# Conditional Lie–Bäcklund Symmetries and Functionally Generalized Separation of Variables to Quasi-Linear Diffusion Equations with Source

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Conditional Lie-Bäcklund Symmetry (6) of Equation (7)

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**1**

**.**Equation (8) admits the CLBS (9) if there exists a function $W(t,x,v,\eta )$ such that

- Case 1. ${a}_{1}\left(x\right)=0$.

- Case 2. ${a}_{1}\left(x\right)\ne 0$.

## 3. Exact Solutions of Equation (7)

**Example**

**1.**

**(i)**For $k\ne -1,$

**(ii)**For $k=-1,$

**Example**

**2.**

**Example**

**3.**

**(i)**For $qs>0,$

**(ii)**For $qs<0,$

**(iii)**For $q=0,$

**Example**

**4.**

**(i)**For $s=4,$

**(ii)**For $s\ne 4,$

**Example**

**5.**

**Example**

**6.**

**(i)**For $s>0$,

**(ii)**For $s<0$,

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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No. | Equation (7) | CLBS (6) |
---|---|---|

1 | ${u}_{t}={\left({u}^{-\frac{1}{2}}{u}_{x}\right)}_{x}$ | $\sigma ={\left({u}^{\frac{1}{2}}\right)}_{xx}-\frac{6}{{x}^{2}}{u}^{\frac{1}{2}}$ |

2 | ${u}_{t}={\left({u}^{\frac{1}{k}}{u}_{x}\right)}_{x}+kq{u}^{\frac{k-1}{k}}$ | $\sigma ={\left({u}^{\frac{1}{k}}\right)}_{xx}-\frac{1}{x}{\left({u}^{\frac{1}{k}}\right)}_{x}$ |

3 | ${u}_{t}={\left[exp\left(u\right){u}_{x}\right]}_{x}+qexp(-u)$ | $\sigma ={\left[exp\left(u\right)\right]}_{xx}-\frac{1}{x}{\left[exp\left(u\right)\right]}_{x}$ |

4 | ${u}_{t}={\left({u}^{-\frac{4}{3}}{u}_{x}\right)}_{x}-\frac{3}{2}q{u}^{\frac{5}{3}}$ | $\sigma ={\left({u}^{-\frac{2}{3}}\right)}_{xx}-\frac{1}{x}{\left({u}^{-\frac{2}{3}}\right)}_{x}$ |

5 | ${u}_{t}={\left({u}^{\frac{1}{k}}{u}_{x}\right)}_{x}+k(k+1)s{u}^{\frac{k+1}{k}}+kq{u}^{\frac{k-1}{k}}$ | $\sigma ={\left({u}^{\frac{1}{k}}\right)}_{xx}+\sqrt{s}tan\left(\sqrt{s}x\right){\left({u}^{\frac{1}{k}}\right)}_{x}$ |

$\sigma ={\left({u}^{\frac{1}{k}}\right)}_{xx}-\sqrt{s}cot\left(\sqrt{s}x\right){\left({u}^{\frac{1}{k}}\right)}_{x}$ | ||

$\sigma ={\left({u}^{\frac{1}{k}}\right)}_{xx}-\sqrt{-s}tanh\left(\sqrt{-s}x\right){\left({u}^{\frac{1}{k}}\right)}_{x}$ | ||

$\sigma ={\left({u}^{\frac{1}{k}}\right)}_{xx}-\sqrt{-s}coth\left(\sqrt{-s}x\right){\left({u}^{\frac{1}{k}}\right)}_{x}$ | ||

6 | ${u}_{t}={\left[exp\left(u\right){u}_{x}\right]}_{x}+sexp\left(u\right)+qexp(-u)$ | $\sigma ={\left[exp\left(u\right)\right]}_{xx}+\sqrt{s}tan\left(\sqrt{s}x\right){\left[exp\left(u\right)\right]}_{x}$ |

$\sigma ={\left[exp\left(u\right)\right]}_{xx}-\sqrt{s}cot\left(\sqrt{s}x\right){\left[exp\left(u\right)\right]}_{x}$ | ||

$\sigma ={\left[exp\left(u\right)\right]}_{xx}-\sqrt{-s}tanh\left(\sqrt{-s}x\right){\left[exp\left(u\right)\right]}_{x}$ | ||

$\sigma ={\left[exp\left(u\right)\right]}_{xx}-\sqrt{-s}coth\left(\sqrt{-s}x\right){\left[exp\left(u\right)\right]}_{x}$ | ||

7 | ${u}_{t}={\left({u}^{-\frac{4}{3}}{u}_{x}\right)}_{x}-\frac{3}{4}s{u}^{-\frac{1}{3}}-\frac{3}{2}q{u}^{\frac{5}{3}}$ | $\sigma ={\left({u}^{-\frac{2}{3}}\right)}_{xx}+\sqrt{s}tan\left(\sqrt{s}x\right){\left({u}^{-\frac{2}{3}}\right)}_{x}$ |

$\sigma ={\left({u}^{-\frac{2}{3}}\right)}_{xx}-\sqrt{s}cot\left(\sqrt{s}x\right){\left({u}^{-\frac{2}{3}}\right)}_{x}$ | ||

$\sigma ={\left({u}^{-\frac{2}{3}}\right)}_{xx}-\sqrt{-s}tanh\left(\sqrt{-s}x\right){\left({u}^{-\frac{2}{3}}\right)}_{x}$ | ||

