# A Numerical Algorithm for Solving Nonlocal Nonlinear Stochastic Delayed Systems with Variable-Order Fractional Brownian Noise

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- An accurate and computationally efficient technique for solving FSDDE-VOFBM with Hurst index was proposed;
- A cubic spline interpolation method for time discretization was adopted;
- Error and convergence analysis of the suggested scheme was performed;
- The proposed numerical technique was applied to fractional stochastic dynamical systems and assessed from the perspective of statistical indicators of stochastic responses.

## 2. Computational Implementation

**Proposition**

**1.**

**Proof.**

**CSM**-algorithm”. It should be noted that, for solving the initial condition problems such as (2), we incorporated the

**CSM**-algorithm with the finite differential quotient stated as:

## 3. Numerical Results and Discussion

**CSM**-algorithm were compared with the

**IQM**[54] and

**BSM**methods [55].

**Example**

**1.**

**IQM**- [54] and $\mathbf{CSM}$-algorithms, for distinct values of $\Delta $, $\gamma =\{0.25,0.5,0.7,0.9\}$, and $t\in [0,10]$. We verified that, for all values of $\gamma $, the errors yielded by the proposed method decrease as $\Delta $ diminishes. Table 2 lists the values of some statistical indicators (SIs) for several fractional orders, with $T=10$. We verified that the values of the median and mean are equal for the fractional orders $\gamma =\{0.55,0.75,0.95\}$. This means that the diagram driven by 50 simulated paths at $T=10$ is symmetric.

**Example**

**2.**

**BSM**- [55] and $\mathbf{CSM}$-algorithms for $\gamma =\{0.55,0.75,0.95\}$, with $\Delta =\{0.02,0.01,0.005\}$ and $\lambda =0.05$ in the time interval $t\in [0,10]$. We verified that a more accurate approximation is obtained when reducing the step size. Table 4 presents the SI values for several $\gamma $ cases, at $T=10$. We verified that the diagram driven by 50 paths for $\gamma =\{0.55,0.75,0.95\}$ at $T=10$ is symmetric. It should be noted that adopting $\sigma =5$ corresponds to having data polluted by large random noise. Still, the stability of the proposed method was verified.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The time evolution of $u\left(t\right)$ for (12) with the proposed algorithm for $\kappa =1.58,\phantom{\rule{3.33333pt}{0ex}}\eta =-1.6$, $\lambda =0.1$, $\gamma =\{0.55,0.75,0.95\}$, $H\left(t\right)=0.95-0.02t$, and $\Delta =0.01$: (

**left side**) $\sigma =0$; (

**right side**) $\sigma =0.01$.

**Figure 3.**The time evolution of $u\left(t\right)$ for (14) with the proposed algorithm for $\kappa =9,\phantom{\rule{3.33333pt}{0ex}}\mu =1$, $\lambda =0.05,\phantom{\rule{3.33333pt}{0ex}}\rho =2$, $\gamma =\{0.55,0.75,0.95\}$, $H\left(t\right)=0.6+0.2exp\left(0.01t\right)$, and $\Delta =0.01$: (

**left side**) $\sigma =0$; (

**right side**) $\sigma =5$.

**Figure 4.**The time evolution of $u\left(t\right)$ (

**left side**) and the SI values (

**right side**) for (14) with the proposed algorithm, with $\kappa =9,\phantom{\rule{3.33333pt}{0ex}}\mu =1,\phantom{\rule{3.33333pt}{0ex}}\lambda =0.05,\phantom{\rule{3.33333pt}{0ex}}\rho =2,\phantom{\rule{3.33333pt}{0ex}}\sigma =5$, $\gamma =0.55$, $H\left(t\right)=0.6+0.2exp\left(0.01t\right)$ and $\Delta =0.01$ over 50 simulated paths.

**Table 1.**Example 1: The values of $\parallel {\overline{\mathcal{E}}}_{\varrho}{\parallel}_{ms}$, $EC{O}_{ms}$, and CPU time (expressed in seconds) for (13) obtained with the $\mathbf{IQM}$- [54] and $\mathbf{CSM}$-algorithms for distinct choices of $\gamma $ and $\Delta $, with $\kappa =1.58$, $\eta =-1.6,\phantom{\rule{3.33333pt}{0ex}}\sigma =0.01,\phantom{\rule{3.33333pt}{0ex}}\lambda =0.1$, and $H\left(t\right)=0.95-0.02\phantom{\rule{0.166667em}{0ex}}t$, in $t\in [0,10]$.

