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\fancyhead[CE]{Maryam Yourdkhany, Mehdi Nadjafikhah} 
\fancyhead[CO]{Symmetry classification and invariance of the Reynolds equation]{Symmetry classification and invariance of the Reynolds equation}



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{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
\vspace*{9mm}

\begin{center}

{\Large \bf 
Symmetry classification and invariance of the Reynolds equation\\}
%{\bf Do You Have a Subtitle? \\ If so, Write It Here} 


\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}

{\bf Maryam Yourdkhany}\vspace*{-2mm}\\
\vspace{2mm} {\small  Karaj Branch, Islamic Azad University } \vspace{2mm}

{\bf Mehdi Nadjafikhah$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small  Iran University of Science and Technology} \vspace{2mm}

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{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} In this essay, extensions to the results of  Lie symmetry classification of Reynolds equation are proposed.
The infinitesimal technique is used to derive symmetry groups of the Reynolds equation.
One-dimensional optimal system is constructed for symmetry sub-algebras
gained through Lie point symmetry. At the end, the general symmetry group of the non-conservative generalized thin-film equation are determined.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 53C10, 53C12, 53A55, 76M60, 58J70

\noindent{\bf Keywords and Phrases:} Symmetry, Reynolds equation, optimal system
\end{quotation}}

\section{Introduction}
\label{intro} % It is advised to give each section and subsection a unique label.
As you know the symmetry property is a natural phenomenon. By using partial differential
equations having symmetry properties, we can describe many physical, biological and chemical processes.
After creating the group classification method by Sophus Lie in 19th century \cite{Nad1}, Lie symmetry analysis has always been an interesting method for mathematicians in dealing with differential equations. The Lie group approach proposes a useful procedure for integrability, reducing equations and finding out the exact solutions of differential equations. Its algorithm is as follows that group of transformations
 transforms solutions of the system of differential equations to other solutions of them \cite{Olv1,Olv2}.
In the first article published on this subject \cite{Lie}, Lie calculated symmetry group of one dimensional heat equation and then reduced this equation by symmetry reduction method to find solution of it. In \cite{Ovs}, partially invariant solutions has extended by Ovsiannikov. In this attempt, the significance of the equivalence group has investigated. An equivalence group or Lie transformation group acts on the generalization space of independent variables, dependent variables, and their derivatives while keeps the class of partial differential equations.


This article is assigned to studying and finding out analytical solutions of the one-dimensional Reynolds equation.
This partial differential equation by describing pressure generation of thin viscous fluid films is one of the important equations in fluid dynamics and lubrication theory. The initial version of this equation was proposed by Osborne Reynolds in 1886. The extension of rupture singularities in this equation is studied in \cite{Fin}. In the one- dimensional Reynolds equation $x$ belongs to  bounded interval $[0,l]$,
\begin{equation}\label{fin}
	u_t=\partial_x\left(u^3p_x\right)-J, 
	\end{equation}
	where $u$ is fluid film thickness and $J=-\gamma p(u) /(u+K_0)$ is the non-conservative flux. $\gamma$ is a scaling constant, $K_{0}>0$ and $p (u) \equiv f (u)-u_{xx}$ is the fluid pressure. $f (u)$ and $u_{xx}$ respectively are disjoining pressure function and the linearised curvature. A physical model stimulates the form of $f (u) $. For $\gamma=0$, we have the fourth-order differential equation  for one dimensional covering flows \cite{Fin} as follows:
	\begin{equation}\label{rey}
	u_t=\partial_x\left (u^{3}\partial_x\left[f (u)-u_{xx}\right]\right). 
	\end{equation}
	Rapture bearing in a non conservative generalized thin film equation 
	will allow to compete evaporation and dewetting and the competition between them conduce finite time rupture.

