---
abstract: 'A preliminary group classification of the class 2D nonlinear heat equations $u_t=f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})$, where $f$ is arbitrary smooth function of the variables $x,y,u,u_x$ and $u_y$ using Lie method, is given. The paper is one of the few applications of an algebraic approach to the problem of group classification: the method of preliminary group classification.'
address: 'School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 1684613114, Iran.'
author:
- 'M. Nadjafikhah'
- 'R. Bakhshandeh-Chamazkoti'
title: |
    Preliminarily group classification of a class of\
    2D nonlinear heat equations 
---

,

,

$2$D Nonlinear heat equation, Optimal system, Preliminarily group classification.

Introduction
============

It is well known that the symmetry group method plays an important role in the analysis of differential equations. The history of group classification methods goes back to Sophus Lie. The first paper on this subject is [@[1]], where Lie proves that a linear two-dimensional second-order PDE may admit at most a three-parameter invariance group (apart from the trivial infinite parameter symmetry group, which is due to linearity). He computed the maximal invariance group of the one-dimensional heat conductivity equation and utilized this symmetry to construct its explicit solutions. Saying it the modern way, he performed symmetry reduction of the heat equation. Nowadays symmetry reduction is one of the most powerful tools for solving nonlinear partial differential equations (PDEs). Recently, there have been several generalizations of the classical Lie group method for symmetry reductions. Ovsiannikov [@[2]] developed the method of partially invariant solutions. His approach is based on the concept of an equivalence group, which is a Lie transformation group acting in the extended space of independent variables, functions and their derivatives, and preserving the class of partial differential equations under study. In an attempt to study nonlinear effects Saied and Hussain [@[3]] gave some new similarity solutions of the (1+1)-nonlinear heat equation. Later Clarkson and Mansfield [@[4]] studied classical and nonclassical symmetries of the (1+1)-heat equation and gave new reductions for the linear heat equation and a catalogue of closed-form solutions for a special choice of the function $f(x,y,u,u_x,u_y)$ that appears in their model. In higher dimensions Servo [@[5]] gave some conditional symmetries for a nonlinear heat equation while Goard et al. [@[6]] studied the nonlinear heat equation in the degenerate case. Nonlinear heat equations in one or higher dimensions are also studied in literature by using both symmetry as well as other methods [@[7]; @[8]].There are a number of papers to study (1+1)-nonlinear heat equations from the point of view of Lie symmetries method. The (2+1)-dimensional nonlinear heat equations $$\begin{aligned}
u_t=f(u)(u_{xx}+u_{yy}),\label{eq:1}\end{aligned}$$ are investigated in [@[9]] and in present paper we studied $$\begin{aligned}
u_t=f(x,y,u,u_x,u_y)(u_{xx}+u_{yy}),\label{eq:2}\end{aligned}$$ Similarity techniques are applied in [@[10]; @[11]; @[12]; @[13]] for (2+1)-dimensional wave equations.

