# A Hybrid Metaheuristic Based on Neurocomputing for Analysis of Unipolar Electrohydrodynamic Pump Flow

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## Abstract

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## 1. Introduction

- In this study, a new design numerical solver is proposed for the solutions of fluid mechanics problem, unipolar EHD model, with help of hybridization of supervised and unsupervised methods, i.e., global search technique sine–cosine algorithm (SCA) and local search technique sequential quadratic programming.
- The precision of the proposed mechanism is analyzed by comparison of solution with Runge–Kutta order four (RK4) technique for each case of the model.
- Reliability, convergence, and validity of the proposed methodology, ANN-SCA-SQP algorithm, assessed through statistical analysis. Interpreted numerically by utilizing mean square error, error in Nash–Sutcliffe efficiency and root-mean-square error. Graphically analyze using convergence plots such as histogram with normal distribution and box plots.
- The designed methodology provides accurate, reliable, valid and robust solutions with defined input grids and promising convergence.

## 2. Dynamic Model of UP-EHD Pump Flow

## 3. Proposed Scheme

#### 3.1. Mathematical Modelling for UP-EHD

#### 3.2. Optimization Procedure

#### 3.3. Performance Operators

## 4. Empirical Simulation

#### 4.1. Problem 1: Base on Variation of Electric Slip ${E}_{sl}$

#### 4.2. Problem 2: Based on the Variation of Reynolds Number ${R}_{{e}_{E}}$

#### 4.3. Problem 3: Based on Variation of Electric Source Number ${E}_{s}$

#### 4.4. Complexity Analysis

#### 4.5. Evaluation

#### 4.6. Analysis Based on Multiple Runs

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MIN | Minimum |

MAX | Maximum |

DC | Direct Current |

E | Electric Field |

$\rho $ | Charge Density |

$\varphi $ | Electric Potential |

STD | Standard Devaition |

SCA | Sine–Cosine Algorithm |

RMSE | Root-Mean-Square Error |

MAD | Mean Absolute Deviation |

ANN | Artificial Neural Network |

UP-EHD | Unipolar Electrohydrodynamic |

ENSE | Error in Nash–Sutcliffe Efficiency |

RK4 | Runge–Kutta order four technique |

SQP | Sequential Quadratic Programming |

## Appendix A

## References

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**Figure 2.**Work flow chartof proposed methodology. Initially, population in SCA is set for generation of solutions, fitness of generated solution is evaluated by SCA. The fittest solution is provided as an initial point to SQP and SQP provides the best solution as weights of ANN.

**Figure 3.**The architecture of ANN model. With input points. It’s the simplest form of ANN. A unidirectional network with no cycle has three layers input, hidden, and output layers. In inputs, $\eta \in [0,1]$ is taken for an initial guess of unknown weights. For the hidden layer, sigmoid function is used as given in Equation (12).

**Figure 5.**Problem 1: (

**a**). Charge density ($\rho $)-Axial coordinate ($\eta $) graph for all cases, (

**b**). Electric field (E)-Axial coordinate ($\eta $) graph for all cases, (

**c**). Electric potential ($\varphi $)-Axial coordinate ($\eta $) graph for all cases, (

**d**). Weights of case 1, (

**e**). Weights of case 2, (

**f**). Weights of case 3, (

**g**). Weights of case 4.

**Figure 6.**Problem 2: (

**a**). Charge density ($\rho $)-Axial coordinate ($\eta $) graph for all cases, (

**b**). Electric field (E)-Axial coordinate ($\eta $) graph for all cases, (

**c**). Electric potential ($\varphi $)-Axial coordinate ($\eta $) graph for all cases, (

**d**). Weights of case 1, (

**e**). Weights of case 2, (

**f**). Weights of case 3, (

**g**). Weights of case 4.

**Figure 7.**Problem 3: (

**a**). Charge density ($\rho $)-Axial coordinate ($\eta $) graph for all cases, (

**b**). Electric field (E)-Axial coordinate ($\eta $) graph for all cases, (

**c**). Electric potential ($\varphi $)-Axial coordinate ($\eta $) graph for all cases, (

**d**). Weights of case 1, (

**e**). Weights of case 2, (

**f**). Weights of case 3, (

**g**). Weights of case 4.

**Figure 8.**(

**a**). Fitness of problem 1 all cases for $\rho $, (

**b**). Fitness of problem 1 all cases for E, (

**c**). Fitness of problem 1 all cases for $\varphi $, (

**d**). Fitness of problem 2 all cases for $\rho $, (

**e**). Fitness of problem 2 all cases for E, (

**f**). Fitness of problem 2 all cases for $\varphi $, (

**g**). Fitness of problem 3 all cases for $\rho $, (

**h**). Fitness of problem 3 all cases for E, (

**i**). Fitness of problem 3 all cases for $\varphi $.

**Figure 9.**(

**a**). MAD of problem 1, all cases, (

**b**). MAD of problem 2, all cases, (

**c**). MAD of problem 3, all cases, (

**d**). RMSE of problem 1, all cases, (

**e**). RMSE of problem 2, all cases, (

**f**). RMSE of problem 3, all cases, (

**g**). ENSE of problem 1, all cases, (

**h**). ENSE of problem 2, all cases, (

**i**). ENSE of problem 3, all cases.

**Figure 10.**(

**a**). MAD of $\rho $, for problem 1 all cases, (

**b**). MAD of E, for problem 1 all cases, (

**c**). MAD of $\varphi $, for problem 1 all cases, (

**d**). RMSE of $\rho $, for problem 1 all cases, (

**e**). RMSE of E, for problem 1 all cases, (

**f**). RMSE of $\varphi $, for problem 1 all cases, (

**g**). ENSE of $\rho $, for problem 1 all cases, (

**h**). ENSE of E, for problem 1 all cases, (

**i**). ENSE of $\varphi $, for problem 1 all cases.

**Figure 11.**(

**a**). MAD of $\rho $, for problem 2 all cases, (

**b**). MAD of E, for problem 2 all cases, (

**c**). MAD of $\varphi $, for problem 2all cases, (

**d**). RMSE of $\rho $, for problem 2 all cases, (

**e**). RMSE of E, for problem 2 all cases, (

**f**). RMSE of $\varphi $, for problem 2 all cases, (

**g**). ENSE of $\rho $, for problem 2 all cases, (

**h**). ENSE of E, for problem 2 all cases, (

**i**). ENSE of $\varphi $, for problem 2 all cases.

**Figure 12.**(

**a**). MAD of $\rho $, for problem 3 all cases, (

**b**). MAD of E, for problem 3 all cases, (

**c**). MAD of $\varphi $, for problem 3, all cases, (

**d**). RMSE of $\rho $, for problem 3 all cases, (

**e**). RMSE of E, for problem 3 all cases, (

**f**). RMSE of $\varphi $, for problem 3 all cases, (

**g**). ENSE of $\rho $, for problem 3 all cases, (

**h**). ENSE of E, for problem 3 all cases, (

**i**). ENSE of $\varphi $, for problem 3 all cases.

**Figure 13.**(

**a**). Fitness of $\rho $, problem 1 all cases, (

**b**). Fitness of E, problem 2 case 1, (

**c**). Fitness of E, problem 2 case 2, (

**d**). Fitness of E, problem 2 case 3, (

**e**). Fitness of E, problem 2 case 4, (

**f**). Fitness of $\varphi $, problem 3 all cases.

**Table 1.**Absolute errors of problem 1 for different inputs in terms of minimum, maximum, mean and standard deviation.

Case | Mode | Absolute Errors for Inputs “$\mathit{\eta}$” | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\eta}$ = 0 | $\mathit{\eta}$ = 0.1 | $\mathit{\eta}$ = 0.2 | $\mathit{\eta}$ = 0.3 | $\mathit{\eta}$ = 0.4 | $\mathit{\eta}$ = 0.5 | $\mathit{\eta}$ = 0.6 | $\mathit{\eta}$ = 0.7 | $\mathit{\eta}$ = 0.8 | $\mathit{\eta}$ = 0.9 | $\mathit{\eta}$ = 1.0 | |||