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\section{Lie symmetry group analysis}
\label{sec:2}
Suppose a partial differential equation including one dependent variable and $p$ independent variables. A one-parameter Lie group of point transformations
	\begin{equation}
	\overline{x}_{i}=x_{i}+\epsilon\xi_{i} (x, u)  +O (\epsilon^{2}) ;\qquad\overline{u}=u+\epsilon\varphi (x, u)  +O (\epsilon^{2}) , 
	\end{equation}
	where $i=1, \cdots, p$, and $\xi_{i}=\partial_\epsilon\overline{x}_{i}|_{\epsilon=0}$, operating on $ (x, u)-space$. 
	\begin{equation}\label{vec}
	\textbf{x}=\xi_{i}\partial_{x_i}+\varphi\partial_u, \qquad i=1, \cdots, p,
	\end{equation}
	is infinitesimal generator.
	Therefore the vector field $\textbf{x}$ have characteristic function $Q (x, u^{1})=\varphi (x, u)-\sum_{i=1}^{p}\xi_{i} (x, u) u_{x_{i}}$. The symmetry generator associated with (\ref{vec}) is presented by 
	\begin{equation}\label{vec1}
	\textbf{x}=\xi\partial_x+\tau\partial_t+\varphi\partial_u. 
	\end{equation}
	The vector field $\textbf{x}^{ (4) }=\textbf{x}+\varphi^x\partial_{u_x}+\cdots+\varphi^{tt}\partial_{u_{tt}}$,is second prolongation of $\textbf{x}$. Its coefficients are:
	\begin{equation}\label{lj}
	\varphi^{l}=D_{l}Q+\xi u_{xl}+\tau u_{tl}, \qquad  \varphi^{lj}=D_{l} (D_{j}Q)  +\xi u_{xlj}+\tau u_{tlj}, 
	\end{equation}
	where $Q=\varphi-\xi u_{x}-\tau u_{t}$ is the charactrestic associated with the presentation vector field by (\ref{vec1}) and the operator $D_{i}$ introduces total derivative and index of $u$ is detail derivative in terms of independent variables. Substituting $x$ and $t$ coordinates into \eqref{lj} cofficient functions are:
% 
\begin{align}
	\varphi^{x}&=D_{x}\varphi-u_{x}D_{x}\xi-u_{t}D_{x}\tau, &\varphi^{t}&=D_{t}\varphi-u_{x}D_{t}\xi-u_{t}D_{t}\tau, \nonumber \\
	\varphi^{xt}&=D_{t}\varphi^{x}-u_{xx}D_{t}\xi-u_{xt}D_{t}\tau,  &\varphi^{xx}&=D_{x}\varphi^{x}-u_{xx}D_{x}\xi-u_{xt}D_{x}\tau,\nonumber \\
\cdots. \nonumber
	\end{align}
	Where the total derivatives operators $D_{x}$ and $D_{t}$ are as follows:
	%
	\begin{align}
	D_{x}&=\partial_x+u_{x}\partial_u+u_{xx}\partial{u_{x}}+u_{xt}\partial_{u_{t}}+\cdots,\nonumber \\
	D_{t}&=\partial_t+u_{t}\partial_u+u_{tt}\partial_{u_{t}}+u_{tx}\partial_{u_{x}}+\cdots. \nonumber
	\end{align}
If vector field \label{vec1} is admitted by Eq.\eqref{rey}, then $\textbf{x}$ must satisfy in the condition $\textbf{x}^{ (4) }[u_{t}-\partial_x (u^3\partial_x (f (u)-u_{xx}) ) ]=0$, whenever \\ $u_{t}=\partial_x (u^3\partial_x[f (u)-u_{xx}])$.
	
	Therefore we obtain the compelet set of determining equations\\
 $\{ \tau_{tt}=\varphi_x=\varphi_t=\xi_t=\xi_u=0, 2u\varphi f_{uu}+uF_tf_u+3\varphi=0, \\
	4u\xi_x=3\varphi+uF_t, u\varphi_u=\varphi\}$, \\
	wherer $\xi, \tau, \varphi$ are depend on $x, t, u$ and $f$ depends on $u$ and $F$ is arbitrary function. With solving determining equation we obtain \\
	$F=(4c_{2}-3c_{1}) t+c_{4}$, $\varphi=c_{1}u$, and $\xi=c_{2}x+c_{3}$. Where $c_{i}$, $i=1, \cdots ,4$ are arbitrary constants. Therefore we have:
	\begin{equation}\label{eq}
	c_1uf_{uu}+2c_2f_u=0. 
	\end{equation}

\section{Lie group classification of Reynolds equation}

In the following section, according to the above discussion
about Lie theory, considering \eqref{eq} and  we classify the symmetries of the Eq.\eqref{rey}. We consider four general Cases(Idea from\cite{Nad1,Nad2,Nad3}). 