Symmetry Methods
================

Let a partial differential equation contains $p$ dependent variables and $q$ independent variables. The one-parameter Lie group of transformations $$\begin{aligned}
x_i\longmapsto x_i+\epsilon\xi^i(x,u)+O(\epsilon^2);\hspace{1cm}
u_{\alpha}\longmapsto
u_{\alpha}+\epsilon\varphi^{\alpha}(x,u)+O(\epsilon^2),\label{eq:3}\end{aligned}$$ where $i=1,\ldots,p$ and $\alpha=1,\ldots,q$. The action of the Lie group can be recovered from that of its associated infinitesimal generators. we consider general vector field $$\begin{aligned}
X=\sum_{i=1}^p\xi^i(x,u)\frac{{\rm \partial}}{{{\rm \partial}}x_i}+
\sum_{\alpha=1}^q\varphi^{\alpha}(x,u)\frac{{\rm \partial}}{{{\rm \partial}}u^{\alpha}}.\label{eq:4}\end{aligned}$$ on the space of independent and dependent variables. The symmetry generator associated with (\[eq:4\]) given by $$\begin{aligned}
X=\xi^1(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}x}+\xi^2(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}y}+
\xi^3(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}t}+\varphi(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}u}.\label{eq:6}\end{aligned}$$ The second prolongation of $X$ is the vector field $$\begin{aligned}
\nonumber
 X^{(2)}=X+\varphi^x\frac{{\rm \partial}}{{{\rm \partial}}u_x}+\varphi^y\frac{{\rm \partial}}{{{\rm \partial}}u_y}+\varphi^t\frac{{\rm \partial}}{{{\rm \partial}}u_t}+
\varphi^{xx}\frac{{\rm \partial}}{{{\rm \partial}}u_{xx}}+\varphi^{xy}\frac{{\rm \partial}}{{{\rm \partial}}u_{xy}}
+\varphi^{xt}\frac{{\rm \partial}}{{{\rm \partial}}u_{xt}}
+\varphi^{yy}\frac{{\rm \partial}}{{{\rm \partial}}u_{yy}}
+\varphi^{yt}\frac{{\rm \partial}}{{{\rm \partial}}u_{yt}}+\varphi^{tt}\frac{{\rm \partial}}{{{\rm \partial}}u_{tt}},\\\label{eq:7}\end{aligned}$$ that its coefficients are obtained with following formulas $$\begin{aligned}
\label{eq:8}
&&\varphi^x={D}_x\varphi-u_x{D}_x\xi^1-u_y{D}_x\xi^2-u_t{D}_x\xi^3,\hspace{2cm}
\varphi^y={D}_y\varphi-u_x{D}_y\xi^1-u_y{D}_y\xi^2-u_t{D}_y\xi^3,\\\nonumber
&&\varphi^t={D}_t\varphi-u_x{D}_t\xi^1-u_y{D}_t\xi^2-u_t{\rm
D}_t\xi^3,\hspace{2.1cm}
\varphi^{xx}={D}_x\varphi^x-u_{xx}{D}_x\xi^1-u_{xy}{D}_x\xi^2-u_{xt}{D}_x\xi^3\\\nonumber
&&\hspace{-2mm}\varphi^{yy}={D}_y\varphi^y-u_{xy}{D}_y\xi^1-u_{yy}{D}_y\xi^2-u_{yt}{D}_y\xi^3\hspace{1.58cm}
\varphi^{tt}={D}_t\varphi^t-u_{xt}{D}_t\xi^1-u_{yt}{D}_t\xi^2-u_{tt}{D}_t\xi^3\\\nonumber
&&\hspace{-2mm}\varphi^{yt}={D}_y\varphi^t-u_{xy}{D}_y\xi^1-u_{yy}{D}_y\xi^2-u_{yt}{D}_y\xi^3\hspace{1.58cm}
\varphi^{xt}={D}_t\varphi^t-u_{xt}{D}_t\xi^1-u_{yt}{D}_t\xi^2-u_{tt}{D}_t\xi^3\end{aligned}$$ where the operators $D_x$, $D_y$ and $D_t$ denote the total derivatives with respect to $x,y$ and $t$: $$\begin{aligned}
\nonumber
D_x&=&\frac{{\rm \partial}}{{{\rm \partial}}x}+u_x\frac{{\rm \partial}}{{{\rm \partial}}u}+u_{xx}\frac{{\rm \partial}}{{{\rm \partial}}u_x}+u_{xy}\frac{{\rm \partial}}{{{\rm \partial}}u_y}+
u_{xt}\frac{{\rm \partial}}{{{\rm \partial}}u_t}+\ldots\\
D_y&=&\frac{{\rm \partial}}{{{\rm \partial}}y}+u_y\frac{{\rm \partial}}{{{\rm \partial}}u}+u_{yy}\frac{{\rm \partial}}{{{\rm \partial}}u_y}+u_{yx}\frac{{\rm \partial}}{{{\rm \partial}}u_x}+
u_{yt}\frac{{\rm \partial}}{{{\rm \partial}}u_t}+\ldots\\\label{eq:9}
D_t&=&\frac{{\rm \partial}}{{{\rm \partial}}t}+u_t\frac{{\rm \partial}}{{{\rm \partial}}u}+u_{tt}\frac{{\rm \partial}}{{{\rm \partial}}u_t}+u_{tx}\frac{{\rm \partial}}{{{\rm \partial}}u_x}+
u_{ty}\frac{{\rm \partial}}{{{\rm \partial}}u_y}+\ldots\nonumber\end{aligned}$$ By theorem 6.5. in [@[14]], $X^{(2)}[u_t-f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})]|_{(6)}=0$ whenever $$\begin{aligned}
u_t-f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})=0.\label{eq:10}\end{aligned}$$ Since $$X^{(2)}[u_t-f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})]=\varphi^t-
(f_x\xi^1+f_y\xi^2+f_u\varphi+f_{u_x}\varphi^x+f_{u_y}\varphi^y)
(u_{xx}+u_{yy})-f(x,y,u,u_x,u_y)(\varphi^{xx}+\varphi^{yy}),$$ therefore we obtain the following determining function: $$\begin{aligned}
\varphi^t-(f_x\xi^1+f_y\xi^2+f_u\varphi+f_{u_x}\varphi^x+f_{u_y}\varphi^y)
(u_{xx}+u_{yy})
-f(x,y,u,u_x,u_y)(\varphi^{xx}+\varphi^{yy})=0.\label{eq:11}\end{aligned}$$ In the case of arbitrary $f$ it follows $$\begin{aligned}
\xi^1=\xi^2=\varphi=0,\label{eq:12}\end{aligned}$$ or $$\begin{aligned}
\xi^1=\xi^2=\varphi=0,\;\;\;\;\;\xi^3=C.\label{eq:13}\end{aligned}$$ Therefore, for arbitrary $f(x,y,u,u_x,u_y)$ Eq. (\[eq:1\]) admits the one-dimensional Lie algebra ${{\goth g}}_1$, with the basis $$\begin{aligned}
X_1=\frac{{\rm \partial}}{{{\rm \partial}}t}.\label{eq:14}\end{aligned}$$ ${{\goth g}}_1$ is called the principle Lie algebra for Eq. (\[eq:1\]). So, the remaining part of the group classification is to specify the coefficient $f$ such that Eq. (\[eq:1\]) admits an extension of the principal algebra ${{\goth g}}_1$. Usually, the group classification is obtained by inspecting the determining equation. But in our case the complete solution of the determining equation (\[eq:11\]) is a wasteful venture. Therefore, we don’t solve the determining equation but, instead we obtain a partial group classification of Eq. (\[eq:1\]) via the so-called method of preliminary group classification. This method was suggested in [@[10]] and applied when an equivalence group is generated by a finite-dimensional Lie algebra ${{\goth g}}_{{\mathscr E}}$. The essential part of the method is the classification of all nonsimilar subalgebras of ${{\goth g}}_{{\mathscr E}}$. Actually, the application of the method is simple and effective when the classification is based on finite-dimensional equivalence algebra ${{\goth g}}_{{\mathscr E}}$.

Equivalence transformations
===========================

An equivalence transformation is a nondegenerate change of the variables $t,x,y,u$ taking any equation of the form (\[eq:1\]) into an equation of the same form, generally speaking, with different $f(x,y,u,u_x,u_y)$. The set of all equivalence transformations forms an equivalence group ${{\mathscr E}}$. We shall find a continuous subgroup ${{\mathscr E}}_C$ of it making use of the infinitesimal method.

We consider an operator of the group ${{\mathscr E}}_C$ in the form $$\begin{aligned}
Y=\xi^1(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}x}+\xi^2(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}y}+
\xi^3(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}t}+\varphi(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}u}
+\mu(x,y,t,u,u_x,u_y,u_t,f)\frac{{\rm \partial}}{{{\rm \partial}}f},\label{eq:15}\end{aligned}$$ from the invariance conditions of Eq. (\[eq:1\]) written as the system: $$\begin{aligned}
\label{eq:3-16}
u_t&-&f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})=0,\\\nonumber
f_t&=&f_{u_t}=0,\end{aligned}$$ where $u$ and $f$ are considered as differential variables: $u$ on the space $(x,y,t)$ and $f$ on the extended space $(x,y,t,u,u_x,u_y)$.

