $n+1$-th
approximate solution of \eqref{3.1} can be read using iteration
correction functional as
\begin{equation} \label{3.2}
u_{n+1}(x,t)=u_{n}(x,t)+\int_0^{t}\lambda(\tau)\big(\mathcal {L}
\tilde{u}(x,\tau)+\mathcal {N}\tilde{u}(x,\tau)-g(x,\tau)\big)d\tau,
\end{equation}
where $\lambda(\tau)$ is a Lagrangian multiplier and $\tilde{u}(x,t
)$ is considered as a restricted variation function, i.e., $\delta
\tilde{u}(x,t)=0$. Extreming the variation of the correction
functional \eqref{3.2} leads to the Lagrangian multiplier
$\lambda (\tau)$. The initial iteration $u_0(x,t)$ can be used as the
initial value $u(x,0)$. As $n$ tends to infinity, the iteration
leads to the solitary wave solution of \eqref{3.1}, i.e.
\[
u(x,t)=\lim_{n\to\infty}u_{n}(x,t).
\]

\section{Time fractional generalized KdV equation} \label{sec4}

The generalized KdV equation in (1+1) dimensions is given as
\begin{equation} \label{4.1}
u_t(x,t)+au^{p}(x,t)u_x(x,t)+bu_{xxx}(x,t)=0,
\end{equation}
where $p>0$ $a,b$ are constants, $u(x,t)$ is a field variable, $x\in
\Omega$ is a space coordinate in the propagation direction of the
field and $t\in T$ is the time. Employing a potential function
$v(x,t)$ on the field variable, set $u(x,t)=v_x(x,t)$ yields the
potential equation of the generalized KdV equation \eqref{4.1} in the
form,
\begin{equation} \label{4.2}
v_{xt}(x,t)+av_x^{p}(x,t)v_{xx}(x,t)+bv_{xxxx}(x,t)=0.
\end{equation}

The Lagrangian of this generalized KdV equation \eqref{4.1} can be
defined using the semi-inverse method \cite{a49,a51} as follows. The
functional of the potential equation \eqref{4.2} can be represented
as
\begin{equation} \label{4.3}
J(v)=\int_{\Omega}dx\int_{T}\Big(v(x,t)\big(c_{1}v_{xt}(x,t)+c_{2}
av_x^{p}(x,t)v_{xx}(x,t)+c_3bv_{xxxx}(x,t)\big)\Big)dt,
\end{equation}
with $c_{i}(i=1,2,3)$ is unknown constant to be determined later.
Integrating \eqref{4.3} by parts and taking $v_t|_{\Omega}=v_x
|_{\Omega}=v_x|_{T}=0$ yield
\begin{equation} \label{4.4}
J(v)=\int_{\Omega}dx\int_{T}\Big(-c_{1}v_x(x,t)v_t(x,t)-\frac
{c_{2}a}{p+1}v_x^{p+2}(x,t)-c_3bv_x(x,t)v_{xxx}(x,t)\Big)dt.
\end{equation}
The constants $c_{i}(i=1,2,\ldots,6)$ can be determined taking the
variation of the functional \eqref{4.4} to make it optimal. By
applying the variation of the functional, integrating each term by
parts, and making use of the variation optimum condition of the
functional $J(v)$, it yields the following expression
\begin{equation} \label{4.5}
-2c_{1}v_{xt}(x,t)-c_{2}a(p+2)v_x^{p}(x,t)v_{xx}(x,t)-2c_3bv_{
xxxx}(x,t)=0.
\end{equation}

We notice that the obtained result \eqref{4.5} is equivalent to
\eqref{4.2}, so one has that the constants $c_{i}(i=1,2, \ldots,6)$
are respectively
\[
c_{1}=-\frac{1}{2},\quad c_{2}=-\frac{1}{p+2},\quad c_3=-\frac{1}{2}.
\]
In addition, the functional expression given by \eqref{4.4} obtains
directly the Lagrangian form of the generalized KdV equation,
\[
L(v_t,v_x,v_{xxx})=\frac{1}{2}v_x(x,t)v_t(x,t)+\frac{a}{(
p+1)(p+2)}v_x^{p+2}(x,t)+\frac{b}{2}v_x(x,t)v_{xxx}(x,t).
\]