\paragraph{Case 1:} If $f^{\prime}=0$, then $f=a$ is constants. Therefore we have
	\begin{equation}\label{a}
	u_t=-3u^2u_xu_{x^3}-u^3u_{x^4}. 
	\end{equation}
	For the equation \eqref{a} the Lie group infinitesimals are $\xi=c_1t+c_2$,\\ 
	$\tau=c_3x+c_4$, $\varphi=u(4c_{3}-c_{1})/3$. Thus the infinitesimal generator of symmetry algebra is resulted as 
	\begin{equation}\label{b}
	\textbf{x}_{1}=\partial_x, \quad \textbf{x}_{2}=\partial_t, \quad \textbf{x}_{3}=t\partial_t-\dfrac{1}{3}u\partial_u, \quad \textbf{x}_{4}=x\partial_x+\dfrac{4}{3}u\partial_u. 
	\end{equation}
	The characteristic equation associated with $\textbf{x}_{3}$ is $dx/0=dt/t=du/(-u/3) $. By integrating we obtain
	\begin{equation}\label{chang}
	r=x, \qquad s=\ln t, \qquad v (r)=ut^{1/3}. 
	\end{equation}
	Substituting \eqref{chang} in \eqref{a} leads to $9vv_{rrr}v_r+3v^2v_{rrrr}=1$. The reduced equation is $ut^{1/3}=0$. Therefore $u=0$. 
	
	The characteristic equation associated with $\textbf{x}_{4}$ is $dx/x=dt/0=du/(4u/3) $.
	By integrating we obtain $r=t$, $s=\ln x$, $v(r)=ux^{-4/3}$. Therefore $u=c_{1}=0$. 
	
	The invariants associated with $\textbf{x}_{1}$ are $t$ and $u$, and its symmetry group is $g_1^\epsilon=(x+\epsilon, t, u) $. 
	
	The invariants associated with $\textbf{x}_{2}$ are $x$ and $u$, and its symmetry group is $g_2^\epsilon=(x, t+\epsilon, u) $. 
	
	The invariants associated with $\textbf{x}_{3}$ are $x$ and $ut^{1/3}$, and its symmetry group is $g_3^\epsilon=(x, e^{\epsilon}t, e^{-\epsilon/3}u) $. 
	
	The invariants associated with $\textbf{x}_{4}$ are $t$ and $x^{-4/3}u$ and its symmetry group is $g_4^\epsilon=(e^{\epsilon}x, t, e^{4\epsilon/3}u) $. 
	\begin{table}
		\caption{Commutator table for Case 1 and Case 3.}
		\label{tab:1}       % Give a unique label
		% For LaTeX tables use
		\begin{center}
\begin{tabular}{c|cccc}
\hline
 $[\textbf{x}_{i}, \textbf{x}_{j}]$ & ${\textbf{x}_{1}}$ & ${\textbf{x}_{2}}$ & ${\textbf{x}_{3}}$ & ${\textbf{x}_{4}}$ \\ \hline $\textbf{x}_{1}$ & 0 & 0 & 0 & $\textbf{x}_{1}$ \\ $\textbf{x}_{2}$ & 0 & 0 & $\textbf{x}_{2}$ & 0 \\ $\textbf{x}_{3}$ & 0 & $-\textbf{x}_{2}$ & 0 & 0 \\ $\textbf{x}_{4}$ & $-\textbf{x}_{1}$ & 0 & 0 & 0 \end{tabular}
%
\begin{tabular}{c|cccc} 
\hline
$[\textbf{x}_{i}, \textbf{x}_{j}]$ & ${\textbf{x}_{1}}$ & ${\textbf{x}_{2}}$ & ${\textbf{x}_{3}}$ \\ \hline $\textbf{x}_{1}$ & 0 & 0 & $-\alpha^{-1}\textbf{x}_{1}$ \\ $\textbf{x}_{2}$ & 0 & 0 & $\textbf{x}_{2}$ \\ $\textbf{x}_{3}$ & $\alpha^{-1}\textbf{x}_{1}$ & $-\textbf{x}_{2}$ & 0
\end{tabular}
\end{center}
	\end{table}