The invariance conditions of the system (\[eq:3-16\]) are $$\begin{aligned}
\label{eq:17}
Y^{(2)}(u_t&-&f(x,y,u,u_x,u_y)(u_{xx}+u_{yy}))=0,\\\nonumber
Y^{(2)}(f_t)&=&Y^{(2)}(f_{u_t})=0,\end{aligned}$$ where $Y^{(2)}$ is the prolongation of the operator (\[eq:15\]): $$\begin{aligned}
\label{eq:3-18}
Y^{(2)}=Y+\varphi^x\frac{{\rm \partial}}{{{\rm \partial}}u_x}+\varphi^y\frac{{\rm \partial}}{{{\rm \partial}}u_y}+\varphi^t\frac{{\rm \partial}}{{{\rm \partial}}u_t}+
\varphi^{xx}\frac{{\rm \partial}}{{{\rm \partial}}u_{xx}}+\varphi^{xy}\frac{{\rm \partial}}{{{\rm \partial}}u_{xy}}
+\varphi^{xt}\frac{{\rm \partial}}{{{\rm \partial}}u_{xt}}&+&\varphi^{yy}\frac{{\rm \partial}}{{{\rm \partial}}u_{yy}}\\\nonumber
&+&\varphi^{yt}\frac{{\rm \partial}}{{{\rm \partial}}u_{yt}}+\varphi^{tt}\frac{{\rm \partial}}{{{\rm \partial}}u_{tt}}+
\mu^t\frac{{\rm \partial}}{{{\rm \partial}}f_{t}}+\mu^{u_t}\frac{{\rm \partial}}{{{\rm \partial}}f_{u_t}}.\end{aligned}$$ The coefficients $\varphi^x, \varphi^y, \varphi^t, \varphi^{xx},
\varphi^{xy}, \varphi^{xt}, \varphi^{yy}, \varphi^{yt},
\varphi^{tt}$ are given in (\[eq:8\]) and the other coefficients of (\[eq:3-18\]) are obtained by applying the prolongation procedure to differential variables $f$ with independent variables $(x,y,t,u,u_x,u_y,u_t)$. we have $$\begin{aligned}
\mu^t&=&\widetilde{D}_t(\mu)-f_x\widetilde{D}_t(\xi^1)-f_y\widetilde{D}_t(\xi^2)
-f_u\widetilde{D}_t(\varphi)-f_{u_x}\widetilde{D}_t(\varphi^x)-f_{u_y}\widetilde{D}_t(\varphi^y),\label{eq:19}\\
\mu^{u_t}&=&\widetilde{D}_{u_t}(\mu)-f_x\widetilde{D}_{u_t}(\xi^1)-f_y\widetilde{D}_{u_t}(\xi^2)
-f_u\widetilde{D}_{u_t}(\varphi)-f_{u_x}\widetilde{D}_{u_t}(\varphi^x)-f_{u_y}\widetilde{D}_{u_t}(\varphi^y),\label{eq:20}\end{aligned}$$ where $$\begin{aligned}
\widetilde{D}_t=\frac{{\rm \partial}}{{{\rm \partial}}t},\hspace{1cm}\widetilde{D}_{u_t}=\frac{{\rm \partial}}{{{\rm \partial}}u_t}.\label{eq:21}\end{aligned}$$ So, we have the following prolongation formulas: $$\begin{aligned}
\label{eq:22}
\mu^t&=&\mu_t-f_x\xi_t^1-f_y\xi_t^2-f_u\varphi_t-f_{u_x}(\varphi^x)_t-f_{u_y}(\varphi^y)_t,\\\nonumber
\mu^{u_t}&=&\mu_{u_t}-f_{u_x}(\varphi^x)_{u_t}-f_{u_y}(\varphi^y)_{u_t},\end{aligned}$$ By the invariance conditions (\[eq:17\]) give rise to $$\begin{aligned}
\mu^t=\mu^{u_t}=0,\label{eq:23}\end{aligned}$$ that is hold for every $f$. Substituting (\[eq:23\]) into (\[eq:22\]), we obtain $$\begin{aligned}
\begin{array}{ll}
\mu_t=\mu_{u_t}=0\\
\xi^1_x=\xi^2_t=\varphi_t=0\\
(\varphi^x)_t=(\varphi^x)_{u_t}=(\varphi^y)_t=(\varphi^y)_{u_t}=0\label{eq:24}
\end{array}\end{aligned}$$ Moreover with substituting (\[eq:3-18\]) into (\[eq:17\]) we obtain $$\begin{aligned}
\varphi^t-f(x,y,u,u_x,u_y)(\varphi^{xx}+\varphi^{yy})-\mu(u_{xx}+u_{yy})=0.\label{eq:25}\end{aligned}$$ We are left with a polynomial equation involving the various derivatives of $u(x,y,t)$ whose coefficients are certain derivatives of $\xi^1,\xi^2,\xi^3$ and $\varphi$. Since $\xi^1,\xi^2,\xi^3,\varphi$ only depend on $x,y,t,u$ we can equate the individual coefficients to zero, leading to the complete set of determining equations: $$\begin{aligned}
\xi^1&=&\xi^1(x,y)\label{eq:26}\\
\xi^2&=&\xi^2(y)\label{eq:27}\\
\xi^3&=&\xi^3(t)\label{eq:28}\\
\varphi_{uu}&=&0\label{eq:29}\\
2\varphi_{xu}&=&\xi^1_{xx}+\xi^1_{yy}\label{eq:30}\\
\varphi_{yu}&=&\xi^2_{xx}+\xi^2_{yy}\label{eq:31}\\
\varphi_u&=&\xi_x^1=\xi_y^2\label{eq:32}\\
\mu&=&(\xi^1_x-\xi_t^3)f\label{eq:33}\\
\varphi_{tt}&=&f(\varphi_{xx}+\varphi_{yy})\label{eq:34}\end{aligned}$$ so, we find that $$\begin{aligned}
\nonumber
&&\xi^1(x)=c_1x+c_2y+c_3,\hspace{1cm}\xi^2(t)=c_1y+c_4,\hspace{1cm}\xi^3(t)=a(t),\\
&&\hspace{1cm}\varphi(x,y,u)=c_1u+\beta(x,y),\hspace{1cm}\mu=(c_1-a'(t))f,\label{eq:35}\end{aligned}$$ with constants $c_1, c_2, c_3$ and $c_4$, also we have $\beta_{xx}=-\beta_{yy}$.

$\;\;\;\;$We summarize: The class of Eq. (\[eq:2\]) has an infinite continuous group of equivalence transformations generated by the following infinitesimal operators: $$\begin{aligned}
Y=(c_1x+c_2y+c_3)\frac{{\rm \partial}}{{{\rm \partial}}x}+ (c_1y+c_4)\frac{{\rm \partial}}{{{\rm \partial}}y}+
a(t)\frac{{\rm \partial}}{{{\rm \partial}}t}+(c_1u+\beta(x,y))\frac{{\rm \partial}}{{{\rm \partial}}u}
+(c_1-a'(t))f\frac{{\rm \partial}}{{{\rm \partial}}f}.\label{eq:36}\end{aligned}$$ Therefore the symmetry algebra of the Burgers’ equation (\[eq:2\]) is spanned by the vector fields $$\begin{aligned}
&Y_1=x\frac{{\rm \partial}}{{{\rm \partial}}x}+y\frac{{\rm \partial}}{{{\rm \partial}}y}+t\frac{{\rm \partial}}{{{\rm \partial}}t}+u\frac{{\rm \partial}}{{{\rm \partial}}u}+f\frac{{\rm \partial}}{{{\rm \partial}}f},
\hspace{1cm} Y_2=y\frac{{\rm \partial}}{{{\rm \partial}}x},\hspace{1cm}
Y_3=\frac{{\rm \partial}}{{{\rm \partial}}x},\hspace{1cm}Y_4=\frac{{\rm \partial}}{{{\rm \partial}}y}&\\\label{eq:37}
&Y_5=a(t)\frac{{\rm \partial}}{{{\rm \partial}}t}-a'(t)f\frac{{\rm \partial}}{{{\rm \partial}}f},\hspace{1cm}
Y_{\beta}=\beta(x,y)\frac{{\rm \partial}}{{{\rm \partial}}u}.&\nonumber\end{aligned}$$

