# Wiener Process Effects on the Solutions of the Fractional (2 + 1)-Dimensional Heisenberg Ferromagnetic Spin Chain Equation

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- $\eta \left(0\right)=0,$
- $\eta \left(t\right)$ is a continuous function for $t\ge 0$,
- For ${t}_{1}<{t}_{2},$$\eta \left({t}_{1}\right)-\eta \left({t}_{2}\right)$ is independent,
- $\eta \left({t}_{2}\right)-\eta \left({t}_{1}\right)$ has a normal distribution with variance ${t}_{2}-{t}_{1}$ and mean 0.

**Definition**

**2.**

- ${\mathbb{T}}_{y}^{\alpha}[a\varphi \left(y\right)+b\Psi \left(y\right)]=a{\mathbb{T}}_{y}^{\alpha}\varphi \left(y\right)+b{\mathbb{T}}_{y}^{\alpha}\Psi \left(y\right),$
- ${\mathbb{T}}_{x}^{\alpha}(\varphi \circ \Psi )\left(y\right)={y}^{1-\alpha}{\Psi}^{\prime}\left(y\right)\varphi (\Psi \left(y\right))$,
- ${\mathbb{T}}_{y}^{\alpha}\left[b\right]=0,$
- ${\mathbb{T}}_{y}^{\alpha}\left[{y}^{b}\right]=b{y}^{b-\alpha},$
- ${\mathbb{T}}_{y}^{\alpha}\Psi \left(y\right)={y}^{1-\alpha}\frac{d\Psi}{dy},$

## 3. The Wave Equation of the SFSHFSCE

## 4. Analytical Solutions of the SFSHFSCE

**Remark**

**1.**

## 5. The Impact of Noise on the SFSHFSCE Solutions

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Profile picture of $\left|U\right|$ given in (14) in three dimension for $\alpha =1$ and ${\theta}_{3}=-5$.

**Figure 2.**Profile picture of $\left|U\right|$ given in (14) in three dimensions for $\alpha =0.5$ and ${\theta}_{3}=-5$.

**Figure 3.**Profile picture of $\left|U\right|$ given in (14) in two dimensions for $\alpha =1$ and different $\rho =0,0.5,1,2$.

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**MDPI and ACS Style**

Mohammed, W.W.; Al-Askar, F.M.; Cesarano, C.; Botmart, T.; El-Morshedy, M.
Wiener Process Effects on the Solutions of the Fractional (2 + 1)-Dimensional Heisenberg Ferromagnetic Spin Chain Equation. *Mathematics* **2022**, *10*, 2043.
https://doi.org/10.3390/math10122043

**AMA Style**

Mohammed WW, Al-Askar FM, Cesarano C, Botmart T, El-Morshedy M.
Wiener Process Effects on the Solutions of the Fractional (2 + 1)-Dimensional Heisenberg Ferromagnetic Spin Chain Equation. *Mathematics*. 2022; 10(12):2043.
https://doi.org/10.3390/math10122043

**Chicago/Turabian Style**

Mohammed, Wael W., Farah M. Al-Askar, Clemente Cesarano, Thongchai Botmart, and M. El-Morshedy.
2022. "Wiener Process Effects on the Solutions of the Fractional (2 + 1)-Dimensional Heisenberg Ferromagnetic Spin Chain Equation" *Mathematics* 10, no. 12: 2043.
https://doi.org/10.3390/math10122043