# The Influence of Noise on the Solutions of Fractional Stochastic Bogoyavlenskii Equation

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Theorem**

**1.**

- ${\mathcal{D}}_{x}^{\alpha}[{c}_{1}\varphi \left(x\right)+{c}_{2}g\left(x\right)]={c}_{1}{\mathcal{D}}_{x}^{\alpha}\varphi \left(x\right)+{c}_{2}{\mathcal{D}}_{x}^{\alpha}g\left(x\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{c}_{1},{c}_{2}\in \mathbb{R}$
- ${\mathcal{D}}_{x}^{\alpha}\left[C\right]=0,\phantom{\rule{4pt}{0ex}}C$ is a constant,
- ${\mathcal{D}}_{x}^{\alpha}\left[{x}^{\gamma}\right]=\gamma {x}^{\gamma -\alpha},\phantom{\rule{4pt}{0ex}}\gamma \in \mathbb{R},$
- ${\mathcal{D}}_{x}^{\alpha}g\left(x\right)={x}^{1-\alpha}\frac{dg}{dx}.$

**Definition**

**2**

## 3. The Wave Equation

## 4. Analytical Solutions of SFSBE

#### 4.1. The $exp(-\mathsf{\Phi}(\eta \left)\right)$-Expansion Method

**Case I:**When $a\ne 0,{b}^{2}-4a>0$, then the solution of Equation (13) is

**Case II:**When $a\ne 0,{b}^{2}-4a<0$, then the solution of Equation (13) is

**Case III:**When $a=0$ and $b\ne 0$, then the solutions of Equation (13) is

**Case IV:**When $a\ne 0,\phantom{\rule{4pt}{0ex}}b\ne 0$ and ${b}^{2}-4a=0$, then the solutions of Equation (13) is

**Case V:**When $a=0,b=0$ and ${b}^{2}-4a=0$, then the solution of Equation (13) is

#### 4.2. Sine–Cosine Method

**First case:**If $\frac{k}{m}>0,$ hence the solutions of Equation (11) take the form:

**Second case:**If $\frac{k}{m}<0,$ then the solutions of Equation (11) are

## 5. Impact of Multiplicative Brownian Motion

- The surface shrank as the order of the fractional operator $\alpha $ decreases,
- At $\rho =0,$ the surface is not completely flat and has some fluctuation,
- After minor transit patterns, the surface becomes considerably flatter when noise is included and its strength is increased $\rho =0.5,\phantom{\rule{4pt}{0ex}}1,\phantom{\rule{4pt}{0ex}}2$.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Al-Askar, F.M.; Mohammed, W.W.; Albalahi, A.M.; El-Morshedy, M.
The Influence of Noise on the Solutions of Fractional Stochastic Bogoyavlenskii Equation. *Fractal Fract.* **2022**, *6*, 156.
https://doi.org/10.3390/fractalfract6030156

**AMA Style**

Al-Askar FM, Mohammed WW, Albalahi AM, El-Morshedy M.
The Influence of Noise on the Solutions of Fractional Stochastic Bogoyavlenskii Equation. *Fractal and Fractional*. 2022; 6(3):156.
https://doi.org/10.3390/fractalfract6030156

**Chicago/Turabian Style**

Al-Askar, Farah M., Wael W. Mohammed, Abeer M. Albalahi, and Mahmoud El-Morshedy.
2022. "The Influence of Noise on the Solutions of Fractional Stochastic Bogoyavlenskii Equation" *Fractal and Fractional* 6, no. 3: 156.
https://doi.org/10.3390/fractalfract6030156