Moreover, in the group of equivalence transformations there are included also discrete transformations, i.e., reflections $$\begin{aligned}
t\longrightarrow-t,\hspace{1.5cm}x\longrightarrow-x,\hspace{1.5cm}u\longrightarrow-u,\hspace{1.5cm}
f\longrightarrow-f.\label{eq:38}\end{aligned}$$

$$\begin{aligned}
\hspace{-0.75cm}\begin{array}{llllll}
\hline
  [\,,\,]&\hspace{2cm}Y_1 &\hspace{2cm}Y_2  &\hspace{2cm}Y_3  &\hspace{2cm}Y_4  &\hspace{2cm}Y_5 \hspace{2cm}Y_6 \\ \hline
  Y_1    &\hspace{2cm} 0  &\hspace{2cm} 0   &\hspace{1.8cm}-Y_3 &\hspace{1.8cm}-Y_4 &\hspace{2cm}0   \hspace{1.6cm}-Y_6 \\
  Y_2    &\hspace{2cm} 0  &\hspace{2cm} 0   &\hspace{2cm} 0   &\hspace{1.8cm}-Y_3 &\hspace{2cm}0   \hspace{2cm}0\\
  Y_3    &\hspace{2cm}Y_3 &\hspace{2cm} 0   &\hspace{2cm} 0   &\hspace{2cm}0    &\hspace{2cm}0   \hspace{2cm}0\\
  Y_4    &\hspace{2cm}Y_4 &\hspace{2cm} Y_3 &\hspace{2cm} 0   &\hspace{2cm}0    &\hspace{2cm}0   \hspace{2cm}0  \\
  Y_5    &\hspace{2cm} 0  &\hspace{2cm} 0   &\hspace{2cm} 0   &\hspace{2cm}0    &\hspace{2cm}0   \hspace{2cm}0\\
  Y_6    &\hspace{2cm}Y_6 &\hspace{2cm} 0   &\hspace{2cm} 0   &\hspace{2cm}0    &\hspace{2cm}0   \hspace{2cm}0\\
  \hline
\end{array}\end{aligned}$$

$$\begin{aligned}
\hspace{-0.75cm}\begin{array}{llllll}
\hline
[\,,\,]&\hspace{2cm}Y_1&\hspace{2cm}Y_2&\hspace{2cm}Y_3&\hspace{2cm}Y_4&\hspace{2cm}Y_5\hspace{2cm}Y_6
\\ \hline
  Y_1  &\hspace{2cm}Y_1&\hspace{2cm}Y_2&\hspace{2cm}e^sY_3&\hspace{2cm}e^sY_4&\hspace{2cm}Y_5\hspace{2cm}e^sY_6     \\
  Y_2  &\hspace{2cm}Y_1&\hspace{2cm}Y_2&\hspace{2cm}Y_3&\hspace{2cm}Y_4+sY_3&\hspace{2cm}Y_5\hspace{2cm}Y_6    \\
  Y_3  &\hspace{1.5cm}Y_1-sY_3&\hspace{2cm}Y_2&\hspace{2cm}Y_3&\hspace{2cm}Y_4 &\hspace{2cm}Y_5\hspace{2cm}Y_6   \\
  Y_4  &\hspace{1.5cm}Y_1-sY_4&\hspace{1.6cm}Y_2-sY_3&\hspace{2cm}Y_3&\hspace{2cm}Y_4&\hspace{2cm}Y_5\hspace{2cm}Y_6    \\
  Y_5  &\hspace{2cm}Y_1&\hspace{2cm}Y_2&\hspace{2cm}Y_3&\hspace{2cm}Y_4&\hspace{2cm}Y_5\hspace{2cm}Y_6  \\
  Y_6  &\hspace{1.5cm}Y_1-sY_6&\hspace{2cm}Y_2&\hspace{2cm}Y_3&\hspace{2cm}Y_4&\hspace{2cm}Y_5\hspace{2cm}Y_6  \\
  \hline
\end{array}\end{aligned}$$