$\rho $ | 1 | MIN | 5.30 × $10{}^{-7}$ | 1.58 × $10{}^{-7}$ | 1.41 × $10{}^{-7}$ | 1.40 × $10{}^{-7}$ | 1.54 × $10{}^{-7}$ | 1.71 × $10{}^{-7}$ | 1.36 × $10{}^{-7}$ | 1.20 × $10{}^{-7}$ | 1.28 × $10{}^{-7}$ | 1.51 × $10{}^{-7}$ | 3.57 × $10{}^{-7}$ |

MAX | 3.42 × $10{}^{-6}$ | 8.17 × $10{}^{-7}$ | 9.06 × $10{}^{-7}$ | 9.05 × $10{}^{-7}$ | 9.01 × $10{}^{-7}$ | 1.11 × $10{}^{-6}$ | 9.44 × $10{}^{-7}$ | 7.08 × $10{}^{-7}$ | 7.68 × $10{}^{-7}$ | 8.73 × $10{}^{-7}$ | 2.19 × $10{}^{-6}$ | ||

MEAN | 3.38 × $10{}^{-7}$ | 1.80 × $10{}^{-7}$ | 9.93 × $10{}^{-8}$ | 1.29 × $10{}^{-7}$ | 1.29 × $10{}^{-7}$ | 8.91 × $10{}^{-8}$ | 1.20 × $10{}^{-7}$ | 1.08 × $10{}^{-7}$ | 8.06 × $10{}^{-8}$ | 1.46 × $10{}^{-7}$ | 2.38 × $10{}^{-7}$ | ||

STD | 5.30 × $10{}^{-7}$ | 1.58 × $10{}^{-7}$ | 1.41 × $10{}^{-7}$ | 1.40 × $10{}^{-7}$ | 1.54 × $10{}^{-7}$ | 1.71 × $10{}^{-7}$ | 1.36 × $10{}^{-7}$ | 1.20 × $10{}^{-7}$ | 1.28 × $10{}^{-7}$ | 1.51 × $10{}^{-7}$ | 3.57 × $10{}^{-7}$ | ||

2 | MIN | 4.95 × $10{}^{-9}$ | 3.29 × $10{}^{-8}$ | 4.82 × $10{}^{-9}$ | 7.00 × $10{}^{-9}$ | 9.74 × $10{}^{-9}$ | 5.55 × $10{}^{-9}$ | 6.61 × $10{}^{-10}$ | 5.95 × $10{}^{-9}$ | 7.25 × $10{}^{-9}$ | 1.11 × $10{}^{-8}$ | 1.51 × $10{}^{-8}$ | |

MAX | 2.13 × $10{}^{-6}$ | 5.76 × $10{}^{-6}$ | 1.10 × $10{}^{-6}$ | 2.21 × $10{}^{-6}$ | 1.77 × $10{}^{-6}$ | 1.33 × $10{}^{-6}$ | 7.35 × $10{}^{-7}$ | 1.55 × $10{}^{-6}$ | 1.96 × $10{}^{-6}$ | 1.08 × $10{}^{-6}$ | 2.46 × $10{}^{-6}$ | ||

MEAN | 3.38 × $10{}^{-7}$ | 1.31 × $10{}^{-6}$ | 2.58 × $10{}^{-7}$ | 3.95 × $10{}^{-7}$ | 4.72 × $10{}^{-7}$ | 2.98 × $10{}^{-7}$ | 1.56 × $10{}^{-7}$ | 3.14 × $10{}^{-7}$ | 5.28 × $10{}^{-7}$ | 2.49 × $10{}^{-7}$ | 6.89 × $10{}^{-7}$ | ||

STD | 3.78 × $10{}^{-7}$ | 1.16 × $10{}^{-6}$ | 2.57 × $10{}^{-7}$ | 3.36 × $10{}^{-7}$ | 4.53 × $10{}^{-7}$ | 2.83 × $10{}^{-7}$ | 1.24 × $10{}^{-7}$ | 3.09 × $10{}^{-7}$ | 4.52 × $10{}^{-7}$ | 2.08 × $10{}^{-7}$ | 5.93 × $10{}^{-7}$ | ||

3 | MIN | 1.76 × $10{}^{-9}$ | 4.97 × $10{}^{-8}$ | 1.10 × $10{}^{-8}$ | 1.61 × $10{}^{-7}$ | 1.74 × $10{}^{-8}$ | 4.27 × $10{}^{-8}$ | 5.28 × $10{}^{-8}$ | 2.12 × $10{}^{-9}$ | 9.66 × $10{}^{-9}$ | 1.68 × $10{}^{-8}$ | 3.05 × $10{}^{-9}$ | |

MAX | 6.52 × $10{}^{-6}$ | 2.78 × $10{}^{-5}$ | 4.27 × $10{}^{-5}$ | 1.34 × $10{}^{-5}$ | 2.25 × $10{}^{-5}$ | 9.32 × $10{}^{-6}$ | 7.61 × $10{}^{-6}$ | 1.51 × $10{}^{-5}$ | 1.41 × $10{}^{-5}$ | 4.43 × $10{}^{-6}$ | 3.00 × $10{}^{-5}$ | ||

MEAN | 8.65 × $10{}^{-7}$ | 4.04 × $10{}^{-6}$ | 2.45 × $10{}^{-6}$ | 2.08 × $10{}^{-6}$ | 1.50 × $10{}^{-6}$ | 1.08 × $10{}^{-6}$ | 1.09 × $10{}^{-6}$ | 1.02 × $10{}^{-6}$ | 1.56 × $10{}^{-6}$ | 9.93 × $10{}^{-7}$ | 2.22 × $10{}^{-6}$ | ||

STD | 1.48 × $10{}^{-6}$ | 6.12 × $10{}^{-6}$ | 5.34 × $10{}^{-6}$ | 2.47 × $10{}^{-6}$ | 2.99 × $10{}^{-6}$ | 1.30 × $10{}^{-6}$ | 1.34 × $10{}^{-6}$ | 2.36 × $10{}^{-6}$ | 2.10 × $10{}^{-6}$ | 8.63 × $10{}^{-7}$ | 3.99 × $10{}^{-6}$ | ||

4 | MIN | 1.39 × $10{}^{-9}$ | 5.35 × $10{}^{-9}$ | 1.48 × $10{}^{-7}$ | 1.17 × $10{}^{-7}$ | 4.31 × $10{}^{-8}$ | 2.11 × $10{}^{-8}$ | 8.81 × $10{}^{-8}$ | 4.67 × $10{}^{-8}$ | 1.24 × $10{}^{-7}$ | 1.63 × $10{}^{-7}$ | 2.44 × $10{}^{-7}$ | |

