# Nonclassical Symmetry Solutions for Non-Autonomous Reaction-Diffusion Equations

## Abstract

**:**

## 1. Introduction

## 2. Nonclassical Symmetries

#### 2.1. $Q(u)$ Cubic

#### 2.2. ${f}_{1}(t,x)=0$

#### 2.2.1. ${f}_{1}(t,x)=0$ and $s(t)={c}_{1}$

#### 2.2.2. ${f}_{1}(t,x)=0$ and $s(t)=\frac{1}{{c}_{1}t+{c}_{2}}$

#### 2.2.3. ${f}_{1}(t,x)=0$ and $s(t)=\frac{1}{{c}_{1}{\mathrm{e}}^{t/{c}_{2}^{2}}+{c}_{2}}$

## 3. Reduced Equations and Some Example Solutions

#### 3.1. $Q(u)=a{u}^{3}$, $D(x)={D}_{0}{x}^{2}$, $r(x)={x}^{2n}$

#### 3.2. $Q(u)$ Arbitrary

#### 3.2.1. $Q(u)$ Arbitrary, $D(x)=-({c}_{1}x-{c}_{2})lnx$, $r(x)=\frac{{c}_{1}x-{c}_{2}}{lnx}$

#### 3.2.2. $Q(u)$ Arbitrary, $D(x)={D}_{0}\gamma {x}^{\alpha +2}$, $r(x)=\gamma {x}^{\alpha}$

#### 3.2.3. $Q(u)$ Arbitrary, $D(x)={D}_{0}{x}^{2}lnx$, $r(x)=\frac{1}{lnx}$

## 4. Discussion and Final Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plot of the spatial dependence and the example solutions arising from Symmetry 2(b). (

**a**) Spatially-dependent diffusion coefficient. (

**b**) Spatially-dependent coefficient of the reaction term. (

**c**) Population density as given by the solution (21) for the quadratic Fisher-type reaction term ($t=0,0.3,\dots ,1.5$). (

**d**) Population density as given by the solution (22) for the cubic Huxley-type reaction term ($t=0,0.5,\dots ,2.5$). In both cases, the arrow shows increasing time. In all plots, the parameter values are $a=0.6,\phantom{\rule{3.33333pt}{0ex}}{c}_{1}=-2,\phantom{\rule{3.33333pt}{0ex}}{c}_{2}=-1.2,\phantom{\rule{3.33333pt}{0ex}}{c}_{3}=-1$.

**Table 1.**Admissible forms of the nonlinear reaction terms and the spatially-dependent diffusion and reaction coefficients when Equation (1) admit classical (c) and nonclassical (n) symmetries. The infinitesimals are also given in each case.

$\mathit{D}(\mathit{x}),\phantom{\rule{3.33333pt}{0ex}}\mathit{r}(\mathit{x}),\phantom{\rule{3.33333pt}{0ex}}\mathit{Q}(\mathit{u})$ | Symmetry | Infinitesimals | |
---|---|---|---|

1 | $D(x)={D}_{0}{x}^{2}$ | n | $T=1$ |

$r(x)={x}^{2n}$ | $X=\left[3\sqrt{{\displaystyle \frac{a{D}_{0}}{2}}}{x}^{n}u+{D}_{0}(2n-1)\right]x$ | ||

$Q(u)=a{u}^{3}$ | $U=-{\textstyle \frac{1}{2}}\left[3a{x}^{2n}{u}^{2}+3\sqrt{2a{D}_{0}}n{x}^{n}u+{D}_{0}n(n+1)\right]u$ | ||

2(a) | $D(x)=-({c}_{1}x-{c}_{2})lnx$ | n | $T=1$ |

$r(x)=\frac{{c}_{1}x-{c}_{2}}{lnx}$ | $X={c}_{1}lnx$ | ||

$Q(u)$ arbitrary | $U=0$ | ||

2(b) | $D(x)={D}_{0}\gamma {x}^{\alpha +2}$ | c | $T=1$ |

$r(x)=\gamma {x}^{\alpha}$ | $X=\frac{x}{\alpha t+{c}_{2}}$ | ||

$Q(u)$ arbitrary | $U=0$ | ||

2(c) | $D(x)={D}_{0}{x}^{2}lnx$ | c | $T=1$ |

$r(x)=\frac{1}{lnx}$ | $X=\frac{{D}_{0}xlnx}{\beta {\mathrm{e}}^{{D}_{0}t}-1}$ | ||

$Q(u)$ arbitrary | $U=0$ |

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Bradshaw-Hajek, B.H.
Nonclassical Symmetry Solutions for Non-Autonomous Reaction-Diffusion Equations. *Symmetry* **2019**, *11*, 208.
https://doi.org/10.3390/sym11020208

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Bradshaw-Hajek BH.
Nonclassical Symmetry Solutions for Non-Autonomous Reaction-Diffusion Equations. *Symmetry*. 2019; 11(2):208.
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Bradshaw-Hajek, Bronwyn H.
2019. "Nonclassical Symmetry Solutions for Non-Autonomous Reaction-Diffusion Equations" *Symmetry* 11, no. 2: 208.
https://doi.org/10.3390/sym11020208