\paragraph{Case 2:} If $f^{\prime\prime}=0$ and $f^{\prime}\neq0$ then $f=au+b$ ($a, b\in\mathbb{R}, a\neq0$) . Therefore we have
	\begin{equation}\label{ab}
	u_t=3u^2u_x(u_x-u_{xxx})+u^3 (au_{xx}-u_{xxxx}). 
	\end{equation}
	The symmetry algebra is generated by the Lie symmetry generators
	$\textbf{x}_1=\partial_x$, $\textbf{x}_2=\partial_t$, $\textbf{x}_3=t\partial_t-(u/3)\partial_u$. 
	The characteristic equation associated with $\textbf{x}_{3}$ is\\
 $dx/0=dt/t=du/(-u/3) $. By integrating we obtain $r=x$, $s=\ln t$, \\
 $v(r)=ut^{1/3}$. Substituting \eqref{chang} in \eqref{ab} leads to 
	\begin{equation*} 
	3v^2v_{r^4}+9vv_rv_{r^3}=3av^2v_{rr}+9avv_r^2+1. 
	\end{equation*}
	The reduced equation is $ut^{1/3}=0$. Therefore $u=0$. The invariants associated and symmetry groups with $\textbf{x}_1$ and $\textbf{x}_2$ and $\textbf{x}_{3}$ aforesaid.
	\paragraph{Case 3:} 
	If $f''\neq0$ then $f''/f'=-2c_2u/c_1$. By integrating we obtain\\
 $f=au^b+c$, where $a,c\in\mathbb{R}$, $b\in\mathbb{N}$, $a\neq0$, and $b>1$. 
	
	The equation is
	\begin{equation*}
	u_t=abu^{b+1}((b+2) u_x^2+uu_{xx})-3u^2u_x u_{x^3}-u^3u_{x^4}. 
	\end{equation*}
	The Lie algebra is generated by the Lie symmetry vectors $\textbf{x}_1=\partial_x$, $\textbf{x}_2=\partial_t$, $\textbf{x}_3=(4b+2)^{-1}\big((b-1)x\partial_x+t(4b+2)\partial_t-2u\partial_u\big)$. 
	By integrating the characteristic equation associated with $\textbf{x}_{3}$ we obtain $r=tx^\alpha$,\\
	$s=-\alpha\ln x$, and $v (r)=ux^{-\alpha/(2b+9)}$, where $\alpha=(4b+2)/(1-b)$. 
	
	The invariants associated and symmetry groups with $\textbf{x}_{1}$ and $\textbf{x}_{2}$ aforesaid. The invariants associated with $\textbf{x}_{3}$ are $tx^\alpha$ and $ux^{2/(b-1)}$. 
	\paragraph{Case 4:} Otherwise $c_{1}=c_{2}=0$. 
Herein, the Lie symmetry algorithm leads us to the generators $\textbf{x}_{1}=\partial_x$ and $\textbf{x}_{2}=\partial_t$. The invariants and the symmetry group associated with $\textbf{x}_{1}$ and $\textbf{x}_{2}$ aforesaid. In this Case $u=0$.\\
The commutatore table for Case 1 and Case 3 are listed in the table \eqref{tab:1}.
\begin{theorem}
The Reynolds equation have maximum four generator \eqref{b} and minimum two generator $\partial_x$ and $\partial_t$.  
\end{theorem}

\section{Optimal control system of the Reynolds equation}
In what follows we perform the optimal system for one dimensional subalgebra of the Reynolds equation. In order to obtain a complete optimal system, we classify the orbits for the adjoint representation. For this purpose, we take an element of the Lie algebra and simplify it by adjoint transformation.
\begin{definition}
{\it Suppose $\mathfrak{g}$ be Lie algebra corresponding to Lie group $G$. 
An optimal system of $r$-parameter subgroups is a list of conjugacy non-equivalent $r$-parameter subalgebras which are not related by transformations that is to say any other subgroup is conjugate to exactly one subgroup in the list. In asimilar way, a list of $r$-parameter subalgebra constitutes an optimal system if  between every $r$-parameter subalgebra of $ \mathfrak{g}$ with a unique element of the list there is an equivalence relation, under some elements of the adjoint representation  $\overline{\mathfrak{h}}=\mathrm{Ad} (g (\mathfrak{h}))$\cite{Olv2}.}
\end{definition}

\begin{theorem} {\normalfont (See \cite{Fin})}
Suppose $G$ be Lie group with corresponding Lie algebra $\mathfrak{g}$ and $H$ and $\overline{H}$ be $s-$dimensional Lie subgroups of the Lie group $G$  that are connected to each other  and corresponding Lie subalgebras are $\mathfrak{h}$ and $\overline{\mathfrak{h}}$ respectively. Then $\overline{H}=gHg^{-1}$ are conjugate subgroups if and only if $\overline{\mathfrak{h}}=\mathrm{Ad} (g (\mathfrak{h}) )$.
\end{theorem}