Preliminary group classification
================================

One can observe in many applications of group analysis that most of extensions of the principal Lie algebra admitted by the equation under consideration are taken from the equivalence algebra ${\goth g}_{{\mathscr E}}$. We call these extensions ${\mathscr E}$-extensions of the principal Lie algebra. The classification of all nonequivalent equations (with respect to a given equivalence group $G_{{\mathscr E}}$,) admitting ${\mathscr E}$-extensions of the principal Lie algebra is called a preliminary group classification. Here, $G_{{\mathscr E}}$ is not necessarily the largest equivalence group but, it can be any subgroup of the group of all equivalence transformations.So, we can take any finite-dimensional subalgebra (desirable as large as possible) of an infinite-dimensional algebra with basis (\[eq:31\]) and use it for a preliminary group classification. We select the subalgebra ${\goth g}_6$ spanned on the following operators: $$\begin{aligned}
&Y_1=x\frac{{\rm \partial}}{{{\rm \partial}}x}+y\frac{{\rm \partial}}{{{\rm \partial}}y}+t\frac{{\rm \partial}}{{{\rm \partial}}t}+u\frac{{\rm \partial}}{{{\rm \partial}}u}+f\frac{{\rm \partial}}{{{\rm \partial}}f},\hspace{1cm}
Y_2=y\frac{{\rm \partial}}{{{\rm \partial}}x},\hspace{1cm}
Y_3=\frac{{\rm \partial}}{{{\rm \partial}}x},\hspace{1cm}
Y_4=\frac{{\rm \partial}}{{{\rm \partial}}y},&\nonumber\\
&Y_5=\frac{{\rm \partial}}{{{\rm \partial}}t}-f\frac{{\rm \partial}}{{{\rm \partial}}f},\hspace{1cm}Y_6=\frac{{\rm \partial}}{{{\rm \partial}}u}.&\label{eq:39}\end{aligned}$$ The communication relations between these vector fields is given in Table 1. To each $s$-parameter subgroup there corresponds a family of group invariant solutions. So, in general, it is quite impossible to determine all possible group-invariant solutions of a PDE. In order to minimize this search, it is useful to construct the optimal system of solutions. It is well known that the problem of the construction of the optimal system of solutions is equivalent to that of the construction of the optimal system of subalgebras [@[2]; @[12]]. Here, we will deal with the construction of the optimal system of subalgebras of ${\goth
g}_5$.Let $G$ be a Lie group, with ${\goth g}$ its Lie algebra. Each element $T\in G$ yields inner automorphism $T_a\longrightarrow
TT_aT^{-1}$ of the group $G$. Every automorphism of the group $G$ induces an automorphism of ${\goth g}$. The set of all these automorphism is a Lie group called [*the adjoint group $G^A$*]{}. The Lie algebra of $G^A$ is the adjoint algebra of ${\goth g}$, defined as follows. Let us have two infinitesimal generators $X,Y\in L$. The linear mapping ${\rm
Ad}X(Y):Y\longrightarrow[X,Y]$ is an automorphism of ${\goth g}$, called [*the inner derivation of ${\goth g}$*]{}. The set of all inner derivations ${\rm ad}X(Y)(X,Y\in{\goth g})$ together with the Lie bracket $[{\rm Ad}X,{\rm Ad}Y]={\rm Ad}[X,Y]$ is a Lie algebra ${\goth g}^A$ called the [*adjoint algebra of ${\goth
g}$*]{}. Clearly ${\goth g}^A$ is the Lie algebra of $G^A$. Two subalgebras in ${\goth g}$ are [*conjugate*]{} (or [*similar*]{}) if there is a transformation of $G^A$ which takes one subalgebra into the other. The collection of pairwise non-conjugate $s$-dimensional subalgebras is the optimal system of subalgebras of order $s$. The construction of the one-dimensional optimal system of subalgebras can be carried out by using a global matrix of the adjoint transformations as suggested by Ovsiannikov [@[2]]. The latter problem, tends to determine a list (that is called an [*optimal system*]{}) of conjugacy inequivalent subalgebras with the property that any other subalgebra is equivalent to a unique member of the list under some element of the adjoint representation i.e. $\overline{{\goth h}}\,{\rm
Ad(g)}\,{\goth h}$ for some ${\rm g}$ of a considered Lie group. Thus we will deal with the construction of the optimal system of subalgebras of ${\goth g}_6$. The adjoint action is given by the Lie series $$\begin{aligned}
{\rm Ad}(\exp(s\,Y_i))Y_j
=Y_j-s\,[Y_i,Y_j]+\frac{s^2}{2}\,[Y_i,[Y_i,Y_j]]-\cdots,\label{eq:40}\end{aligned}$$ where $s$ is a parameter and $i,j=1,\cdots,6$. The adjoint representations of ${\goth g}_6$ is listed in Table 2, it consists the separate adjoint actions of each element of ${\goth g}_6$ on all other elements. [**Theorem 4.1.**]{} [*An optimal system of one-dimensional Lie subalgebras of general Burgers’ equation (\[eq:2\]) is provided by those generated by*]{} $$\begin{aligned}
&1)&Y^1=Y_1=x{{\rm \partial}}_x+y{{\rm \partial}}_y+t{{\rm \partial}}_t+u{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{2.2cm}2)~Y^2=Y_2=y{{\rm \partial}}_x,\\
&3)&Y^3=-Y_4=-{{\rm \partial}}_y,\hspace{5.7cm}4)~Y^4=Y_1+Y_5=x{{\rm \partial}}_x+y{{\rm \partial}}_y+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u,\\
&5)&Y^5=Y_1-Y_2=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+t{{\rm \partial}}_t+u{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{0.5cm}6)~Y^6=Y_2-Y_4=y{{\rm \partial}}_x-{{\rm \partial}}_y,\\
&7)&Y^7=-Y_4+Y_6=-{{\rm \partial}}_y+{{\rm \partial}}_u,\hspace{4.1cm}8)~Y^{8}=-Y_4-Y_6=-{{\rm \partial}}_y-{{\rm \partial}}_u,\\
&9)&Y^9=Y_2+Y_5=y{{\rm \partial}}_x+{{\rm \partial}}_t-f{{\rm \partial}}_f,\hspace{3.3cm}10)~Y^{10}=Y_2-Y_5=y{{\rm \partial}}_x-{{\rm \partial}}_t+f{{\rm \partial}}_f,\\
&11)&Y^{11}=Y_2+Y_6=y{{\rm \partial}}_x+{{\rm \partial}}_u,\hspace{4.1cm}12)~Y^{12}=Y_2-Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_u,\\
&13)&Y^{13}=Y_1+Y_2=(x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+t{{\rm \partial}}_t+u{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{1mm}14)~Y^{14}=-Y_4+Y_5+Y_6=-{{\rm \partial}}_y+{{\rm \partial}}_t+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
&15)&Y^{15}=Y_2-Y_4-Y_5+Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_t+{{\rm \partial}}_u+f{{\rm \partial}}_f,16)~Y^{16}=Y_2-Y_4+Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_u,\\
&17)&Y^{17}=Y_2-Y_4+Y_5-Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_t-{{\rm \partial}}_u-f{{\rm \partial}}_f,18)~Y^{18}=Y_2-Y_4-Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u,\\
&19)&Y^{19}=Y_1+Y_2+Y_5=(x+y){{\rm \partial}}_x+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u,\hspace{0.5cm}20)~Y^{20}=Y_2+Y_5+Y_6=y{{\rm \partial}}_x+{{\rm \partial}}_t+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
&21)&Y^{21}=Y_2+Y_5-Y_6=y{{\rm \partial}}_x+{{\rm \partial}}_t-{{\rm \partial}}_u-f{{\rm \partial}}_f,\hspace{1.6cm}22)~Y^{22}=Y_2-Y_5-Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_t-{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
&23)&Y^{23}=Y_2-Y_5+Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_t+{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{1.6cm}24)~Y^{24}=-Y_4-Y_5-Y_6=-{{\rm \partial}}_y-{{\rm \partial}}_t-{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
&25)&Y^{25}=-Y_4-Y_5+Y_6=-{{\rm \partial}}_y-{{\rm \partial}}_t+{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{1.2cm}26)~Y^{26}=-Y_4+Y_5-Y_6=-{{\rm \partial}}_y+{{\rm \partial}}_t-{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
&27)&Y^{27}=Y_2-Y_4+Y_5+Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_t+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
&28)&Y^{28}=Y_1+Y_2-Y_5=(x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u+2f{{\rm \partial}}_f,\\
&29)&Y^{29}=Y_1-Y_2-Y_5=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u+2f{{\rm \partial}}_f,\\
&30)&Y^{31}=Y_1-Y_2+Y_5=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u,\\
&31)&Y^{31}=Y_1-Y_5=x{{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u+2f{{\rm \partial}}_f\\
&32)&Y^{32}=Y_2-Y_4-Y_5-Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_t-{{\rm \partial}}_u+f{{\rm \partial}}_f,\end{aligned}$$ [**Proof.**]{} Let ${\goth g}_6$ is the symmetry algebra of Eq. (\[eq:2\]) with adjoint representation determined in Table 2 and $$\begin{aligned}
Y=a_1Y_1+a_2Y_2+a_3Y_3+a_4Y_4+a_5Y_5+a_6Y_6,\end{aligned}$$ is a nonzero vector field of ${\goth g}_6$. We will simplify as many of the coefficients $a_i;i=1,\ldots,6$, as possible through proper adjoint applications on $Y$. We follow our aim in the below easy cases:[*Case 1:*]{} At first, assume that $a_1\neq 0$. Scaling $Y$ if necessary, we can assume that $a_1=1$ and so we get $$\begin{aligned}
Y=Y_1+a_2Y_2+a_3Y_3+a_4Y_4+a_5Y_5+a_6Y_6.\end{aligned}$$ Using the table of adjoint (Table 2) , if we act on $Y$ with ${\rm
Ad}(\exp(a_3Y_3))$, the coefficient of $Y_3$ can be vanished: $$\begin{aligned}
Y'=Y_1+a_2Y_2+a_4Y_4+a_5Y_5+a_6Y_6.\end{aligned}$$ Then we apply ${\rm Ad}(\exp(a_4Y_4))$ on $Y'$ to cancel the coefficient of $Y_4$: $$\begin{aligned}
Y''=Y_1+a_2Y_2+a_5Y_5+a_6Y_6.\end{aligned}$$ At last, we apply ${\rm Ad}(\exp(a_6Y_6))$ on $Y''$ to cancel the coefficient of $Y_6$: $$\begin{aligned}
Y'''=Y_1+a_2Y_2+a_5Y_5.\end{aligned}$$ [*Case 1a:*]{}\
If $a_2,a_5\neq 0$ then we can make the coefficient of $Y_2$ and $Y_5$ either $+1$ or $-1$. Thus any one-dimensional subalgebra generated by $Y$ with $a_2,a_5\neq 0$ is equivalent to one generated by $Y_1\pm Y_2\pm Y_5$ which introduce parts 19), 28), 29) and 30) of the theorem.[*Case 1b:*]{} For $a_2=0, a_5\neq0$ we can see that each one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $
Y_1\pm Y_5$ which introduce parts 4) and 31) of the theorem.[*Case 1c:*]{} For $a_2\neq0, a_5=0$, each one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $Y_1\pm Y_2$ which introduce parts 5) and 13) of the theorem.[*Case 1d:*]{} For $a_2=0, a_5=0$, each one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $Y_1$ which introduce parts 1) of the theorem.[*Case 2:*]{} The remaining one-dimensional subalgebras are spanned by vector fields of the form $Y$ with $a_1=0$. [*Case 2a:*]{} If $a_4\neq 0$ then by scaling $Y$, we can assume that $a_4=-1$. Now by the action of ${\rm Ad}(\exp a_3Y_3))$ on $Y$, we can cancel the coefficient of $Y_3$: $$\begin{aligned}
\overline{Y}=a_2Y_2-Y_4+a_5Y_5+a_6Y_6.\end{aligned}$$ Let $a_2\neq0$ then by scaling $Y$, we can assume that $a_2=1$, and we have $$\begin{aligned}
\overline{Y}'=Y_2-Y_4+a_5Y_5+a_6Y_6.\end{aligned}$$ [*Case 2a-1:*]{} Suppose $a_5=a_6=0$, then the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $Y_2-Y_4$ which introduce parts 6). [*Case 2a-2:*]{} Suppose $a_5=0, a_6\neq0$, all of the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $Y_2-Y_4\pm
Y_6$ which introduce parts 16) and 18).[*Case 2a-3:*]{} Suppose $a_5\neq0, a_6\neq0$, all of the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $Y_2-Y_4\pm Y_5\pm Y_6$ which introduce parts 15), 17), 27), and 32). Now if $a_2=0$, we have $$\begin{aligned}
\overline{Y}''=-Y_4+a_5Y_5+a_6Y_6.\end{aligned}$$ [*Case 2a-4:*]{} Suppose $a_5=a_6=0$, then the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $-Y_4$ which introduce parts 3). [*Case 2a-5:*]{} Suppose $a_5=0, a_6\neq0$, all of the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $-Y_4\pm Y_6$ which introduce parts 7) and 8).[*Case 2a-6:*]{} Suppose $a_5\neq0, a_6\neq0$, all of the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $-Y_4\pm Y_5\pm Y_6$ which introduce parts 14), 24), 25) and 26).