IQM-Algorithm [54] | $\mathbf{CSM}$-Algorithm | ||||||
---|---|---|---|---|---|---|---|

$\gamma $ | $\Delta $ | $\parallel {\overline{\mathcal{E}}}_{\varrho}{\parallel}_{ms}$ | $EC{O}_{ms}$ | $CPU\phantom{\rule{4pt}{0ex}}Time$ | $\parallel {\overline{\mathcal{E}}}_{\varrho}{\parallel}_{ms}$ | $EC{O}_{ms}$ | $CPU\phantom{\rule{4pt}{0ex}}Time$ |

$0.02\phantom{\rule{3.33333pt}{0ex}}$ | $2.18\times {10}^{-4}$ | − | 35.703 | $7.31\times {10}^{-5}$ | − | 26.344 | |

0.55 | $0.01\phantom{\rule{3.33333pt}{0ex}}$ | $1.12\times {10}^{-4}$ | $0.96$ | 151.516 | $6.31\times {10}^{-6}$ | $3.53$ | 114.860 |

$0.005$ | $6.04\times {10}^{-5}$ | $0.89$ | 708.140 | $1.13\times {10}^{-6}$ | $2.48$ | 494.953 | |

$0.02\phantom{\rule{3.33333pt}{0ex}}$ | $2.49\times {10}^{-4}$ | − | 35.578 | $1.47\times {10}^{-4}$ | − | 25.719 | |

0.75 | $0.01\phantom{\rule{3.33333pt}{0ex}}$ | $1.36\times {10}^{-4}$ | $0.87$ | 147.922 | $1.01\times {10}^{-5}$ | $3.86$ | 111.859 |

$0.005$ | $7.39\times {10}^{-5}$ | $0.88$ | 704.922 | $5.02\times {10}^{-6}$ | $2.32$ | 490.843 | |

$0.02\phantom{\rule{3.33333pt}{0ex}}$ | $5.65\times {10}^{-4}$ | − | 35.641 | $2.84\times {10}^{-4}$ | − | 25.438 | |

0.95 | $0.01\phantom{\rule{3.33333pt}{0ex}}$ | $3.21\times {10}^{-4}$ | $0.85$ | 151.937 | $1.60\times {10}^{-5}$ | $4.13$ | 109.656 |

$0.005$ | $1.57\times {10}^{-4}$ | $1.00$ | 705.917 | $4.04\times {10}^{-6}$ | $1.98$ | 499.797 |

**Table 2.**The approximated SI values concerning the 50 simulated paths for (13), with $\gamma =\{0.55,0.75,0.95\}$, $\kappa =1.58,\phantom{\rule{3.33333pt}{0ex}}\eta =-1.6,\phantom{\rule{3.33333pt}{0ex}}\lambda =0.1,\phantom{\rule{3.33333pt}{0ex}}\sigma =0.01,\phantom{\rule{3.33333pt}{0ex}}H\left(t\right)=0.95-0.02\phantom{\rule{0.166667em}{0ex}}t$, and step size $\Delta =0.01$, at $T=10$.

$\mathbf{SI}$ | $\mathit{\gamma}=0.55$ | $\mathit{\gamma}=0.75$ | $\mathit{\gamma}=0.95$ |
---|---|---|---|

Mean | $0.093$ | $0.089$ | $0.083$ |

Median | $0.093$ | $0.089$ | $0.083$ |

First quartile | $0.091$ | $0.088$ | $0.081$ |

Third quartile | $0.094$ | $0.092$ | $0.087$ |

Kurtosis | $2.350$ | $2.582$ | $2.944$ |

Skewness | $0.200$ | $-0.038$ | $-0.197$ |

Standard deviation | $2.268\times {10}^{-3}$ | $3.661\times {10}^{-3}$ | $5.975\times {10}^{-3}$ |

95% Confidence interval | $[0.089,0.097]$ | $[0.083,0.095]$ | $[0.073,0.093]$ |

**Table 3.**Example 2: The values of $\parallel {\overline{\mathcal{E}}}_{\varrho}{\parallel}_{ms}$, $EC{O}_{ms}$ and CPU time (expressed in seconds) for (14) obtained with the

**BSM**- [55] and $\mathbf{CSM}$-algorithms for different values of $\gamma $ and $\Delta $, with $\kappa =9$, $\mu =1,\phantom{\rule{3.33333pt}{0ex}}\lambda =0.05,\phantom{\rule{3.33333pt}{0ex}}\rho =2,\phantom{\rule{3.33333pt}{0ex}}\sigma =5$, and $H\left(t\right)=0.6+0.2exp\left(0.01t\right)$, in $t\in [0,10]$.