We apply the following Lie series to computing the adjoint representation \\
	$\mathrm{Ad} (\exp (\epsilon\textbf{x}_i) \textbf{x}_j)=\textbf{x}_j-\epsilon[\textbf{x}_i, \textbf{x}_j]+(\epsilon^2/2)[\textbf{x}_i, [\textbf{x}_i, \textbf{x}_j]]-\cdots$,\\
 where $[\textbf{x}_{i}, \textbf{x}_{j}]$ is the Lie bracket for the Lie algebra, $\epsilon$ is a parameter, $i, j=1, 2, 3, 4$. \\
 The adjoint actions of the symmetry generators for Case 1 are listed in the table \eqref{tab:2}.
 \begin{table}
		\caption{Adjoint representation table of the symmetry generators for Case 1.}
	\label{tab:2}
	\begin{center}
	\begin{tabular}{c|cccc} 
	\hline
$\mathrm{Ad} (\exp (\epsilon\textbf{x}_{i}) \textbf{x}_{j})$ & ${\textbf{x}_{1}}$ & ${\textbf{x}_{2}}$ & ${\textbf{x}_{3}}$ & ${\textbf{x}_{4}}$ \\ 
\hline $\textbf{x}_{1}$ & $\textbf{x}_{1}$ & $\textbf{x}_{2}$ & $\textbf{x}_{3}$ & $\textbf{x}_{4}-\epsilon\textbf{x}_{1}$\\
$\textbf{x}_{2}$ & $\textbf{x}_{1}$ & $\textbf{x}_{2}$ & $\textbf{x}_{3}-\epsilon\textbf{x}_{2}$ & $\textbf{x}_{4}$\\
$\textbf{x}_{3}$ & $\textbf{x}_{1}$ & $e^{\epsilon}\textbf{x}_{2}$ & $\textbf{x}_{3}$ & $\textbf{x}_{4}$\\
$\textbf{x}_{4}$ & $e^{\epsilon}\textbf{x}_{1}$ & $\textbf{x}_{2}$ & $\textbf{x}_{3}$ & $\textbf{x}_{4}$
\end{tabular}
\end{center}	
		\end{table}
		
\begin{theorem}
The one-dimensional optimal system of Lie subalgebras of the equation (\ref{a}) is as $\{\textbf{x}_1$, $\beta\textbf{x}_{1}+\textbf{x}_3$, $\gamma\textbf{x}_1+\textbf{x}_2$, $\delta\textbf{x}_3+\textbf{x}_4\}$.
\end{theorem}		
\begin{proof}
A nonzero vector $\textbf{x}=a_1\textbf{x}_1+a_2\textbf{x}_2+a_3\textbf{x}_3+a_4\textbf{x}_4$ is given. 
We start by Simplification of the coefficient $a_i$ as far as possible through judicious applications of adjoint maps to $\textbf{x}$. 
	
	Let $a_4\neq0$. Scaling $\textbf{x}$ if necessary, we let $a_4=1$. Considering table 3 and the vanishing coefficients $\textbf{x}_1$, $\textbf{x}_2$ then vector $\textbf{x}$ is equivalent to $\delta\textbf{x}_3+\textbf{x}_4$. 
	
	If $a_4=0$, $a_3\neq0$, then we can consider that $a_3=1$  and then the coefficients of $\textbf{x}_2$ vanish. Thus the vector $\textbf{x}$ is equivalent to $\beta\textbf{x}_1+\textbf{x}_3$. 
	
	If $a_4=a_3=0$, $a_2\neq0$, then we can assume that $a_2=1$. Then the vector $\textbf{x}$ is equivalent to $\gamma\textbf{x}_1+\textbf{x}_2$. 
	
	If $a_4=a_3=a_2=0$, then we can assume that $a_1=1$ and the vector $\textbf{x}$ is equivalent to $\textbf{x}_1$. 
\end{proof}

\section{Generalization}
	 Consider the following generalized thin-film equation describing the effect of surface tension
	\begin{equation}\label{gen}
	u_{t}=-(u^nu_{xxx}) _{x}.
	\end{equation}
	If we let $n=3$ in the above equation we get to Case1. Bernis and Friedman in their work obtained the solutions of Eq.\eqref{gen} for $n\geq4$ on bounded domains \cite{Ber}. In \cite{Fin} influence of the non-conservative to create rupture to dominate the intermolecular forces has been investigated.

 
	
	The behavior of solution of the non-conservative generalized thin-film equation
	\begin{equation}\label{gene}
	u_{t}=-\partial_x(u^n\partial_x[u^{-4}+u_{xx}])-u^{-(m+4) }+u^{-m}u_{xx},
	\end{equation} 
depend on the values of the parameters $m$, $n$ that control the competing non-conservative influences and arranging conservative surface tension influences. The dynamism coefficients $(n,m)$ are the conservative and evaporative terms respectively \cite{Fin1}.