[*Case 2b:*]{} $~~~~$ Let $a_4=0$ then $Y$ is in the form $$\begin{aligned}
\widehat{Y}=a_2Y_2+a_5Y_5+a_6Y_6.\end{aligned}$$ Suppose that $a_2\neq 0$ then if necessary we can let it equal to $1$ and we obtain $$\begin{aligned}
\widehat{Y}'=Y_2+a_5Y_5+a_6Y_6.\end{aligned}$$ [*Case 2b-1:*]{} Let $a_5=a_6=0$, then $Y_2$ is remained and find 2) section of the theorem.[*Case 2b-2:*]{} If $a_5\neq0, a_6\neq0$, then $\widehat{Y}'$ is equal to $Y_2\pm Y_5\pm Y_6$. Hence this case suggests part 20), 21), 22) and 23).[*Case 2b-3:*]{} If $a_5\neq0, a_6=0$, then $\widehat{Y}'=Y_2\pm Y_5$ . Hence this case suggests part 9) and 10).[*Case 2b-4:*]{} If $a_5=0, a_6\neq0$, then $Y_2\pm Y_6$ is obtained. So, this case suggests part 11) and 12). There is not any more possible case for studying and the proof is complete. $\Box$ The coefficients $f$ of Eq. (\[eq:2\]) depend on the variables $x,y,u,u_x,u_y$. Therefore, we take their optimal system’s projections on the space $(x,y,u,u_x,u_y,f)$. we have $$\begin{aligned}
\hspace{-0.7cm}
\begin{array}{rlrl}
1)&Z^1=Y^1=x{{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+f{{\rm \partial}}_f, \hspace{1cm}&17)&Z^{17}=Y^{17}=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+2f{{\rm \partial}}_f,\\
2)&Z^2=Y^2=y{{\rm \partial}}_x,\hspace{1cm}&18)&Z^{18}=Y^{18}=(x+y){{\rm \partial}}_x+u{{\rm \partial}}_u,\\
3)&Z^3=Y^3=-{{\rm \partial}}_y,\hspace{1cm}&19)&Z^{19}=Y^{19}=y{{\rm \partial}}_x+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
4)&Z^4=Y^4=(x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{1cm}&20)&Z^{20}=Y^{20}=y{{\rm \partial}}_x-{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
5)&Z^5=Y^5=x{{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u,\hspace{1cm}&21)&Z^{21}=Y^{21}=y{{\rm \partial}}_x-{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
6)&Z^6=Y^6=x{{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+2f{{\rm \partial}}_f,\hspace{1cm} &22)&Z^{22}=Y^{22}=y{{\rm \partial}}_x+{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
7)&Z^7=Y^7=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{1cm} &23)&Z^{23}=Y^{23}=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_u,\\
8)&Z^8=Y^8=y{{\rm \partial}}_x-{{\rm \partial}}_y,\hspace{1cm}&24)&Z^{24}=Y^{24}=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u,\\
9)&Z^9=Y^9=-{{\rm \partial}}_y+{{\rm \partial}}_u,\hspace{1cm}&25)&Z^{25}=Y^{25}=-{{\rm \partial}}_y+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
10&Z^{10}=Y^{10}=-{{\rm \partial}}_y-{{\rm \partial}}_u,\hspace{1cm}&26)&Z^{26}=Y^{26}=-{{\rm \partial}}_y-{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
11&Z^{11}=Y^{11}=y{{\rm \partial}}_x-f{{\rm \partial}}_f,\hspace{1cm}&27)&Z^{27}=Y^{27}=-{{\rm \partial}}_y+{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
12)&Z^{12}=Y^{12}=y{{\rm \partial}}_x+f{{\rm \partial}}_f,\hspace{1cm}&28)&Z^{28}=Y^{28}=-{{\rm \partial}}_y-{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
\end{array}\end{aligned}$$ $$\begin{aligned}
\hspace{-0.7cm}
\begin{array}{rlrl}
13)&Z^{13}=Y^{13}=y{{\rm \partial}}_x+{{\rm \partial}}_u,\hspace{1cm}&29)&Z^{29}=Y^{29}=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
14)&Z^{14}=Y^{14}=y{{\rm \partial}}_x-{{\rm \partial}}_u,\hspace{1cm}&30)&Z^{30}=Y^{30}=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
15)&Z^{15}=Y^{15}=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u,\hspace{1cm}&31)&Z^{31}=Y^{31}=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
16)&Z^{16}=Y^{16}=(x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+2f{{\rm \partial}}_f,\hspace{1cm}&32)&Z^{32}=Y^{32}=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u-f{{\rm \partial}}_f,
\end{array}\end{aligned}$$ [**Proposition 4.2.**]{} [*Let ${\goth g}_m:=\langle Y_1, \ldots,
Y_m\rangle$, be an $m$-dimensional algebra. Denote by $Y^i (i=1,
\ldots, r, 0<r\leq m, r\in{\Bbb N})$ an optimal system of one-dimensional subalgebras of ${\goth g}_m$ and by $Z^i\, (i =
1,\cdots, t, 0<t\leq r, t\in{\Bbb N})$ the projections of $Y^i$, i.e., $Z^i = {\rm pr}(Y^i)$. If equations $$\begin{aligned}
f = \Phi(x,y,u,u_x,u_y),\label{eq:18}\end{aligned}$$ are invariant with respect to the optimal system $Z^i$ then the equation $$\begin{aligned}
u_t = \Phi(x,y,u,u_x,u_y)(u_{xx}+u_{yy}),\label{eq:19}\end{aligned}$$ admits the operators $X^i=$ projection of $Y^i$ on $(t,x,y,u,u_x,u_y)$.*]{} [**Proposition 4.3.**]{} [*Let Eq. (\[eq:19\]) and the equation $$\begin{aligned}
u_t =  \Phi'(x,y,u,u_x,u_y)(u_{xx}+u_{yy}),\label{eq:20}\end{aligned}$$ be constructed according to Proposition 4.2. via optimal systems $Z^i$ and ${Z^i}'$, respectively. If the subalgebras spanned on the optimal systems $Z^i$ and ${Z^i}'$, respectively, are similar in ${\goth g}_m$, then Eqs. (\[eq:19\]) and (\[eq:20\]) are equivalent with respect to the equivalence group $G_m$, generated by ${\goth g}_m$.* ]{} Now we apply Proposition 4.2. and Proposition 4.3. to the optimal system (\[eq:17\]) and obtain all nonequivalent Eq. (\[eq:2\]) admitting ${\mathscr E}$-extensions of the principal Lie algebra ${\goth
g}$, by one dimension, i.e., equations of the form (\[eq:2\]) such that they admit, together with the one basic operators (\[eq:21\]) of ${\goth g}$, also a second operator $X^{(2)}$. For every case, when this extension occurs, we indicate the corresponding coefficients $f, g$ and the additional operator $X^{(2)}$.

