# Lie Symmetry Analysis, Explicit Solutions and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation

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## Abstract

**:**

## 1. Introduction

## 2. Lie Symmetry Analysis

#### 2.1. Transformed Solutions

#### 2.2. Optimal System of Subalgebras

## 3. Symmetry Reduction

#### Symmetry Reductions for Optimal System

- (i)
- For the linear combination $a{X}_{1}+{X}_{2}$, the invariants can be obtained by solving the characteristic equation $\frac{dt}{a}=\frac{dx}{1}=\frac{dy}{0}=\frac{du}{0}$, giving$$s=ax-t,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}r=y,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}\Psi =u.$$Using,$$u=\Psi (s,r),$$$${a}^{2}\frac{{\partial}^{2}\Psi}{\partial {s}^{2}}+\frac{{\partial}^{2}\Psi}{\partial {r}^{2}}+(1+a\Psi )\frac{\partial \Psi}{\partial s}+\Psi \frac{\partial \Psi}{\partial r}+\Psi (k-\Psi )(\Psi -1)=0.$$By the application of the similarity transformation method on reduced Equation (11) again, we have$$\xi ={c}_{1},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}\zeta ={c}_{2},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}\phi =0,$$$$\frac{ds}{1}=\frac{dr}{1}=\frac{d\Psi}{0}.$$Hence, $\Psi $ can be written as:$$\Psi =\beta (\alpha ),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}\alpha =s-r.$$
- (ii)
- For the linear combination $a{X}_{1}+b{X}_{2}+{X}_{3}$, the invariants can be obtained by solving the characteristic equation$$\frac{dt}{a}=\frac{dx}{b}=\frac{dy}{1}=\frac{du}{0},$$$$s=ax-bt,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}r=ay-t,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}\Psi =u.$$Using,$$u=\Psi (s,r),$$$${a}^{2}\frac{{\partial}^{2}\Psi}{\partial {s}^{2}}+{a}^{2}\frac{{\partial}^{2}\Psi}{\partial {r}^{2}}+a\Psi \frac{\partial \Psi}{\partial s}+a\Psi \frac{\partial \Psi}{\partial r}+b\frac{\partial \Psi}{\partial s}+\frac{\partial \Psi}{\partial r}+\Psi (k-\Psi )(\Psi -1)=0.$$By the application of the similarity transformation method on reduced Equation (17) again, we obtain$$\xi ={c}_{1},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}\zeta ={c}_{2},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}\phi =0,$$$$\frac{ds}{1}=\frac{dr}{1}=\frac{d\Psi}{0}.$$Therefore, $\Psi $ can be written as:$$\Psi =\beta (\alpha ),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}\alpha =s-r.$$Substituting Equation (20) into (17), we have following ODE:$$2{a}^{2}\frac{{d}^{2}\beta}{d{\alpha}^{2}}+(b-1)\frac{d\beta}{d\alpha}+\beta (\beta -1)(k-\beta )=0.$$Reductions corresponding to the remaining vectors occuring in the optimal system can be obtained in a similar way, hence, we omit the details here.