	The Lie symmetry algebra of \eqref{gene} is generated by vector fields $\partial_x$ and $\partial_t$. By useing the characteristic equation associated with $\partial_x$ and integrating and substituting in \eqref{gene} we obtain $v-{r}=-v^{-m-4}$ that $r=t$, $s=x$, $v(r)=u$. The reduced equation is $u=(-5t-tm+c_{1}) ^{-m-5}$. $u$ is independent on the values of the parameter $n$.

\section{Conclusion and motivation}
	In what presented, we consider the Reynolds equations in the 4 Cases, then using the classical Lie symmetric method we determined Lie symmetries group and their invariants associated to the Reynolds equation $u_{t}=\partial_x (u^3\partial_x[f (u)-u_{xx}])$ where $f (u)$ is a arbitrary smooth function on $u$, in the different Cases.  The commutator table and adjoint representation table of the Lie symmetry generator is constructed. Ultimately, using the optimal control theory  we achieve the optimal system of the Reynolds equation. 
	
	Classical and nonclassical symmetries for similar and generalized equations can be obtained, for example $u_{t}=\partial_x (u^3\partial_x[f (x)-u_{xx}])$, with $f=f(t)$, $f=f (x,t)$ or $f=f (x,t,u)$.


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%\vspace{4mm}\noindent{\bf Acknowledgements}\\
%\noindent If you'd like to thank anyone, place your comments here.


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\begin{thebibliography}{99} % Enter references in alphabetical order and according to the following format.

%
\bibitem{Ber} 
%
F.~Bernis, A.~Friedman,{\it Higher order nonlinear degenerate parabolic equations} Journal of
Differential Equations,  83(1990) no.1, 179–206.

\bibitem{Fin}
% Format for Journal Reference
J.~Hangjie, P.~T.~Witelski, {\it Finite-time thin rupture driven by modified evaporative loss} Duke University, United States (2016).

\bibitem{Fin1}
% Format for Journal Reference
J.~Hangjie, P.~T.~ Witelski, {\it Finite-time thin film rupture driven by generalized evaporative loss} Duke University, United States (2015).

\bibitem{Ib}
%
N.~H.~Ibragimov, {\it Lie group analysis of differential equations-symmetries, exact solutions and conservation low} CRC, Boka Raton, FL (1994). 

\bibitem{Lie}
%
M.~S.~Lie, J.~Merker, {\it Theory of transformation groups : general properties of continuous transformation groups.A contemporary approach and translation}, Springer (2015).


\bibitem{Nad1} M.~Nadjafikhah, R.~Bakhshandeh-Chamazkoti,and A.~Mahdipour-Shirayeh, {\it A symmetry classification for a class of $(2+1)$-nonlinear wave  equation} Nonlinear Anal., 71(2009), no. 11, 5164--5169. 
	
\bibitem{Nad2} M.~Nadjafikhah, R.~Bakhshandeh-Chamazkoti {\it Symmetry group classification for general Burgers' equation} Commun. Nonlinear Sci. Numer. Simul., 15(2010), no. 9, 2303--2310. 
	
\bibitem{Nad3} M.~Nadjafikhah, V.~Shirvani-Sh, {\it Lie symmetries and conservation laws of the Hirota-Ramani equation} Commun. Nonlinear Sci. Numer. Simul., 17 (2012) , no. 11, 4064--4073.
	
	\bibitem{Olv1} P.~J.~Olver,{\it  Equivalence, invariants, and symmetry} Cambridge University Press, Cambridge (1995). 
	
		\bibitem{Olv2} P.~J.~Olver,{\it Applications of Lie groups to differential equations} Lecture Notes. Oxford University, Mathematical Institute, Oxford (1980). 
	
		\bibitem{Ovs} L.~V.~Ovsiannikov, {\it Group analysis of differential equations} New York, Academic Press (1982).
	


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{\small

\noindent{\bf Maryam Yourdkhany}

\noindent Department of Mathematics

\noindent Ph.D Student of Mathematics

\noindent  Karaj Branch, Islamic Azad University

\noindent Karaj, Iran

\noindent E-mail:maryam.yourdkhany@kiau.ac.ir}\\

{\small
\noindent{\bf  Second Author  }

\noindent  School of Mathematics

\noindent  Professor of Mathematics

\noindent  Iran University of Science and Technology


\noindent Narmak, Tehran, Iran

\noindent E-mail: m\_nadjafikhah@iust.ac.ir}\\



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