Similarly, the Lagrangian of the time-fractional version of the
generalized KdV equation could be read as
\begin{equation} \label{4.6}
\begin{aligned}
&F({}_0D_t^{\alpha}v,v_x,v_{xxx})\\
&=\frac{1}{2}v_x(x,t)
{}_0D_t^{\alpha}v(x,t)+\frac{a}{(p+1)(p+2)}v_x^{p+2}(x,t)
+\frac{b}{2}v_x(x,t)v_{xxx}(x,t),
\end{aligned}
\end{equation}
where $ \alpha\in]0,1]$.
Then the functional of the time-fractional generalized KdV equation
will take the expression
\begin{equation} \label{4.7}
J(v)=\int_{\Omega}dx\int_{T}F({}_0D_t^{\alpha}v,v_x,v_{xxx})dt,
\end{equation}
where the time-fractional Lagrangian $F({}_0D_t^{\alpha}v,
v_x,v_{xxx})$ is given by \eqref{4.6}. Following Agrawal's method
\cite{a32,a33,a34}, the variation of functional \eqref{4.7} with
respect to $v(x,t)$ leads to
\begin{equation} \label{4.8}
\delta J(v)=\int_{\Omega}dx\int_{T}\Big(\frac{\partial F}{\partial
{}_0D_t^{\alpha}v}\delta({}_0D_t^{\alpha}v(x,t))+
\frac{\partial F}{\partial v_x}\delta v_x(x,t)+\frac{\partial F}
{\partial v_{xxx}}\delta v_{xxx}(x,t)\Big)dt.
\end{equation}
By Lemma \ref{lem01}, upon integrating the right-hand side of
\eqref{4.8}, one has
\[
\delta J(v)=\int_{\Omega}dx\int_{T}\Big({}_tD_{T}^{\alpha}\big(
\frac{\partial F}{\partial{}_0D_t^{\alpha}v}\big)-\frac
{\partial}{\partial x}\big(\frac{\partial F}{\partial v_x}\big)
-\frac{\partial^{3}}{\partial x^{3}}\big(\frac{\partial F}{\partial
v_{xxx}}\big)\Big)\delta vdt,
\]
noting that $\delta v|_{T}=\delta v|_{\Omega}=\delta v_x|_{\Omega}
=0$.

Obviously, optimizing the variation of the functional $J(v)$, i.e.,
$\delta J(v)=0$, yields the Euler-Lagrange equation for
time-fractional generalized KdV equation in the following expression
\begin{equation} \label{4.9}
{}_tD_{T}^{\alpha}\big(\frac{\partial F}{\partial{}_0
D_t^{\alpha}v}\big)-\frac{\partial}{\partial x}\big(\frac{\partial
F}{\partial v_x}\big)-\frac{\partial^{3}}{\partial x^{3}}\big(
\frac{\partial F}{\partial v_{xxx}}\big)=0.
\end{equation}
Substituting the Lagrangian of the time-fractional generalized KdV
equation \eqref{4.6} into Euler-Lagrange formula \eqref{4.9} obtains
\[
{}_tD_{T}^{\alpha}\big(\frac{1}{2}v_x(x,t)\big)-{}_0
D_t^{\alpha}\big(\frac{1}{2}v_x(x,t)\big)-av_x^{p}(x,t)v_{xx}
(x,t)-bv_{xxxx}(x,t)=0.
\]

Once again, substituting the potential function $v_x(x,t)$ for
$u(x,t)$, yields the time-fractional generalized KdV equation for the
state function $u(x,t)$ as
\begin{equation} \label{4.10}
\begin{split}
\frac{1}{2}\big({}_0D_t^{\alpha}u(x,t)-{}_tD_{T}^{
\alpha}u(x,t)\big)+au^{p}(x,t)u_x(x,t)+bu_{xxx}(x,t)=0.
\end{split}
\end{equation}