We perform the algorithm passing from operators $Z^i\,(i=1,\cdots,32)$ to $f$ and $X^{(2)}$ via the following example. Let consider the vector field $$\begin{aligned}
Z^{32}=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u-f{{\rm \partial}}_f,\label{eq:21}\end{aligned}$$ then the characteristic equation corresponding to $Z^6$ is $$\begin{aligned}
{dx\over y}={dy\over-1}=\frac{du}{-1}=\frac{df}{-f},\end{aligned}$$ and can be taken in the form $$\begin{aligned}
I_1=u+{x\over y},\hspace{5mm}I_2=e^{x\over y}f.\end{aligned}$$ From the invariance equations we can write $$\begin{aligned}
I_2=\Phi(I_1),\end{aligned}$$ it follows that $$\begin{aligned}
f=e^{-{x\over y}}\Phi(\lambda),\end{aligned}$$ where $\lambda=I_1$.

From Proposition 4.2. applied to the operator $Z^6$ we obtain the additional operator $X^{(2)}$ $$\begin{aligned}
y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_t-{{\rm \partial}}_u.\end{aligned}$$ After similar calculations applied to all operators (\[eq:17\]) we obtain the following result (Table 3) for the preliminary group classification of Eq. (\[eq:2\]) admitting an extension ${\goth
g}_3$ of the principal Lie algebra ${\goth g}_1$.