MAX | 1.57 × $10{}^{-4}$ | 2.20 × $10{}^{-5}$ | 9.88 × $10{}^{-5}$ | 1.12 × $10{}^{-4}$ | 4.53 × $10{}^{-5}$ | 5.21 × $10{}^{-5}$ | 3.61 × $10{}^{-5}$ | 4.65 × $10{}^{-5}$ | 5.71 × $10{}^{-5}$ | 1.94 × $10{}^{-5}$ | 6.67 × $10{}^{-5}$ | ||

MEAN | 5.58 × $10{}^{-6}$ | 1.56 × $10{}^{-5}$ | 1.34 × $10{}^{-5}$ | 1.36 × $10{}^{-5}$ | 4.55 × $10{}^{-6}$ | 7.26 × $10{}^{-6}$ | 4.74 × $10{}^{-6}$ | 4.35 × $10{}^{-6}$ | 7.49 × $10{}^{-6}$ | 3.23 × $10{}^{-6}$ | 1.05 × $10{}^{-5}$ | ||

STD | 1.96 × $10{}^{-5}$ | 2.98 × $10{}^{-5}$ | 1.81 × $10{}^{-5}$ | 1.82 × $10{}^{-5}$ | 5.74 × $10{}^{-6}$ | 9.32 × $10{}^{-6}$ | 7.07 × $10{}^{-6}$ | 7.82 × $10{}^{-6}$ | 8.91 × $10{}^{-6}$ | 3.57 × $10{}^{-6}$ | 1.40 × $10{}^{-5}$ | ||

E | 1 | MIN | 1.59 × $10{}^{-8}$ | 3.54 × $10{}^{-9}$ | 4.85 × $10{}^{-9}$ | 6.65 × $10{}^{-9}$ | 2.90 × $10{}^{-9}$ | 6.18 × $10{}^{-10}$ | 1.98 × $10{}^{-9}$ | 2.49 × $10{}^{-9}$ | 1.36 × $10{}^{-9}$ | 1.03 × $10{}^{-9}$ | 3.89 × $10{}^{-9}$ |

MAX | 3.42 × $10{}^{-6}$ | 8.17 × $10{}^{-7}$ | 9.06 × $10{}^{-7}$ | 9.05 × $10{}^{-7}$ | 9.01 × $10{}^{-7}$ | 1.11 × $10{}^{-6}$ | 9.44 × $10{}^{-7}$ | 7.08 × $10{}^{-7}$ | 7.68 × $10{}^{-7}$ | 8.73 × $10{}^{-7}$ | 2.19 × $10{}^{-6}$ | ||

MEAN | 3.38 × $10{}^{-7}$ | 1.80 × $10{}^{-7}$ | 9.93 × $10{}^{-8}$ | 1.29 × $10{}^{-7}$ | 1.29 × $10{}^{-7}$ | 8.91 × $10{}^{-8}$ | 1.20 × $10{}^{-7}$ | 1.08 × $10{}^{-7}$ | 8.06 × $10{}^{-8}$ | 1.46 × $10{}^{-7}$ | 2.38 × $10{}^{-7}$ | ||

STD | 5.30 × $10{}^{-7}$ | 1.58 × $10{}^{-7}$ | 1.41 × $10{}^{-7}$ | 1.40 × $10{}^{-7}$ | 1.54 × $10{}^{-7}$ | 1.71 × $10{}^{-7}$ | 1.36 × $10{}^{-7}$ | 1.20 × $10{}^{-7}$ | 1.28 × $10{}^{-7}$ | 1.51 × $10{}^{-7}$ | 3.57 × $10{}^{-7}$ | ||

2 | MIN | 4.95 × $10{}^{-9}$ | 3.29 × $10{}^{-8}$ | 4.82 × $10{}^{-9}$ | 7.00 × $10{}^{-9}$ | 9.74 × $10{}^{-9}$ | 5.55 × $10{}^{-9}$ | 6.61 × $10{}^{-10}$ | 5.95 × $10{}^{-9}$ | 7.25 × $10{}^{-9}$ | 1.11 × $10{}^{-8}$ | 1.51 × $10{}^{-8}$ | |

MAX | 2.13 × $10{}^{-6}$ | 5.76 × $10{}^{-6}$ | 1.10 × $10{}^{-6}$ | 2.21 × $10{}^{-6}$ | 1.77 × $10{}^{-6}$ | 1.33 × $10{}^{-6}$ | 7.35 × $10{}^{-7}$ | 1.55 × $10{}^{-6}$ | 1.96 × $10{}^{-6}$ | 1.08 × $10{}^{-6}$ | 2.46 × $10{}^{-6}$ | ||

MEAN | 3.38 × $10{}^{-7}$ | 1.31 × $10{}^{-6}$ | 2.58 × $10{}^{-7}$ | 3.95 × $10{}^{-7}$ | 4.72 × $10{}^{-7}$ | 2.98 × $10{}^{-7}$ | 1.56 × $10{}^{-7}$ | 3.14 × $10{}^{-7}$ | 5.28 × $10{}^{-7}$ | 2.49 × $10{}^{-7}$ | 6.89 × $10{}^{-7}$ | ||

STD | 3.78 × $10{}^{-7}$ | 1.16 × $10{}^{-6}$ | 2.57 × $10{}^{-7}$ | 3.36 × $10{}^{-7}$ | 4.53 × $10{}^{-7}$ | 2.83 × $10{}^{-7}$ | 1.24 × $10{}^{-7}$ | 3.09 × $10{}^{-7}$ | 4.52 × $10{}^{-7}$ | 2.08 × $10{}^{-7}$ | 5.93 × $10{}^{-7}$ | ||

3 | MIN | 1.76 × $10{}^{-9}$ | 4.97 × $10{}^{-8}$ | 1.10 × $10{}^{-8}$ | 1.61 × $10{}^{-7}$ | 1.74 × $10{}^{-8}$ | 4.27 × $10{}^{-8}$ | 5.28 × $10{}^{-8}$ | 2.12 × $10{}^{-9}$ | 9.66 × $10{}^{-9}$ | 1.68 × $10{}^{-8}$ | 3.05 × $10{}^{-9}$ | |

MAX | 6.52 × $10{}^{-6}$ | 2.78 × $10{}^{-5}$ | 4.27 × $10{}^{-5}$ | 1.34 × $10{}^{-5}$ | 2.25 × $10{}^{-5}$ | 9.32 × $10{}^{-6}$ | 7.61 × $10{}^{-6}$ | 1.51 × $10{}^{-5}$ | 1.41 × $10{}^{-5}$ | 4.43 × $10{}^{-6}$ | 3.00 × $10{}^{-5}$ | ||

MEAN | 8.65 × $10{}^{-7}$ | 4.04 × $10{}^{-6}$ | 2.45 × $10{}^{-6}$ | 2.08 × $10{}^{-6}$ | 1.50 × $10{}^{-6}$ | 1.08 × $10{}^{-6}$ | 1.09 × $10{}^{-6}$ | 1.02 × $10{}^{-6}$ | 1.56 × $10{}^{-6}$ | 9.93 × $10{}^{-7}$ | 2.22 × $10{}^{-6}$ | ||

