## 1. Introduction

Mathematical modeling of dynamical systems results in nonlinear partial differential equations (PDEs). In reality, most complicated phenomena—namely diffusion, reaction, conservation, and many more—can be illustrated by means of partial differential equations. Due to their quintessence, PDEs are studied profusely in science and engineering. Various peculiar methods are designed for obtaining their exact and approximate solutions, which, in turn, help us in quantitative and qualitative analysis of these PDEs. The interested reader can see some of these methods in [

1,

2].

Lie symmetry analysis is a powerful and influential tool for mathematically analyzing partial differential equations. It can be used in securing analytic solutions or in switching PDEs into solvable ordinary differential equations (ODEs). Diverse symmetry vectors are also discovered for the considered system in some cases, however, sometimes there emerges a chance of linear combination of these vectors. To avoid this, an optimal system is constructed. Each member of this system is used in lessening independent variables of the system until analytic solutions are obtained or PDEs are switched to solvable ODEs [

3,

4,

5,

6,

7]. The system also analyzes solutions of PDEs of different kinds as well as opens many fields [

8].

The generalized Burgers–Huxley equation, which has many utilizations in the fields of biology, metallurgy, chemistry, mathematics, and engineering is of the following type,

This is a non-linear equation that has secured much importance due to its appearance in many physical phenomena and its scientific utilization. The parameters

$a,b\ge 0$ are real constants,

n is a positive integer and

$k\in [0,1]$. When

$a=0$ and

$n=1,$ Equation (

1) reduces to the Huxley equation and with

$b=0,\phantom{\rule{4pt}{0ex}}n=1$, it becomes the Burgers equation. Some exact numerical and traveling wave solutions to (

1) were reported in [

9,

10,

11]. However, the spline collocation method for the Burgers–Huxley equation was discussed in a book by Schiesser [

12]. In addition, many other analytical and numerical methods for generalized Burgers–Huxley equations have been developed in the past, see for example [

13,

14,

15,

16,

17,

18,

19,

20,

21,

22,

23,

24,

25,

26,

27,

28,

29].

In this paper, we will analyze the Burgers–Huxley equation in two spatial dimensions, which is of the following form,

Equation (

2) couples both the assets of the Burgers equation (this is one of the basic models in fluid mechanics and is used to catch some of the properties of turbulent flow in a channel, which occurs due to the interaction of the reverse outcome of convection and diffusion and also describes the format of shock waves, traffic flow, and acoustic transmission [

30]) and the Huxley equation (which is used for nerve proliferation in neurophysics and wall proliferation in liquid crystals [

31]). So, we will designate this equation as simply a two dimensional Burgers–Huxley equation. The combined Burgers–Huxley equation shows a prototypical imitation that specifies the interaction between the reaction gadget, diffusion transport and convection effects, nerve proliferation, and motion in liquid crystals [

32].

In

Section 2, the vector fields and optimal systems for (

2) are obtained by using the Lie symmetry method. In

Section 3, we computed the similarity diminution for one and two dimensional subalgebras and hence obtained the group invariant solutions for (

2). Employing the power series method, certain power series solutions are achieved in

Section 4. Finally, in

Section 5, conservation laws are derived using optimal systems.

## 2. Lie Symmetry Analysis

In this section, we will study the Lie symmetries and optimal systems of the Burgers–Huxley equation. Consider the one-parameter Lie group of transformation:

where

$\u03f5$ is the group parameter. The infinitesimal operator associated with the above transformations is:

The coefficient functions

$\xi (x,y,t;u)$,

$\eta (x,y,t;u)$,

$\zeta (x,y,t;u)$, and

$\phi (x,y,t;u)$ are to be determined and the vector field

X satisfies the Lie symmetry condition,

where

The second prolongation of the infinitesimal generator is given by

with

where the the operator

${D}_{i}$ is defined as:

Coupling Equations (

5) and (

6), we can obtain the compatible condition for Equation (

5). Substituting extended transformations into the obtained compatible conditions and making the coefficients of several monomials in partial derivatives and numerous powers of

u equal, we get the following over determining system of PDEs:

By solving the over determining system of PDEs (

8), we obtain the coefficient functions

$\zeta ,\xi $, and

$\eta $ as:

where

${c}_{1}$,

${c}_{2}$, and

${c}_{3}$ are random constants. The Lie algebra of infinitesimal symmetry of Equation (

2) with

$k\ne 0$ is given by,

.

#### 2.1. Transformed Solutions

One can acquire the group transformation initiated by the Lie point symmetry operator

${X}_{i}$$(i=1,2,3)$ by solving the following ODEs

The one-parameter Lie symmetry groups generated by infinitesimals

${X}_{1}$,

${X}_{2}$, and

${X}_{3}$ are given by

where

${\u03f5}_{1}$,

${\u03f5}_{2}$, and

${\u03f5}_{3}$ are group parameters.

Depending on the values of

${r}_{1}$,

${r}_{2}$, and

${r}_{3}$, if

$f(x,y,t)$ is any confidential solutions of Equation (

2), then the new solutions can be given by

#### 2.2. Optimal System of Subalgebras

For the optimal system, we will first construct the tables for commutation relations and adjoint action of the obtained symmetries. For the sake of obtaining the adjoint representation we will use the Lie series in the form:

The commutation relations between basis elements satisfies:

whereas, the adjoint representation

Table 1 is given as:

Following the method given in [

8], consider the vector

X with random coefficients

${a}_{1},{a}_{2},{a}_{3},$ such that

suppose

${a}_{3}\ne 0$ and set up

${a}_{3}=1,$ so that

The scheme involves simplifying the coefficients as much as possible. To abolish the coefficient of ${X}_{2}$, we will use it to act on X. It is easy to see that the vector form cannot be reduced much more because commutation relations are zero.

Further, suppose

${a}_{3}$ = 0 and establish

${a}_{2}$ = 1 so that,

Repeating the same process and normalizing the coefficients we have the following one-dimensional optimal system of subalgebras,

where

a and

b are arbitrary constants.