According to the Riesz fractional derivative ${}_0^{R}D_t^
{\alpha}u(x,t)$, the time-fractional generalized KdV equation
represented in \eqref{4.10} can write as
\begin{equation} \label{4.11}
{}_0^{R}D_t^{\alpha}u(x,t)+au^{p}(x,t)u_x(x,t)+bu_{xxx}
(x,t)=0.
\end{equation}
Acting from left-hand side by the Riesz fractional operator
${}_0^{R}D_t^{1-\alpha}$ on \eqref{4.11} leads to
\begin{equation} \label{4.12}
\begin{split}
&\frac{\partial}{\partial t}u(x,t)-{}_0^{R}D_t^{\alpha-1}
u(x,t)|_{t=0}\frac{t^{\alpha-2}}{\Gamma(\alpha-1)}\\
&+{}_0^{R}
D_t^{1-\alpha}\Big(au^{p}(x,t)\frac{\partial}{\partial x}u(x,t)
+b\frac{\partial^{3}}{\partial x^{3}}u(x,t)\Big)=0,
\end{split}
\end{equation}
from Lemma \ref{lem02}. In view of the variational iteration method,
combining with \eqref{4.12}, the $n+1$-th approximate solution of
\eqref{4.11} can be read using iteration correction functional as
\begin{equation} \label{4.13}
\begin{split}
u_{n+1}(x,t)
&=u_{n}(x,t)+\int_0^{t}\lambda(\tau)\Big[\frac{\partial}
{\partial \tau}u_{n}(x,\tau)-{}_0^{R}D_{\tau}^{\alpha-1}u_{n}(x,
\tau)|_{\tau=0}\frac{\tau^{\alpha-2}}{\Gamma(\alpha-1)} \\
&\quad+{}_0^{R}D_{\tau}^{1-\alpha}\Big(a\tilde{u}_{n}^{p}(x,\tau)
\frac{\partial}{\partial x}\tilde{u}_{n}(x,\tau)+b\frac{\partial^{3}}
{\partial x^{3}}\tilde{u}_{n}(x,\tau)\Big)\Big]d\tau, \quad n\geq 0,
\end{split}
\end{equation}
where the function $\tilde{u}_{n}(x,t)$ is considered as a
restricted variation function, i.e., $\delta\tilde{u}_{n}(x,t)=0$.
The extreme of the variation of \eqref{4.13} subject to the
restricted variation function straightforwardly yields
\begin{align*}
\delta u_{n+1}(x,t)
&=\delta u_{n}(x,t)+\int_0^{t}\lambda(\tau)
\delta\frac{\partial}{\partial \tau}u_{n}(x,\tau)d\tau \\
&=\delta u_{n}(x,t)+\lambda(\tau)\delta u_{n}(x,\tau)|_{\tau=t}-
\int_0^{t}\frac{\partial}{\partial\tau}\lambda(\tau)\delta u_{n}
(x,\tau)d\tau
=0.
\end{align*}
This expression reduces to the  stationary conditions
\[
\frac{\partial}{\partial \tau}\lambda(\tau)=0, \quad 1+\lambda(\tau
)=0,
\]
which converted to the Lagrangian multiplier at $\lambda(\tau)=-1$.
Therefore, the correction functional \eqref{4.13} takes the
following form
\begin{equation} \label{4.14}
\begin{split}
u_{n+1}(x,t)
&=u_{n}(x,t)-\int_0^{t}\Big[\frac{\partial}{\partial
\tau}u_{n}(x,\tau)-{}_0^{R}I_{\tau}^{1-\alpha}u_{n}(x,\tau)|
_{\tau=0}\frac{\tau^{\alpha-2}}{\Gamma(\alpha-1)} \\
&\quad+{}_0^{R}D_{\tau}^{1-\alpha}\Big(au_{n}^{p}(x,\tau)
\frac{\partial}{\partial x}u_{n}(x,\tau)+b\frac{\partial^{3}}
{\partial x^{3}}u_{n}(x,\tau)\Big)\Big]d\tau, \quad n\geq0,
\end{split}
\end{equation}
since $\alpha-1<0$, the fractional derivative operator ${}_0^
{R}D_t^{\alpha-1}$ reduces to fractional integral operator $
{}_0^{R}I_t^{1-\alpha}$ by Remark \ref{rem01}.

In view of the right-hand side Riemann-Liouville fractional
derivative is interpreted as a future state of the process in
physics. For this reason, the right-derivative is usually neglected
in applications, when the present state of the process does not
depend on the results of the future development, and so the
right-derivative is used equal to zero in the following
calculations. The zero order solitary wave solution can be taken as
the initial value of the state variable, which is taken in this case
as
\[
u_0(x,t)=u(x,0)=k\operatorname{sech} ^{2/p}(\frac{p}{2\sqrt{b}}(x
+\eta_0)),
\]
where $k=\big(\frac{(p+1)(p+2)}{2a}\big)^{\frac{1}{p}}$, $\eta_0$
is a constant.

