# Analysis of the Stochastic Quarter-Five Spot Problem Using Polynomial Chaos

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Statement of the Quarter-Five Spot Problem

#### 2.1. The Deterministic Problem

#### 2.2. The Stochastic Problem

## 3. Solution Implementation

#### 3.1. The Deterministic Problem

#### Numerical Example

#### 3.2. The Stochastic Quarter-Five Spot Problem

- First: Using PCE

- Second: Using KL expansion

## 4. Results

#### 4.1. Case of PCE Only

#### 4.2. Case of KL with PCE

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Solution of the quarter-five spot problem on a $32\times 32\text{}$ uniform grid. The plot shows the pressure distribution on each point in the grid, where the highest pressure values are in the bottom left corner and the pressure decreases gradually until it reaches the lowest value at the top right corner.

**Figure 3.**(

**a**) The pressure mean solution using MC method. (

**b**) The percentage of difference of solutions between MC and PCE methods.

**Table 1.**$\text{}{\psi}_{i}\left(\eta \right)$ used to evaluate the PC coefficients ${k}_{i}\left(x\right)$ [11].

${\mathit{\psi}}_{\mathit{i}}\left(\mathit{\xi}\right)$ | ${\mathit{\psi}}_{\mathit{i}}\left(\mathit{\eta}\right)$ | ${\mathit{\psi}}_{\mathit{i}}\left(\mathit{\eta}\right)$ |
---|---|---|

${\xi}_{i}$ | ${\eta}_{i}+{z}_{i}$ | ${z}_{i}$ |

${\xi}_{i}{\xi}_{j}-{\delta}_{ij}$ | $\left({\eta}_{i}+{z}_{i}\right)\left({\eta}_{j}+{z}_{j}\right)\_-{\delta}_{ij}$ | ${z}_{i}{z}_{j}$ |

${\xi}_{i}{\xi}_{j}{\xi}_{k}-{\xi}_{i}{\delta}_{jk}-{\xi}_{j}{\delta}_{ik}-{\xi}_{k}{\delta}_{ij}$ | $\left({\eta}_{i}+{z}_{i}\right)\left({\eta}_{j}+{z}_{j}\right)\left({\eta}_{j}+{z}_{j}\right)\_-{z}_{i}{\delta}_{jk}-{z}_{j}{\delta}_{ik}-{z}_{k}{\delta}_{ij}$ | ${z}_{i}{z}_{j}{z}_{k}$ |

${\mathit{k}}_{1}=0$ | ${\mathit{k}}_{2}=0$ | ${\mathit{k}}_{1}\&{\mathit{k}}_{2}\text{}\mathit{n}\mathit{o}\mathit{n}\text{}\mathit{z}\mathit{e}\mathit{r}\mathit{o}$ | |
---|---|---|---|

Mean difference percentage | 0.0035 | 0.0039 | 0.0031 |

Variance difference percentage | 2.56 | 2.61 | 2.52 |

$\mathbf{Correlation}\text{}\mathbf{Length}\text{}\mathit{l}$ | $\mathit{l}=0.1$ | $\mathit{l}=0.3$ | $\mathit{l}=0.5$ | $\mathit{l}=1$ |
---|---|---|---|---|

Number of terms for convergence | $>12$ | 12 | 8 | 5 |

Method | $\mathbf{MSE}\%\text{}\mathbf{at}\text{}\mathit{\sigma}=0.1$ | $\mathbf{MSE}\%\text{}\mathbf{at}\text{}\mathit{\sigma}=0.5$ | $\mathbf{MSE}\%\text{}\mathbf{at}\text{}\mathit{\sigma}=1$ |
---|---|---|---|

PCE | $0.02$ | 0.4 | 1 |

KL-PCE $l=0.1$ | 0.01 | 0.5 | 0.1 |

KL-PCE $l=0.3$ | 0.02 | 0.3 | 0.9 |

KL-PCE $l=0.5$ | 0.1 | 0.8 | 4 |

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**MDPI and ACS Style**

AbdelFattah, H.; Al-Johani, A.; El-Beltagy, M. Analysis of the Stochastic Quarter-Five Spot Problem Using Polynomial Chaos. *Molecules* **2020**, *25*, 3370.
https://doi.org/10.3390/molecules25153370

**AMA Style**

AbdelFattah H, Al-Johani A, El-Beltagy M. Analysis of the Stochastic Quarter-Five Spot Problem Using Polynomial Chaos. *Molecules*. 2020; 25(15):3370.
https://doi.org/10.3390/molecules25153370

**Chicago/Turabian Style**

AbdelFattah, Hesham, Amnah Al-Johani, and Mohamed El-Beltagy. 2020. "Analysis of the Stochastic Quarter-Five Spot Problem Using Polynomial Chaos" *Molecules* 25, no. 15: 3370.
https://doi.org/10.3390/molecules25153370