STD | 1.48 × $10{}^{-6}$ | 6.12 × $10{}^{-6}$ | 5.34 × $10{}^{-6}$ | 2.47 × $10{}^{-6}$ | 2.99 × $10{}^{-6}$ | 1.30 × $10{}^{-6}$ | 1.34 × $10{}^{-6}$ | 2.36 × $10{}^{-6}$ | 2.10 × $10{}^{-6}$ | 8.63 × $10{}^{-7}$ | 3.99 × $10{}^{-6}$ | ||

4 | MIN | 1.39 × $10{}^{-9}$ | 5.35 × $10{}^{-9}$ | 1.48 × $10{}^{-7}$ | 1.17 × $10{}^{-7}$ | 4.31 × $10{}^{-8}$ | 2.11 × $10{}^{-8}$ | 8.81 × $10{}^{-8}$ | 4.67 × $10{}^{-8}$ | 1.24 × $10{}^{-7}$ | 1.63 × $10{}^{-7}$ | 2.44 × $10{}^{-7}$ | |

MAX | 1.57 × $10{}^{-4}$ | 2.20 × $10{}^{-4}$ | 9.88 × $10{}^{-5}$ | 1.12 × $10{}^{-4}$ | 4.53 × $10{}^{-5}$ | 5.21 × $10{}^{-5}$ | 3.61 × $10{}^{-5}$ | 4.65 × $10{}^{-5}$ | 5.71 × $10{}^{-5}$ | 1.94 × $10{}^{-5}$ | 6.67 × $10{}^{-5}$ | ||

MEAN | 5.58 × $10{}^{-6}$ | 1.56 × $10{}^{-5}$ | 1.34 × $10{}^{-5}$ | 1.36 × $10{}^{-5}$ | 4.55 × $10{}^{-6}$ | 7.26 × $10{}^{-6}$ | 4.74 × $10{}^{-6}$ | 4.35 × $10{}^{-6}$ | 7.49 × $10{}^{-6}$ | 3.23 × $10{}^{-6}$ | 1.05 × $10{}^{-5}$ | ||

STD | 1.96 × $10{}^{-5}$ | 2.98 × $10{}^{-5}$ | 1.81 × $10{}^{-5}$ | 1.82 × $10{}^{-5}$ | 5.74 × $10{}^{-6}$ | 9.32 × $10{}^{-6}$ | 7.07 × $10{}^{-6}$ | 7.82 × $10{}^{-6}$ | 8.91 × $10{}^{-6}$ | 3.57 × $10{}^{-6}$ | 1.40 × $10{}^{-5}$ | ||

$\varphi $ | 1 | MIN | 1.59 × $10{}^{-8}$ | 3.54 × $10{}^{-9}$ | 4.85 × $10{}^{-9}$ | 6.65 × $10{}^{-9}$ | 2.90 × $10{}^{-9}$ | 6.18 × $10{}^{-10}$ | 1.98 × $10{}^{-9}$ | 2.49 × $10{}^{-9}$ | 1.36 × $10{}^{-9}$ | 1.03 × $10{}^{-9}$ | 3.89 × $10{}^{-9}$ |

MAX | 3.42 × $10{}^{-6}$ | 8.17 × $10{}^{-7}$ | 9.06 × $10{}^{-7}$ | 9.05 × $10{}^{-7}$ | 9.01 × $10{}^{-7}$ | 1.11 × $10{}^{-6}$ | 9.44 × $10{}^{-7}$ | 7.08 × $10{}^{-7}$ | 7.68 × $10{}^{-7}$ | 8.73 × $10{}^{-7}$ | 2.19 × $10{}^{-6}$ | ||

MEAN | 3.38 × $10{}^{-7}$ | 1.80 × $10{}^{-7}$ | 9.93 × $10{}^{-8}$ | 1.29 × $10{}^{-7}$ | 1.29 × $10{}^{-7}$ | 8.91 × $10{}^{-8}$ | 1.20 × $10{}^{-7}$ | 1.08 × $10{}^{-7}$ | 8.06 × $10{}^{-8}$ | 1.46 × $10{}^{-7}$ | 2.38 × $10{}^{-7}$ | ||

STD | 5.30 × $10{}^{-7}$ | 1.58 × $10{}^{-7}$ | 1.41 × $10{}^{-7}$ | 1.40 × $10{}^{-7}$ | 1.54 × $10{}^{-7}$ | 1.71 × $10{}^{-7}$ | 1.36 × $10{}^{-7}$ | 1.20 × $10{}^{-7}$ | 1.28 × $10{}^{-7}$ | 1.51 × $10{}^{-7}$ | 3.57 × $10{}^{-7}$ | ||

2 | MIN | 4.95 × $10{}^{-9}$ | 3.29 × $10{}^{-8}$ | 4.82 × $10{}^{-9}$ | 7.00 × $10{}^{-9}$ | 9.74 × $10{}^{-9}$ | 5.55 × $10{}^{-9}$ | 6.61 × $10{}^{-10}$ | 5.95 × $10{}^{-9}$ | 7.25 × $10{}^{-9}$ | 1.11 × $10{}^{-8}$ | 1.51 × $10{}^{-8}$ | |

MAX | 2.13 × $10{}^{-6}$ | 5.76 × $10{}^{-6}$ | 1.10 × $10{}^{-6}$ | 2.21 × $10{}^{-6}$ | 1.77 × $10{}^{-6}$ | 1.33 × $10{}^{-6}$ | 7.35 × $10{}^{-7}$ | 1.55 × $10{}^{-6}$ | 1.96 × $10{}^{-6}$ | 1.08 × $10{}^{-6}$ | 2.46 × $10{}^{-6}$ | ||

MEAN | 3.38 × $10{}^{-7}$ | 1.31 × $10{}^{-6}$ | 2.58 × $10{}^{-7}$ | 3.95 × $10{}^{-7}$ | 4.72 × $10{}^{-7}$ | 2.98 × $10{}^{-7}$ | 1.56 × $10{}^{-7}$ | 3.14 × $10{}^{-7}$ | 5.28 × $10{}^{-7}$ | 2.49 × $10{}^{-7}$ | 6.89 × $10{}^{-7}$ | ||

STD | 3.78 × $10{}^{-7}$ | 1.16 × $10{}^{-6}$ | 2.57 × $10{}^{-7}$ | 3.36 × $10{}^{-7}$ | 4.53 × $10{}^{-7}$ | 2.83 × $10{}^{-7}$ | 1.24 × $10{}^{-7}$ | 3.09 × $10{}^{-7}$ | 4.52 × $10{}^{-7}$ | 2.08 × $10{}^{-7}$ | 5.93 × $10{}^{-7}$ | ||

3 | MIN | 1.76 × $10{}^{-9}$ | 4.97 × $10{}^{-8}$ | 1.10 × $10{}^{-8}$ | 1.61 × $10{}^{-7}$ | 1.74 × $10{}^{-8}$ | 4.27 × $10{}^{-8}$ | 5.28 × $10{}^{-8}$ | 2.12 × $10{}^{-9}$ | 9.66 × $10{}^{-9}$ | 1.68 × $10{}^{-8}$ | 3.05 × $10{}^{-9}$ | |