Substituting this zero order approximate solitary wave solution into
\eqref{4.14} and using the Definition \ref{def} leads to the first
order approximate solitary wave solution
\[
u_{1}(x,t)=k\operatorname{sech} ^{2/p}(\frac{p}{2\sqrt{b}}(x+\eta_0))
\Big(1+\frac{t^{\alpha}}{\sqrt{b}\Gamma(\alpha+1)}\tanh(\frac{p}
{2\sqrt{b}}(x+\eta_0))\Big).
\]

Substituting the first order approximate solitary wave solution into
\eqref{4.14}, using the Definition \ref{def} then leads to the
second order approximate solitary wave solution in the following
form
\begin{align*}
&u_{2}(x,t)\\
&=k\operatorname{sech} ^{2/p}(\frac{p}{2\sqrt{b}}(x+\eta_0)
)+\frac{t^{\alpha}}{\Gamma(\alpha+1)}\Big[\frac{ak^{p+1}}{\sqrt{b}}
\operatorname{sech} ^{\frac{2}{p}+2}(\frac{p}{2\sqrt{b}}(x+\eta_0))\\
&\quad\times \tanh\big(\frac{p}{2\sqrt{b}}(x+\eta_0)\big)
 +\frac{k}{\sqrt{b}}\operatorname{sech} ^{2/p}(\frac{p}{2\sqrt{b}}
(x+\eta_0))\tanh^{3}(\frac{p}{2\sqrt{b}}(x+\eta_0))
\\
&\quad -\frac{3kp}{2\sqrt{b}}
\operatorname{sech} ^{\frac{2}{p}+2}(\frac{p}{2\sqrt{b}}(x+\eta_0))
-\frac{kp^{2}}{2\sqrt{b}}\operatorname{sech} ^{\frac{2}{p}+2}(\frac{p}
{2\sqrt{b}}(x+\eta_0))\\
&\quad\times \tanh(\frac{p}{2\sqrt{b}}(x+\eta_0))\Big] \\
&\quad -\frac{t^{2\alpha}}{\Gamma(2\alpha+1)}\Big[\frac{ak^{p+1}}{b}
\operatorname{sech} ^{\frac{2}{p}+2}(\frac{p}{2\sqrt{b}}(x+\eta_0))\big(\frac
{p}{2}-\frac{3p+2}{2}\tanh^{2}(\frac{p}{2\sqrt{b}}(x+\eta_0))\big)
\\
&\quad-\frac{3kp^{2}(p+2)}{8b}\operatorname{sech} ^{2/p}(\frac{p}{2\sqrt{b}}
(x+\eta_0))\tanh^{2}(\frac{p}{2\sqrt{b}}(x+\eta_0))
\\
&\quad -\frac{3kp^{2}(p+2)} {8b}\operatorname{sech} ^{\frac{2}{p}+2}
 (\frac{p}{2\sqrt{b}}(x+\eta_0))
 +\frac{3kp(p+2)^{2}}{8b}\operatorname{sech} ^{2/p}(\frac{p}{2\sqrt{b}}
(x+\eta_0))\\
&\quad\times \tanh^{2}(\frac{p}{2\sqrt{b}}(x+\eta_0))
 +\frac{kp^{3}}{8b}
 \operatorname{sech} ^{2/p}(\frac{p}{2\sqrt{b}}(x+\eta_0)) \\
&\quad -\frac{k(p+2)^{3}}{8b}\operatorname{sech} ^{2/p}(\frac{p}{2\sqrt{b}}
(x+\eta_0))\tanh^{4}(\frac{p}{2\sqrt{b}}(x+\eta_0)) \\
&\quad+\frac{3k(p+2)^{2}}{8b}\operatorname{sech} ^{\frac{2}{p}+2}(\frac{p}{2\sqrt{b}}
(x+\eta_0))\tanh^{2}(\frac{p}{2\sqrt{b}}(x+\eta_0)) \\
&\quad +\frac{kp^{2}(p+2)}{4b}\operatorname{sech} ^{\frac{2}{p}+2}(\frac{p}{2\sqrt{b}}
(x+\eta_0))\tanh^{2}(\frac{p}{2\sqrt{b}}(x+\eta_0))\Big] \\
&\quad -\frac{\Gamma(2\alpha+1)t^{3\alpha}} {\Gamma(3\alpha+1)(\Gamma(\alpha
+1))^{2}}\Big[\frac{apk^{p+1}}{2b\sqrt{b}}\operatorname{sech} ^{\frac{2}{p}+2}(\frac
{p}{2\sqrt{b}}(x+\eta_0))\tanh(\frac{p}{2\sqrt{b}}(x+\eta_0)) \\
&\quad \times\big(p-\frac{p+2}{2}\tanh^{2}(\frac{p}{2\sqrt{b}}(x+\eta_0))
\big)\Big].
\end{align*}
Using Definition \ref{def} and the Maple package or
Mathematics, we obtain $u_3(x,t)$, $u_4(x,t)$ and so on,
substituting $n-1$ order approximate solitary wave solution into
\eqref{4.14}, there leads to the $n$ order approximate solitary wave
solution. As $n$ tends to infinity, the iteration leads to the
solitary wave solution of the time-fractional generalized KdV equation
$$
u(x,t)=k\operatorname{sech} ^{2/p}\Big(\frac{p}{2\sqrt{b}}(x-t+\eta_0)\Big).
$$

