# Exact Solutions and Conservation Laws of Time-Fractional Levi Equation

## Abstract

**:**

## 1. Introductions

## 2. Preliminaries on Fractional Calculus

## 3. Symmetry Reduction

**Theorem**

**1.**

#### 3.1. Reduction under Space-Translation

#### 3.2. Reduction under Scaling

## 4. Conservation Law

## 5. Invariant Subspace

**Theorem**

**2.**

- (i)
- The operators ${F}^{1}[u,v]$ and ${F}^{2}[u,v]$ in (45) admit no invariant subspace, if $({n}_{1},{n}_{2})$ is an element of the list (53) with ${n}_{1}\ge 2$.
- (ii)
- When $({n}_{1},{n}_{2})=(1,1)$, the operators ${F}^{1}[u,v]$ and ${F}^{2}[u,v]$ in (45) admit two invariant subspaces, given by$$\begin{array}{c}{W}_{1}^{1}\times {W}_{1}^{2}=\mathfrak{L}\left\{1\right\}\times \mathfrak{L}\left\{1\right\};\hfill \end{array}$$$$\begin{array}{c}{W}_{1}^{1}\times {W}_{1}^{2}=\mathfrak{L}\left\{x\right\}\times \mathfrak{L}\left\{1\right\}.\hfill \end{array}$$

#### 5.1. Invariant Subspace ${W}_{1}^{1}\times {W}_{1}^{2}=\mathfrak{L}\left\{1\right\}\times \mathfrak{L}\left\{1\right\}$

#### 5.2. Invariant Subspace ${W}_{1}^{1}\times {W}_{1}^{2}=\mathfrak{L}\left\{x\right\}\times \mathfrak{L}\left\{1\right\}$

**Proposition**

**1.**

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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Symm. | $({\mathit{W}}^{1},{\mathit{W}}^{2})$ | ${\mathit{C}}^{\mathit{t}}$ | ${\mathit{C}}^{\mathit{x}}$ |
---|---|---|---|

${\mathrm{X}}_{1}$ | $(-{u}_{x},-{v}_{x})$ | $-{D}_{t}^{\alpha -1}\left({u}_{x}\right)-{D}_{t}^{\alpha -1}\left({v}_{x}\right)$ | $\begin{array}{c}2(u+v){u}_{x}+2(u+1){v}_{x}\hfill \\ \phantom{\rule{1.em}{0ex}}-{u}_{xx}+{v}_{xx}\hfill \end{array}$ |

${\mathrm{X}}_{2}$ | $\begin{array}{c}(-t{u}_{t}-\frac{\alpha}{2}(x{u}_{x}+u),\hfill \\ \phantom{\rule{1.em}{0ex}}-t{v}_{t}-\alpha v-\frac{\alpha}{2}x{u}_{x})\hfill \end{array}$ | $\begin{array}{c}-\alpha x{D}_{t}^{\alpha -1}\left({u}_{x}\right)-2{D}_{t}^{\alpha -1}\left(t{u}_{t}\right)\hfill \\ \phantom{\rule{1.em}{0ex}}-\alpha {D}_{t}^{\alpha -1}v-\frac{\alpha}{2}{D}_{t}^{\alpha -1}u\hfill \end{array}$ | $\begin{array}{c}(u+v)(\alpha u+\alpha x{u}_{x}+2t{u}_{t})\hfill \\ \phantom{\rule{1.em}{0ex}}+(1+u)(2\alpha v+\alpha x{u}_{x}+2t{u}_{t})\hfill \\ \phantom{\rule{1.em}{0ex}}-\alpha x{u}_{xx}-\alpha {v}_{x}-2t{u}_{tx}-\frac{3\alpha}{2}{u}_{x}\hfill \end{array}$ |

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Feng, W.
Exact Solutions and Conservation Laws of Time-Fractional Levi Equation. *Symmetry* **2020**, *12*, 1074.
https://doi.org/10.3390/sym12071074

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Feng W.
Exact Solutions and Conservation Laws of Time-Fractional Levi Equation. *Symmetry*. 2020; 12(7):1074.
https://doi.org/10.3390/sym12071074

**Chicago/Turabian Style**

Feng, Wei.
2020. "Exact Solutions and Conservation Laws of Time-Fractional Levi Equation" *Symmetry* 12, no. 7: 1074.
https://doi.org/10.3390/sym12071074