MAX | 6.52 × $10{}^{-6}$ | 2.78 × $10{}^{-5}$ | 4.27 × $10{}^{-5}$ | 1.34 × $10{}^{-5}$ | 2.25 × $10{}^{-5}$ | 9.32 × $10{}^{-6}$ | 7.61 × $10{}^{-6}$ | 1.51 × $10{}^{-5}$ | 1.41 × $10{}^{-5}$ | 4.43 × $10{}^{-6}$ | 3.00 × $10{}^{-5}$ | ||

MEAN | 8.65 × $10{}^{-7}$ | 4.04 × $10{}^{-6}$ | 2.45 × $10{}^{-6}$ | 2.08 × $10{}^{-6}$ | 1.50 × $10{}^{-6}$ | 1.08 × $10{}^{-6}$ | 1.09 × $10{}^{-6}$ | 1.02 × $10{}^{-6}$ | 1.56 × $10{}^{-6}$ | 9.93 × $10{}^{-7}$ | 2.22 × $10{}^{-6}$ | ||

STD | 1.48 × $10{}^{-6}$ | 6.12 × $10{}^{-6}$ | 5.34 × $10{}^{-6}$ | 2.47 × $10{}^{-6}$ | 2.99 × $10{}^{-6}$ | 1.30 × $10{}^{-6}$ | 1.34 × $10{}^{-6}$ | 2.36 × $10{}^{-6}$ | 2.10 × $10{}^{-6}$ | 8.63 × $10{}^{-7}$ | 3.99 × $10{}^{-6}$ | ||

4 | MIN | 1.39 × $10{}^{-9}$ | 5.35 × $10{}^{-9}$ | 1.48 × $10{}^{-7}$ | 1.17 × $10{}^{-7}$ | 4.31 × $10{}^{-8}$ | 2.11 × $10{}^{-8}$ | 8.81 × $10{}^{-8}$ | 4.67 × $10{}^{-8}$ | 1.24 × $10{}^{-7}$ | 1.63 × $10{}^{-7}$ | 2.44 × $10{}^{-7}$ | |

MAX | 1.57 × $10{}^{-4}$ | 2.20 × $10{}^{-4}$ | 9.88 × $10{}^{-5}$ | 1.12 × $10{}^{-4}$ | 4.53 × $10{}^{-5}$ | 5.21 × $10{}^{-5}$ | 3.61 × $10{}^{-5}$ | 4.65 × $10{}^{-5}$ | 5.71 × $10{}^{-5}$ | 1.94 × $10{}^{-5}$ | 6.67 × $10{}^{-5}$ | ||

MEAN | 5.58 × $10{}^{-6}$ | 1.56 × $10{}^{-5}$ | 1.34 × $10{}^{-5}$ | 1.36 × $10{}^{-5}$ | 4.55 × $10{}^{-6}$ | 7.26 × $10{}^{-6}$ | 4.74 × $10{}^{-6}$ | 4.35 × $10{}^{-6}$ | 7.49 × $10{}^{-6}$ | 3.23 × $10{}^{-6}$ | 1.05 × $10{}^{-5}$ | ||

STD | 1.96 × $10{}^{-5}$ | 2.98 × $10{}^{-5}$ | 1.81 × $10{}^{-5}$ | 1.82 × $10{}^{-5}$ | 5.74 × $10{}^{-6}$ | 9.32 × $10{}^{-6}$ | 7.07 × $10{}^{-6}$ | 7.82 × $10{}^{-6}$ | 8.91 × $10{}^{-6}$ | 3.57 × $10{}^{-6}$ | 1.40 × $10{}^{-5}$ |

**Table 2.**Absolute errors of problem 2 for different inputs in terms of minimum, maximum, mean and standard deviation.

Case | Mode | Absolute Errors for Inputs “$\mathit{\eta}$” | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\eta}$ = 0 | $\mathit{\eta}$ = 0.1 | $\mathit{\eta}$ = 0.2 | $\mathit{\eta}$ = 0.3 | $\mathit{\eta}$ = 0.4 | $\mathit{\eta}$ = 0.5 | $\mathit{\eta}$ = 0.6 | $\mathit{\eta}$ = 0.7 | $\mathit{\eta}$ = 0.8 | $\mathit{\eta}$ = 0.9 | $\mathit{\eta}$ = 1.0 | |||

$\rho $ | 1 | MIN | 5.920 × $10{}^{-9}$ | 8.090 × $10{}^{-8}$ | 8.020 × $10{}^{-9}$ | 5.010 × $10{}^{-9}$ | 5.680 × $10{}^{-9}$ | 1.040 × $10{}^{-8}$ | 2.340 × $10{}^{-9}$ | 1.620 × $10{}^{-9}$ | 1.430 × $10{}^{-8}$ | 1.860 × $10{}^{-8}$ | 2.630 × $10{}^{-8}$ |

MAX | 2.700 × $10{}^{-6}$ | 1.550 × $10{}^{-5}$ | 7.040 × $10{}^{-6}$ | 5.210 × $10{}^{-6}$ | 5.370 × $10{}^{-6}$ | 4.250 × $10{}^{-6}$ | 1.950 × $10{}^{-6}$ | 4.310 × $10{}^{-6}$ | 5.920 × $10{}^{-6}$ | 2.520 × $10{}^{-6}$ | 8.060 × $10{}^{-6}$ | ||

MEAN | 2.760 × $10{}^{-7}$ | 1.440 × $10{}^{-6}$ | 6.470 × $10{}^{-7}$ | 4.990 × $10{}^{-7}$ | 5.100 × $10{}^{-7}$ | 4.120 × $10{}^{-7}$ | 2.510 × $10{}^{-7}$ | 3.730 × $10{}^{-7}$ | 6.230 × $10{}^{-7}$ | 3.030 × $10{}^{-7}$ | 8.630 × $10{}^{-7}$ | ||

STD | 4.930 × $10{}^{-7}$ | 2.350 × $10{}^{-6}$ | 1.080 × $10{}^{-6}$ | 8.260 × $10{}^{-7}$ | 8.540 × $10{}^{-7}$ | 6.980 × $10{}^{-7}$ | 3.250 × $10{}^{-7}$ | 7.370 × $10{}^{-7}$ | 1.020 × $10{}^{-6}$ | 3.760 × $10{}^{-7}$ | 1.470 × $10{}^{-6}$ | ||

2 | MIN | 5.500 × $10{}^{-10}$ | 1.040 × $10{}^{-8}$ | 8.840 × $10{}^{-9}$ | 7.560 × $10{}^{-9}$ | 4.840 × $10{}^{-9}$ | 4.830 × $10{}^{-9}$ | 9.580 × $10{}^{-9}$ | 1.200 × $10{}^{-9}$ | 9.190 × $10{}^{-9}$ | 1.890 × $10{}^{-9}$ | 1.650 × $10{}^{-8}$ | |

MAX | 1.410 × $10{}^{-5}$ | 5.000 × $10{}^{-5}$ | 1.440 × $10{}^{-6}$ | 1.450 × $10{}^{-5}$ | 1.870 × $10{}^{-5}$ | 3.040 × $10{}^{-6}$ | 2.720 × $10{}^{-6}$ | 1.480 × $10{}^{-5}$ | 1.380 × $10{}^{-5}$ | 1.070 × $10{}^{-6}$ | 2.450 × $10{}^{-5}$ | ||