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig1a} % 45.eps
\includegraphics[width=0.45\textwidth]{fig1b} % 12.eps
\end{center}
\caption{The function $u$ as a
3-dimensions graph for order $\alpha$:
 (A1) $\alpha=4/5$, (A2) $\alpha=1/2$}
\label{fig1}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig2a} % B1.eps
\includegraphics[width=0.45\textwidth]{fig2b} % B2.eps
\end{center}
\caption{The function $u$ as a
function of space $x$ at time $t=1$ for order $\alpha$:
(B1) 3-dimensions graph, (B2) 2-dimensions graph}
\label{fig2}
\end{figure}


\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig3a} % C1.eps
\includegraphics[width=0.45\textwidth]{fig3b} % C2.eps
\end{center}
\caption{The amplitude of the
function $u$ as a function of time $t$ at space $x=1$ for order
$\alpha$: (C1) 3-dimensions graph, (C2) 2-dimensions graph}
\label{fig3}
\end{figure}


\section{Discussion} \label{sec5}

The target of present work is to explore the effect of the
fractional order derivative on the structure and propagation of the
resulting solitary waves obtained from time-fractional generalized
KdV equation. We derive the Lagrangian of the generalized KdV
equation by the semiinverse method, then take a similar form of
Lagrangian to the time-fractional generalized KdV equation. Using
the Euler-Lagrange variational technique, we continue our
calculations until the three-order iteration. During this period,
our approximate calculations are carried out concerning the solution
of the time-fractional generalized KdV equation taking into account
the values of the coefficients and some meaningful values namely,
$\frac{4}{5}$, $\frac{1}{2}$ and $p=3, a=10, b=1, \eta_0=0$. The
solitary wave solution of time-fractional generalized KdV equation
are obtained. In addition, 3-dimensional representation of the
solution $u(x,t)$ for the time-fractional generalized KdV equation
with space $x$ and time $t$ for different values of the order
$\alpha$ is presented respectively in Figure \ref{fig1}, the solution $u$ is
still a single soliton wave solution for all values of the order
$\alpha$. It shows that the balancing scenario between nonlinearity
and dispersion is still valid. Figure \ref{fig2} presents the change of
amplitude and width of the soliton due to the variation of the order
$\alpha$, 2- and 3-dimensional graphs depicted the behavior of the
solution $u(x,t)$ at time $t=1$ corresponding to different values of
the order $\alpha$. This behavior indicates that the increasing of
the value $\alpha$ is uniform both the height and the width of the
solitary wave solution. That is, the order $\alpha$ can be used to
modify the shape of the solitary wave without change of the
nonlinearity and the dispersion effects in the medium. Figure \ref{fig3}
devoted to study the expression between the amplitude of the soliton
and the fractional order at different time values. These figures
show that at the same time, the increasing of the fractional
$\alpha$ increases the amplitude of the solitary wave to some value
of $\alpha$.

\subsection*{Acknowledgements}
The author is very grateful to the anonymous referees for their
careful reading of this manuscript and for suggesting invaluable
changes.

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\end{document}