$\sigma ={\left({u}^{-\frac{2}{3}}\right)}_{xx}-\sqrt{-s}coth\left(\sqrt{-s}x\right){\left({u}^{-\frac{2}{3}}\right)}_{x}$ | ||

8 | ${u}_{t}={\left({u}^{-\frac{4}{3}}{u}_{x}\right)}_{x}$ | $\sigma ={\left({u}^{-\frac{2}{3}}\right)}_{xx}-\frac{2}{x}{\left({u}^{-\frac{2}{3}}\right)}_{x}+\frac{2}{{x}^{2}}{u}^{-\frac{2}{3}}$ |

$\sigma ={\left({u}^{-\frac{4}{3}}\right)}_{xx}-\frac{6}{x}{\left({u}^{-\frac{4}{3}}\right)}_{x}+\frac{12}{{x}^{2}}{u}^{-\frac{4}{3}}$ | ||

9 | ${u}_{t}={\left({u}^{\frac{s}{2-s}}{u}_{x}\right)}_{x}$ | $\sigma ={\left({u}^{\frac{s}{2-s}}\right)}_{xx}-\frac{s+2}{2x}{\left({u}^{\frac{s}{2-s}}\right)}_{x}+\frac{s}{{x}^{2}}{u}^{\frac{s}{2-s}}$ |

10 | ${u}_{t}={\left[exp\left(u\right){u}_{x}\right]}_{x}$ | $\sigma ={\left[exp\left(u\right)\right]}_{xx}-\frac{2}{x}{\left[exp\left(u\right)\right]}_{x}+\frac{2}{{x}^{2}}exp\left(u\right)$ |

11 | ${u}_{t}={\left({u}^{-\frac{4}{3}}{u}_{x}\right)}_{x}-\frac{3}{4}q{u}^{\frac{7}{3}}$ | $\sigma ={\left({u}^{-\frac{4}{3}}\right)}_{xxx}-\frac{3}{x}{\left({u}^{-\frac{4}{3}}\right)}_{xx}+\frac{3}{{x}^{2}}{\left({u}^{-\frac{4}{3}}\right)}_{x}$ |

12 | ${u}_{t}={\left({u}^{-\frac{4}{3}}{u}_{x}\right)}_{x}$ | $\sigma ={\left({u}^{-\frac{4}{3}}\right)}_{xxx}-\frac{6}{x}{\left({u}^{-\frac{4}{3}}\right)}_{xx}+\frac{18}{{x}^{2}}{\left({u}^{-\frac{4}{3}}\right)}_{x}-\frac{24}{{x}^{3}}{u}^{-\frac{4}{3}}$ |

13 | ${u}_{t}={\left({u}^{-\frac{4}{3}}{u}_{x}\right)}_{x}-\frac{3}{4}s{u}^{-\frac{1}{3}}-\frac{3}{4}q{u}^{\frac{7}{3}}$ | $\begin{array}{ll}\sigma =& {\left({u}^{-\frac{4}{3}}\right)}_{xxx}+3\sqrt{s}tan\left(\sqrt{s}x\right){\left({u}^{-\frac{4}{3}}\right)}_{xx}\\ & +\left[1+3{tan}^{2}\left(\sqrt{s}x\right)\right]s{\left({u}^{-\frac{4}{3}}\right)}_{x}\end{array}$ |

$\begin{array}{ll}\sigma =& {\left({u}^{-\frac{4}{3}}\right)}_{xxx}-3\sqrt{s}cot\left(\sqrt{s}x\right){\left({u}^{-\frac{4}{3}}\right)}_{xx}\\ & +\left[1+3{cot}^{2}\left(\sqrt{s}x\right)\right]s{\left({u}^{-\frac{4}{3}}\right)}_{x}\end{array}$ | ||

$\begin{array}{ll}\sigma =& {\left({u}^{-\frac{4}{3}}\right)}_{xxx}-3\sqrt{-s}tanh\left(\sqrt{-s}x\right){\left({u}^{-\frac{4}{3}}\right)}_{xx}\\ & +\left[1-3{tanh}^{2}\left(\sqrt{-s}x\right)\right]s{\left({u}^{-\frac{4}{3}}\right)}_{x}\end{array}$ | ||

$\begin{array}{ll}\sigma =& {\left({u}^{-\frac{4}{3}}\right)}_{xxx}-3\sqrt{-s}coth\left(\sqrt{-s}x\right){\left({u}^{-\frac{4}{3}}\right)}_{xx}\\ & +\left[1-3{coth}^{2}\left(\sqrt{-s}x\right)\right]s{\left({u}^{-\frac{4}{3}}\right)}_{x}\end{array}$ |

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**MDPI and ACS Style**

Wang, R.; Ji, L.
Conditional Lie–Bäcklund Symmetries and Functionally Generalized Separation of Variables to Quasi-Linear Diffusion Equations with Source. *Symmetry* **2020**, *12*, 844.
https://doi.org/10.3390/sym12050844

**AMA Style**

Wang R, Ji L.
Conditional Lie–Bäcklund Symmetries and Functionally Generalized Separation of Variables to Quasi-Linear Diffusion Equations with Source. *Symmetry*. 2020; 12(5):844.
https://doi.org/10.3390/sym12050844

**Chicago/Turabian Style**

Wang, Rui, and Lina Ji.
2020. "Conditional Lie–Bäcklund Symmetries and Functionally Generalized Separation of Variables to Quasi-Linear Diffusion Equations with Source" *Symmetry* 12, no. 5: 844.
https://doi.org/10.3390/sym12050844