Conclusion
==========

In this paper, following the classical Lie method, the preliminary group classification for the class of heat equation (\[eq:2\]) and investigated the algebraic structure of the symmetry groups for this equation, is obtained. The classification is obtained by constructing an optimal system with the aid of Propositions 4.2. and 4.3.. The result of the work is summarized in Table 3. Of course it is also possible to obtain the corresponding reduced equations for all the cases in the classification reported in Table 3.

\[table:3\] $$\begin{aligned}
\hspace{-0.75cm}\begin{array}{l l l l l l} \hline
  N       &\hspace{1cm} Z     &\hspace{1.1cm} \mbox{Invariant}  &\hspace{1cm} \mbox{Equation}
  &\hspace{1cm} \mbox{Additional operator}\,X^{(2)} \\ \hline
  1       &\hspace{1cm} Z^1   &\hspace{1.1cm} {u\over x}  &\hspace{1cm}u_t=x\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} x{{\rm \partial}}_x+y{{\rm \partial}}_y+t{{\rm \partial}}_t+u{{\rm \partial}}_u \\
  2       &\hspace{1cm} Z^2   &\hspace{1.1cm} u   &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
          &\hspace{1cm} y{{\rm \partial}}_x \\
  3       &\hspace{1cm} Z^3   &\hspace{1.1cm} u  &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} -{{\rm \partial}}_y\\
  4        &\hspace{1cm} Z^4    &\hspace{1.1cm} {u\over x+y}  &\hspace{1cm}u_t=y\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} (x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u \\
  5        &\hspace{1cm} Z^5    &\hspace{1.1cm} {u\over x}&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
          &\hspace{1cm} x{{\rm \partial}}_x+y{{\rm \partial}}_y+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
  6        &\hspace{1cm} Z^6   &\hspace{1.1cm} {u\over x}&\hspace{1cm}u_t=x^2\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} x{{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
  7        &\hspace{1cm} Z^7   &\hspace{1.1cm} {u\over x-y}&\hspace{1cm}u_t=(x-y)\Phi(u_{xx}+u_{yy})
  &\hspace{1cm}(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+t{{\rm \partial}}_t+u{{\rm \partial}}_u \\
  8        &\hspace{1cm} Z^8   &\hspace{1.1cm} u &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y \\
  9        &\hspace{1cm} Z^9   &\hspace{1.1cm}  u&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} -{{\rm \partial}}_y+{{\rm \partial}}_u \\
  10        &\hspace{1cm} Z^{10}   &\hspace{1.1cm} x &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} -{{\rm \partial}}_y-{{\rm \partial}}_u \\
  11       &\hspace{1cm} Z^{11}   &\hspace{1.1cm} u &\hspace{1cm}u_t=e^{-{x\over y}}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x+{{\rm \partial}}_t \\
  12       &\hspace{1cm} Z^{12}   &\hspace{1.1cm} u &\hspace{1cm}u_t=e^{x\over y}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_t \\
  13       &\hspace{1cm} Z^{13}   &\hspace{1.1cm}  u-{x\over y} &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x+{{\rm \partial}}_u \\
  14       &\hspace{1cm} Z^{14}   &\hspace{1.1cm} u+{x\over y} &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_u \\
  15       &\hspace{1cm} Z^{15}   &\hspace{1.1cm} {u\over x-y}&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} (x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
  16       &\hspace{1cm} Z^{16}   &\hspace{1.1cm} {u\over x+y}&\hspace{1cm}u_t=(x+y)^2\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} (x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
  17       &\hspace{1cm} Z^{17}   &\hspace{1.1cm} {u\over x-y}&\hspace{1cm}u_t=(x-y)^2\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} (x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
  18       &\hspace{1cm} Z^{18}   &\hspace{1.1cm} {u\over x+y}&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} (x+y){{\rm \partial}}_x+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
  19       &\hspace{1cm} Z^{19}   &\hspace{1.1cm}u-{x\over y}&\hspace{1cm}u_t=e^{-{x\over y}}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x+{{\rm \partial}}_t+{{\rm \partial}}_u\\
  20       &\hspace{1cm} Z^{20}   &\hspace{1.1cm} u+{x\over y}&\hspace{1cm}u_t=e^{-{x\over y}}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x+{{\rm \partial}}_t-{{\rm \partial}}_u \\
  21       &\hspace{1cm} Z^{21}   &\hspace{1.1cm} u+{x\over y}&\hspace{1cm}u_t=e^{x\over y}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_t-{{\rm \partial}}_u\\
  22       &\hspace{1cm} Z^{22}   &\hspace{1.1cm} u-{x\over y}&\hspace{1cm}u_t=e^{x\over y}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_t+{{\rm \partial}}_u\\
  23       &\hspace{1cm} Z^{23}   &\hspace{1.1cm} u-{x\over y}&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_u\\
  24       &\hspace{1cm} Z^{24}   &\hspace{1.1cm}u+{x\over y}&\hspace{1cm}u_t=e^y\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u \\
  25       &\hspace{1cm} Z^{25}   &\hspace{1.1cm} u+y&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} -{{\rm \partial}}_y+{{\rm \partial}}_t+{{\rm \partial}}_u \\
  26       &\hspace{1cm} Z^{26}   &\hspace{1.1cm} u-y&\hspace{1cm}u_t=e^{-y}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} -{{\rm \partial}}_y-{{\rm \partial}}_t-{{\rm \partial}}_u \\
  27       &\hspace{1cm} Z^{27}   &\hspace{1.1cm} u+y&\hspace{1cm}u_t=e^{-y}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} -{{\rm \partial}}_y-{{\rm \partial}}_t+{{\rm \partial}}_u \\
  28       &\hspace{1cm} Z^{28}   &\hspace{1.1cm} u-y &\hspace{1cm}u_t=e^y\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} -{{\rm \partial}}_y+{{\rm \partial}}_t-{{\rm \partial}}_u \\
  29       &\hspace{1cm} Z^{29}   &\hspace{1.1cm} u+{x\over y}&\hspace{1cm}u_t=e^{-{x\over y}}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_t+{{\rm \partial}}_u \\
  30       &\hspace{1cm} Z^{30}   &\hspace{1.1cm} u+{x\over y}&\hspace{1cm}u_t=e^{x\over y}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_t-{{\rm \partial}}_u \\
  31       &\hspace{1cm} Z^{31}   &\hspace{1.1cm} u-{x\over y}&\hspace{1cm}u_t=e^{x\over y}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_t+{{\rm \partial}}_u \\
  32       &\hspace{1cm} Z^{32}   &\hspace{1.1cm} u+{x\over y}&\hspace{1cm}u_t=e^{-{x\over y}}\Phi(u_{xx}+u_{yy})
  &\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_t-{{\rm \partial}}_u \\
  \hline
\end{array}\end{aligned}$$

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