MEAN | 4.360 × $10{}^{-7}$ | 1.750 × $10{}^{-6}$ | 2.560 × $10{}^{-7}$ | 5.440 × $10{}^{-7}$ | 6.300 × $10{}^{-7}$ | 2.730 × $10{}^{-7}$ | 1.800 × $10{}^{-7}$ | 4.550 × $10{}^{-7}$ | 5.900 × $10{}^{-7}$ | 2.050 × $10{}^{-7}$ | 8.590 × $10{}^{-7}$ | ||

STD | 1.420 × $10{}^{-6}$ | 5.020 × $10{}^{-6}$ | 2.800 × $10{}^{-7}$ | 1.440 × $10{}^{-6}$ | 1.890 × $10{}^{-6}$ | 3.730 × $10{}^{-7}$ | 2.790 × $10{}^{-7}$ | 1.480 × $10{}^{-6}$ | 1.410 × $10{}^{-6}$ | 1.640 × $10{}^{-7}$ | 2.460 × $10{}^{-6}$ | ||

3 | MIN | 6.710 × $10{}^{-9}$ | 3.990 × $10{}^{-8}$ | 1.180 × $10{}^{-9}$ | 2.350 × $10{}^{-8}$ | 1.360 × $10{}^{-8}$ | 1.040 × $10{}^{-8}$ | 3.080 × $10{}^{-9}$ | 3.450 × $10{}^{-9}$ | 1.860 × $10{}^{-8}$ | 2.720 × $10{}^{-8}$ | 3.100 × $10{}^{-8}$ | |

MAX | 1.930 × $10{}^{-6}$ | 6.980 × $10{}^{-6}$ | 1.840 × $10{}^{-6}$ | 3.280 × $10{}^{-6}$ | 2.220 × $10{}^{-6}$ | 1.940 × $10{}^{-6}$ | 1.350 × $10{}^{-6}$ | 2.170 × $10{}^{-6}$ | 2.720 × $10{}^{-6}$ | 9.430 × $10{}^{-7}$ | 4.260 × $10{}^{-6}$ | ||

MEAN | 3.840 × $10{}^{-7}$ | 1.390 × $10{}^{-6}$ | 3.160 × $10{}^{-7}$ | 4.100 × $10{}^{-7}$ | 5.050 × $10{}^{-7}$ | 3.460 × $10{}^{-7}$ | 1.750 × $10{}^{-7}$ | 3.130 × $10{}^{-7}$ | 5.560 × $10{}^{-7}$ | 2.900 × $10{}^{-7}$ | 7.020 × $10{}^{-7}$ | ||

STD | 3.520 × $10{}^{-7}$ | 1.060 × $10{}^{-6}$ | 3.500 × $10{}^{-7}$ | 3.660 × $10{}^{-7}$ | 4.300 × $10{}^{-7}$ | 3.310 × $10{}^{-7}$ | 1.630 × $10{}^{-7}$ | 2.880 × $10{}^{-7}$ | 4.810 × $10{}^{-7}$ | 2.140 × $10{}^{-7}$ | 6.130 × $10{}^{-7}$ | ||

4 | MIN | 4.950 × $10{}^{-9}$ | 3.290 × $10{}^{-8}$ | 4.820 × $10{}^{-9}$ | 7.000 × $10{}^{-9}$ | 9.740 × $10{}^{-9}$ | 5.550 × $10{}^{-9}$ | 6.610 × $10{}^{-10}$ | 5.950 × $10{}^{-9}$ | 7.250 × $10{}^{-9}$ | 1.110 × $10{}^{-8}$ | 1.510 × $10{}^{-8}$ | |

MAX | 2.130 × $10{}^{-6}$ | 5.760 × $10{}^{-6}$ | 1.100 × $10{}^{-6}$ | 2.210 × $10{}^{-6}$ | 1.770 × $10{}^{-6}$ | 1.330 × $10{}^{-6}$ | 7.350 × $10{}^{-7}$ | 1.550 × $10{}^{-6}$ | 1.960 × $10{}^{-6}$ | 1.080 × $10{}^{-6}$ | 2.460 × $10{}^{-6}$ | ||

MEAN | 3.380 × $10{}^{-7}$ | 1.310 × $10{}^{-6}$ | 2.580 × $10{}^{-7}$ | 3.950 × $10{}^{-7}$ | 4.720 × $10{}^{-7}$ | 2.980 × $10{}^{-7}$ | 1.560 × $10{}^{-7}$ | 3.140 × $10{}^{-7}$ | 5.280 × $10{}^{-7}$ | 2.490 × $10{}^{-7}$ | 6.890 × $10{}^{-7}$ | ||

STD | 3.780 × $10{}^{-7}$ | 1.160 × $10{}^{-6}$ | 2.570 × $10{}^{-7}$ | 3.360 × $10{}^{-7}$ | 4.530 × $10{}^{-7}$ | 2.830 × $10{}^{-7}$ | 1.240 × $10{}^{-7}$ | 3.090 × $10{}^{-7}$ | 4.520 × $10{}^{-7}$ | 2.080 × $10{}^{-7}$ | 5.930 × $10{}^{-7}$ | ||

E | 1 | MIN | 5.920 × $10{}^{-9}$ | 8.090 × $10{}^{-8}$ | 8.020 × $10{}^{-9}$ | 5.010 × $10{}^{-9}$ | 5.680 × $10{}^{-9}$ | 1.040 × $10{}^{-8}$ | 2.340 × $10{}^{-9}$ | 1.620 × $10{}^{-9}$ | 1.430 × $10{}^{-8}$ | 1.860 × $10{}^{-8}$ | 2.630 × $10{}^{-8}$ |

MAX | 2.700 × $10{}^{-6}$ | 1.550 × $10{}^{-5}$ | 7.040 × $10{}^{-6}$ | 5.210 × $10{}^{-6}$ | 5.370 × $10{}^{-6}$ | 4.250 × $10{}^{-6}$ | 1.950 × $10{}^{-6}$ | 4.310 × $10{}^{-6}$ | 5.920 × $10{}^{-6}$ | 2.520 × $10{}^{-6}$ | 8.060 × $10{}^{-6}$ | ||

MEAN | 2.760 × $10{}^{-7}$ | 1.440 × $10{}^{-6}$ | 6.470 × $10{}^{-7}$ | 4.990 × $10{}^{-7}$ | 5.100 × $10{}^{-7}$ | 4.120 × $10{}^{-7}$ | 2.510 × $10{}^{-7}$ | 3.730 × $10{}^{-7}$ | 6.230 × $10{}^{-7}$ | 3.030 × $10{}^{-7}$ | 8.630 × $10{}^{-7}$ | ||

STD | 4.930 × $10{}^{-7}$ | 2.350 × $10{}^{-6}$ | 1.080 × $10{}^{-6}$ | 8.260 × $10{}^{-7}$ | 8.540 × $10{}^{-7}$ | 6.980 × $10{}^{-7}$ | 3.250 × $10{}^{-7}$ | 7.370 × $10{}^{-7}$ | 1.020 × $10{}^{-6}$ | 3.760 × $10{}^{-7}$ | 1.470 × $10{}^{-6}$ | ||

2 | MIN | 5.500 × $10{}^{-10}$ | 1.040 × $10{}^{-8}$ | 8.840 × $10{}^{-9}$ | 7.560 × $10{}^{-9}$ | 4.840 × $10{}^{-9}$ | 4.830 × $10{}^{-9}$ | 9.580 × $10{}^{-9}$ | 1.200 × $10{}^{-9}$ | 9.190 × $10{}^{-9}$ | 1.890 × $10{}^{-9}$ | 1.650 × $10{}^{-8}$ | |