$\mathbf{BSM}$-Algorithm [55] | $\mathbf{CSM}$-Algorithm | ||||||
---|---|---|---|---|---|---|---|

$\gamma $ | $\Delta $ | $\parallel {\overline{\mathcal{E}}}_{\varrho}{\parallel}_{ms}$ | $EC{O}_{ms}$ | $CPU\phantom{\rule{4pt}{0ex}}Time$ | $\parallel {\overline{\mathcal{E}}}_{\varrho}{\parallel}_{ms}$ | $EC{O}_{ms}$ | $CPU\phantom{\rule{4pt}{0ex}}Time$ |

$0.02\phantom{\rule{3.33333pt}{0ex}}$ | $1.58\times {10}^{-3}$ | − | 10.140 | $9.54\times {10}^{-4}$ | − | 25.907 | |

0.55 | $0.01\phantom{\rule{3.33333pt}{0ex}}$ | $9.88\times {10}^{-4}$ | $0.68$ | 44.828 | $4.71\times {10}^{-4}$ | $1.02$ | 114.094 |

$0.005$ | $7.28\times {10}^{-4}$ | $0.44$ | 194.265 | $8.68\times {10}^{-5}$ | $2.44$ | 495.860 | |

$0.02\phantom{\rule{3.33333pt}{0ex}}$ | $5.80\times {10}^{-4}$ | − | 9.937 | $3.98\times {10}^{-4}$ | − | 25.719 | |

0.75 | $0.01\phantom{\rule{3.33333pt}{0ex}}$ | $2.91\times {10}^{-4}$ | $0.99$ | 4.328 | $1.85\times {10}^{-4}$ | $1.10$ | 115.375 |

$0.005$ | $2.07\times {10}^{-4}$ | $0.49$ | 191.828 | $5.43\times {10}^{-5}$ | $1.77$ | 508.531 | |

$0.02\phantom{\rule{3.33333pt}{0ex}}$ | $4.05\times {10}^{-4}$ | − | 10.687 | $3.96\times {10}^{-4}$ | − | 26.391 | |

0.95 | $0.01\phantom{\rule{3.33333pt}{0ex}}$ | $2.26\times {10}^{-4}$ | $0.84$ | 45.094 | $1.87\times {10}^{-4}$ | $1.08$ | 120.328 |

$0.005$ | $9.42\times {10}^{-5}$ | $1.26$ | 185.047 | $5.89\times {10}^{-5}$ | $1.66$ | 494.093 |

**Table 4.**The approximated SI values concerning the 50 simulated paths for (14), with $\gamma =\{0.55,0.75,0.95\}$, $\kappa =9,\phantom{\rule{3.33333pt}{0ex}}\mu =1,\phantom{\rule{3.33333pt}{0ex}}\lambda =0.05,\phantom{\rule{3.33333pt}{0ex}}\rho =2,\phantom{\rule{3.33333pt}{0ex}}\sigma =5,\phantom{\rule{3.33333pt}{0ex}}H\left(t\right)=0.6+0.2exp\left(0.01t\right)$ and step size $\Delta =0.01$, at $T=10$.

$\mathit{S}\mathit{I}$ | $\mathit{\gamma}=0.55$ | $\mathit{\gamma}=0.75$ | $\mathit{\gamma}=0.95$ |
---|---|---|---|

Mean | $1.496$ | $1.501$ | $1.503$ |

Median | $1.496$ | $1.501$ | $1.503$ |

First quartile | $1.495$ | $1.500$ | $1.503$ |

Third quartile | $1.497$ | $1.502$ | $1.504$ |

Kurtosis | $3.145$ | $2.670$ | $2.567$ |

Skewness | $-0.834$ | $-0.067$ | $0.215$ |

Standard deviation | $1.490\times {10}^{-3}$ | $4.645\times {10}^{-4}$ | $5.198\times {10}^{-5}$ |

95% Confidence interval | $[1.493,1.498]$ | $[1.500,1.502]$ | $[1.503,1.504]$ |

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**MDPI and ACS Style**

Moghaddam, B.P.; Pishbin, M.; Mostaghim, Z.S.; Iyiola, O.S.; Galhano, A.; Lopes, A.M. A Numerical Algorithm for Solving Nonlocal Nonlinear Stochastic Delayed Systems with Variable-Order Fractional Brownian Noise. *Fractal Fract.* **2023**, *7*, 293.
https://doi.org/10.3390/fractalfract7040293

**AMA Style**

Moghaddam BP, Pishbin M, Mostaghim ZS, Iyiola OS, Galhano A, Lopes AM. A Numerical Algorithm for Solving Nonlocal Nonlinear Stochastic Delayed Systems with Variable-Order Fractional Brownian Noise. *Fractal and Fractional*. 2023; 7(4):293.
https://doi.org/10.3390/fractalfract7040293

**Chicago/Turabian Style**

Moghaddam, Behrouz Parsa, Maryam Pishbin, Zeinab Salamat Mostaghim, Olaniyi Samuel Iyiola, Alexandra Galhano, and António M. Lopes. 2023. "A Numerical Algorithm for Solving Nonlocal Nonlinear Stochastic Delayed Systems with Variable-Order Fractional Brownian Noise" *Fractal and Fractional* 7, no. 4: 293.
https://doi.org/10.3390/fractalfract7040293