## 4. Explicit Power Series Solution

#### 4.1. Series Solution of Reduced Equation (15)

#### 4.2. Series Solution of Reduced Equation (21)

## 5. Conservation Laws

#### 5.1. Preliminaries

**Theorem**

**1.**

**Proof.**

#### 5.2. Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation

- (i)
- For ${X}_{1}$, we have $W=-{u}_{t}$ and ${\xi}^{t}=1.$ Substituting these values in Equation (35), we find$$\begin{array}{c}{C}^{t}=-v{u}_{xx}-v{u}_{yy}-vu{u}_{x}-vu{u}_{y}-kv{u}^{2}+kuv+{u}^{3}v-v{u}^{2},\hfill \\ {C}^{x}=uv{u}_{t}-{v}_{x}{u}_{t}+v{u}_{xt},\hfill \\ {C}^{y}=uv{u}_{t}-{v}_{y}{u}_{t}+v{u}_{yt}.\hfill \end{array}$$
- (ii)
- For ${X}_{2}$, W =$-{u}_{x}$ and ${\xi}^{x}=1.$ Substituting in Equation (35), we obtain$$\begin{array}{c}{C}^{t}=-v{u}_{x},\hfill \\ {C}^{x}=v{u}_{t}-v{u}_{yy}-uv{u}_{y}-kv{u}^{2}+kuv+{u}^{3}v-v{u}^{2}-{u}_{x}{v}_{x},\hfill \\ {C}^{y}=uv{u}_{x}-{v}_{y}{u}_{x}+v{u}_{xy}.\hfill \end{array}$$
- (iii)
- For the generator ${X}_{3}$, W =$-{u}_{y}$ and ${\xi}^{y}$= 1. So, in this case, we have the following conserved vector:$$\begin{array}{c}{C}^{t}=-v{u}_{y},\hfill \\ {C}^{x}={u}_{y}uv-{u}_{y}{v}_{x}+v{u}_{xy},\hfill \\ {C}^{y}=v{u}_{t}-v{u}_{xx}-uv{u}_{x}-kv{u}^{2}+kuv+v{u}^{3}-v{u}^{2}-{u}_{y}{v}_{y}.\hfill \end{array}$$
- (iv)
- For the generator $a{X}_{1}+{X}_{2}$, W =$-a{u}_{t}-{u}_{x}$ and ${\xi}^{t}$= a, ${\xi}^{x}$= 1. Substituting into (35), we have$$\begin{array}{c}{C}^{t}=-av{u}_{xx}-av{u}_{yy}-auv{u}_{x}-auv{u}_{y}-akv{u}^{2}+akuv+av{u}^{3}-av{u}^{2}\hfill \\ -v{u}_{x},\hfill \\ {C}^{x}=v{u}_{t}-v{u}_{yy}-uv{u}_{y}-{u}^{2}kv+kuv+v{u}^{3}-v{u}^{2}+auv{u}_{t}-a{u}_{t}{v}_{x}\hfill \\ -{u}_{x}{v}_{x}+av{u}_{xt},\hfill \\ {C}^{y}=auv{u}_{t}-a{u}_{t}{v}_{y}+uv{u}_{x}-{u}_{x}{v}_{y}+av{u}_{yt}+v{u}_{xy}.\hfill \end{array}$$
- (v)
- For the generator $a{X}_{1}+b{X}_{2}+{X}_{3}$, W =$-a{u}_{t}-b{u}_{x}-{u}_{y}$ and ${\xi}^{t}$ = a, ${\xi}^{x}$ = b and ${\xi}^{y}$ = 1. Using these values in Equation (35), we have$$\begin{array}{c}{C}^{t}=-av{u}_{xx}-av{u}_{yy}-auv{u}_{x}-auv{u}_{y}-akv{u}^{2}+akuv+av{u}^{3}-av{u}^{2}\hfill \\ -bv{u}_{x}-v{u}_{y},\hfill \\ {C}^{x}=v{u}_{t}-bv{u}_{yy}-buv{u}_{y}-b{u}^{2}kv+bkuv+bv{u}^{3}-bv{u}^{2}+auv{u}_{t}\hfill \\ +bv{u}_{t}+uv{u}_{y}-a{u}_{t}{v}_{x}-b{u}_{x}{v}_{x}-{v}_{x}{u}_{y}+av{u}_{xt}+vuxy,\hfill \\ {C}^{y}=v{u}_{t}-v{u}_{xx}-vu{u}_{x}-kv{u}^{2}+kuv+v{u}^{3}+auv{u}_{t}+buv{u}_{x}-a{v}_{y}{u}_{t}\hfill \\ -b{u}_{x}{v}_{y}-{v}_{y}{u}_{y}+av{u}_{yt}+bv{u}_{xy}.\hfill \end{array}$$The conserved vectors comprise random solutions of the adjoint equation, thereby implying the interminable number of conservation laws. Conservation laws play a compelling role in the solution process of an equation or system of equations.

## 6. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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$\mathit{Ad}$ | ${\mathit{X}}_{1}$ | ${\mathit{X}}_{2}$ | ${\mathit{X}}_{3}$ |
---|---|---|---|

${X}_{1}$ | ${X}_{1}$ | ${X}_{2}$ | ${X}_{3}$ |

${X}_{2}$ | ${X}_{1}$ | ${X}_{2}$ | ${X}_{3}$ |

${X}_{3}$ | ${X}_{1}$ | ${X}_{2}$ | ${X}_{3}$ |

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**MDPI and ACS Style**

Hussain, A.; Bano, S.; Khan, I.; Baleanu, D.; Sooppy Nisar, K.
Lie Symmetry Analysis, Explicit Solutions and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation. *Symmetry* **2020**, *12*, 170.
https://doi.org/10.3390/sym12010170

**AMA Style**

Hussain A, Bano S, Khan I, Baleanu D, Sooppy Nisar K.
Lie Symmetry Analysis, Explicit Solutions and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation. *Symmetry*. 2020; 12(1):170.
https://doi.org/10.3390/sym12010170

**Chicago/Turabian Style**

Hussain, Amjad, Shahida Bano, Ilyas Khan, Dumitru Baleanu, and Kottakkaran Sooppy Nisar.
2020. "Lie Symmetry Analysis, Explicit Solutions and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation" *Symmetry* 12, no. 1: 170.
https://doi.org/10.3390/sym12010170