MAX | 1.410 × $10{}^{-5}$ | 5.000 × $10{}^{-5}$ | 1.440 × $10{}^{-6}$ | 1.450 × $10{}^{-5}$ | 1.870 × $10{}^{-5}$ | 3.040 × $10{}^{-6}$ | 2.720 × $10{}^{-6}$ | 1.480 × $10{}^{-5}$ | 1.380 × $10{}^{-5}$ | 1.070 × $10{}^{-6}$ | 2.450 × $10{}^{-5}$ | ||

MEAN | 4.360 × $10{}^{-7}$ | 1.750 × $10{}^{-6}$ | 2.560 × $10{}^{-7}$ | 5.440 × $10{}^{-7}$ | 6.300 × $10{}^{-7}$ | 2.730 × $10{}^{-7}$ | 1.800 × $10{}^{-7}$ | 4.550 × $10{}^{-7}$ | 5.900 × $10{}^{-7}$ | 2.050 × $10{}^{-7}$ | 8.590 × $10{}^{-7}$ | ||

STD | 1.420 × $10{}^{-6}$ | 5.020 × $10{}^{-6}$ | 2.800 × $10{}^{-7}$ | 1.440 × $10{}^{-6}$ | 1.890 × $10{}^{-6}$ | 3.730 × $10{}^{-7}$ | 2.790 × $10{}^{-7}$ | 1.480 × $10{}^{-6}$ | 1.410 × $10{}^{-6}$ | 1.640 × $10{}^{-7}$ | 2.460 × $10{}^{-6}$ | ||

3 | MIN | 6.710 × $10{}^{-9}$ | 3.990 × $10{}^{-8}$ | 1.180 × $10{}^{-9}$ | 2.350 × $10{}^{-8}$ | 1.360 × $10{}^{-8}$ | 1.040 × $10{}^{-8}$ | 3.080 × $10{}^{-9}$ | 3.450 × $10{}^{-9}$ | 1.860 × $10{}^{-8}$ | 2.720 × $10{}^{-8}$ | 3.100 × $10{}^{-8}$ | |

MAX | 1.930 × $10{}^{-6}$ | 6.980 × $10{}^{-6}$ | 1.840 × $10{}^{-6}$ | 3.280 × $10{}^{-6}$ | 2.220 × $10{}^{-6}$ | 1.940 × $10{}^{-6}$ | 1.350 × $10{}^{-6}$ | 2.170 × $10{}^{-6}$ | 2.720 × $10{}^{-6}$ | 9.430 × $10{}^{-7}$ | 4.260 × $10{}^{-6}$ | ||

MEAN | 3.840 × $10{}^{-7}$ | 1.390 × $10{}^{-6}$ | 3.160 × $10{}^{-7}$ | 4.100 × $10{}^{-7}$ | 5.050 × $10{}^{-7}$ | 3.460 × $10{}^{-7}$ | 1.750 × $10{}^{-7}$ | 3.130 × $10{}^{-7}$ | 5.560 × $10{}^{-7}$ | 2.900 × $10{}^{-7}$ | 7.020 × $10{}^{-7}$ | ||

STD | 3.520 × $10{}^{-7}$ | 1.060 × $10{}^{-6}$ | 3.500 × $10{}^{-7}$ | 3.660 × $10{}^{-7}$ | 4.300 × $10{}^{-7}$ | 3.310 × $10{}^{-7}$ | 1.630 × $10{}^{-7}$ | 2.880 × $10{}^{-7}$ | 4.810 × $10{}^{-7}$ | 2.140 × $10{}^{-7}$ | 6.130 × $10{}^{-7}$ | ||

4 | MIN | 4.950 × $10{}^{-9}$ | 3.290 × $10{}^{-8}$ | 4.820 × $10{}^{-9}$ | 7.000 × $10{}^{-9}$ | 9.740 × $10{}^{-9}$ | 5.550 × $10{}^{-9}$ | 6.610 × $10{}^{-10}$ | 5.950 × $10{}^{-9}$ | 7.250 × $10{}^{-9}$ | 1.110 × $10{}^{-8}$ | 1.510 × $10{}^{-8}$ | |

MAX | 2.130 × $10{}^{-6}$ | 5.760 × $10{}^{-6}$ | 1.100 × $10{}^{-6}$ | 2.210 × $10{}^{-6}$ | 1.770 × $10{}^{-6}$ | 1.330 × $10{}^{-6}$ | 7.350 × $10{}^{-7}$ | 1.550 × $10{}^{-6}$ | 1.960 × $10{}^{-6}$ | 1.080 × $10{}^{-6}$ | 2.460 × $10{}^{-6}$ | ||

MEAN | 3.380 × $10{}^{-7}$ | 1.310 × $10{}^{-6}$ | 2.580 × $10{}^{-7}$ | 3.950 × $10{}^{-7}$ | 4.720 × $10{}^{-7}$ | 2.980 × $10{}^{-7}$ | 1.560 × $10{}^{-7}$ | 3.140 × $10{}^{-7}$ | 5.280 × $10{}^{-7}$ | 2.490 × $10{}^{-7}$ | 6.890 × $10{}^{-7}$ | ||

STD | 3.780 × $10{}^{-7}$ | 1.160 × $10{}^{-6}$ | 2.570 × $10{}^{-7}$ | 3.360 × $10{}^{-7}$ | 4.530 × $10{}^{-7}$ | 2.830 × $10{}^{-7}$ | 1.240 × $10{}^{-7}$ | 3.090 × $10{}^{-7}$ | 4.520 × $10{}^{-7}$ | 2.080 × $10{}^{-7}$ | 5.930 × $10{}^{-7}$ | ||

$\varphi $ | 1 | MIN | 5.920 × $10{}^{-9}$ | 8.090 × $10{}^{-8}$ | 8.020 × $10{}^{-9}$ | 5.010 × $10{}^{-9}$ | 5.680 × $10{}^{-9}$ | 1.040 × $10{}^{-8}$ | 2.340 × $10{}^{-9}$ | 1.620 × $10{}^{-9}$ | 1.430 × $10{}^{-8}$ | 1.860 × $10{}^{-8}$ | 2.630 × $10{}^{-8}$ |

MAX | 2.700 × $10{}^{-6}$ | 1.550 × $10{}^{-5}$ | 7.040 × $10{}^{-6}$ | 5.210 × $10{}^{-6}$ | 5.370 × $10{}^{-6}$ | 4.250 × $10{}^{-6}$ | 1.950 × $10{}^{-6}$ | 4.310 × $10{}^{-6}$ | 5.920 × $10{}^{-6}$ | 2.520 × $10{}^{-6}$ | 8.060 × $10{}^{-6}$ | ||

MEAN | 2.760 × $10{}^{-7}$ | 1.440 × $10{}^{-6}$ | 6.470 × $10{}^{-7}$ | 4.990 × $10{}^{-7}$ | 5.100 × $10{}^{-7}$ | 4.120 × $10{}^{-7}$ | 2.510 × $10{}^{-7}$ | 3.730 × $10{}^{-7}$ | 6.230 × $10{}^{-7}$ | 3.030 × $10{}^{-7}$ | 8.630 × $10{}^{-7}$ | ||

STD | 4.930 × $10{}^{-7}$ | 2.350 × $10{}^{-6}$ | 1.080 × $10{}^{-6}$ | 8.260 × $10{}^{-7}$ | 8.540 × $10{}^{-7}$ | 6.980 × $10{}^{-7}$ | 3.250 × $10{}^{-7}$ | 7.370 × $10{}^{-7}$ | 1.020 × $10{}^{-6}$ | 3.760 × $10{}^{-7}$ | 1.470 × $10{}^{-6}$ | ||

2 | MIN | 5.500 × $10{}^{-10}$ | 1.040 × $10{}^{-8}$ | 8.840 × $10{}^{-9}$ | 7.560 × $10{}^{-9}$ | 4.840 × $10{}^{-9}$ | 4.830 × $10{}^{-9}$ | 9.580 × $10{}^{-9}$ | 1.200 × $10{}^{-9}$ | 9.190 × $10{}^{-9}$ | 1.890 × $10{}^{-9}$ | 1.650 × $10{}^{-8}$ | |

MAX | 1.410 × $10{}^{-5}$ | 5.000 × $10{}^{-5}$ | 1.440 × $10{}^{-6}$ | 1.450 × $10{}^{-5}$ | 1.870 × $10{}^{-5}$ | 3.040 × $10{}^{-6}$ | 2.720 × $10{}^{-6}$ | 1.480 × $10{}^{-5}$ | 1.380 × $10{}^{-5}$ | 1.070 × $10{}^{-6}$ | 2.450 × $10{}^{-5}$ | ||

MEAN | 4.360 × $10{}^{-7}$ | 1.750 × $10{}^{-6}$ | 2.560 × $10{}^{-7}$ | 5.440 × $10{}^{-7}$ | 6.300 × $10{}^{-7}$ | 2.730 × $10{}^{-7}$ | 1.800 × $10{}^{-7}$ | 4.550 × $10{}^{-7}$ | 5.900 × $10{}^{-7}$ | 2.050 × $10{}^{-7}$ | 8.590 × $10{}^{-7}$ | ||

STD | 1.420 × $10{}^{-6}$ | 5.020 × $10{}^{-6}$ | 2.800 × $10{}^{-7}$ | 1.440 × $10{}^{-6}$ | 1.890 × $10{}^{-6}$ | 3.730 × $10{}^{-7}$ | 2.790 × $10{}^{-7}$ | 1.480 × $10{}^{-6}$ | 1.410 × $10{}^{-6}$ | 1.640 × $10{}^{-7}$ | 2.460 × $10{}^{-6}$ | ||

3 | MIN | 6.710 × $10{}^{-9}$ | 3.990 × $10{}^{-8}$ | 1.180 × $10{}^{-9}$ | 2.350 × $10{}^{-8}$ | 1.360 × $10{}^{-8}$ | 1.040 × $10{}^{-8}$ | 3.08 × $10{}^{-9}$ | 3.450 × $10{}^{-9}$ | 1.860 × $10{}^{-8}$ | 2.720 × $10{}^{-8}$ | 3.100 × $10{}^{-8}$ | |

MAX | 1.930 × $10{}^{-6}$ | 6.980 × $10{}^{-6}$ | 1.840 × $10{}^{-6}$ | 3.280 × $10{}^{-6}$ | 2.220 × $10{}^{-6}$ | 1.940 × $10{}^{-6}$ | 1.350 × $10{}^{-6}$ | 2.170 × $10{}^{-6}$ | 2.720 × $10{}^{-6}$ | 9.430 × $10{}^{-7}$ | 4.260 × $10{}^{-6}$ | ||

MEAN | 3.840 × $10{}^{-7}$ | 1.390 × $10{}^{-6}$ | 3.160 × $10{}^{-7}$ | 4.100 × $10{}^{-7}$ | 5.050 × $10{}^{-7}$ | 3.460 × $10{}^{-7}$ | 1.750 × $10{}^{-7}$ | 3.130 × $10{}^{-7}$ | 5.560 × $10{}^{-7}$ | 2.900 × $10{}^{-7}$ | 7.020 × $10{}^{-7}$ | ||

STD | 3.520 × $10{}^{-7}$ | 1.060 × $10{}^{-6}$ | 3.500 × $10{}^{-7}$ | 3.660 × $10{}^{-7}$ | 4.300 × $10{}^{-7}$ | 3.310 × $10{}^{-7}$ | 1.630 × $10{}^{-7}$ | 2.880 × $10{}^{-7}$ | 4.810 × $10{}^{-7}$ | 2.140 × $10{}^{-7}$ | 6.130 × $10{}^{-7}$ | ||

4 | MIN | 4.950 × $10{}^{-9}$ | 3.290 × $10{}^{-8}$ | 4.820 × $10{}^{-9}$ | 7.000 × $10{}^{-9}$ | 9.740 × $10{}^{-9}$ | 5.550 × $10{}^{-9}$ | 6.610 × $10{}^{-10}$ | 5.950 × $10{}^{-9}$ | 7.250 × $10{}^{-9}$ | 1.110 × $10{}^{-8}$ | 1.510 × $10{}^{-8}$ | |

MAX | 2.130 × $10{}^{-6}$ | 5.760 × $10{}^{-6}$ | 1.100 × $10{}^{-6}$ | 2.210 × $10{}^{-6}$ | 1.770 × $10{}^{-6}$ | 1.330 × $10{}^{-6}$ | 7.350 × $10{}^{-7}$ | 1.550 × $10{}^{-6}$ | 1.960 × $10{}^{-6}$ | 1.080 × $10{}^{-6}$ | 2.460 × $10{}^{-6}$ | ||

MEAN | 3.380 × $10{}^{-7}$ | 1.310 × $10{}^{-6}$ | 2.580 × $10{}^{-7}$ | 3.950 × $10{}^{-7}$ | 4.720 × $10{}^{-7}$ | 2.980 × $10{}^{-7}$ | 1.560 × $10{}^{-7}$ | 3.140 × $10{}^{-7}$ | 5.280 × $10{}^{-7}$ | 2.490 × $10{}^{-7}$ | 6.890 × $10{}^{-7}$ | ||

STD | 3.780 × $10{}^{-7}$ | 1.160 × $10{}^{-6}$ | 2.570 × $10{}^{-7}$ | 3.360 × $10{}^{-7}$ | 4.530 × $10{}^{-7}$ | 2.830 × $10{}^{-7}$ | 1.240 × $10{}^{-7}$ | 3.090 × $10{}^{-7}$ | 4.520 × $10{}^{-7}$ | 2.080 × $10{}^{-7}$ | 5.930 × $10{}^{-7}$ |

**Table 3.**Absolute errors of problem 3 for different inputs in terms of minimum, maximum, mean and standard deviation.

Case | Mode | Absolute Errors for Inputs “$\mathit{\eta}$” | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\eta}$ = 0 | $\mathit{\eta}$ = 0.1 | $\mathit{\eta}$ = 0.2 | $\mathit{\eta}$ = 0.3 | $\mathit{\eta}$ = 0.4 | $\mathit{\eta}$ = 0.5 | $\mathit{\eta}$ = 0.6 | $\mathit{\eta}$ = 0.7 | $\mathit{\eta}$ = 0.8 | $\mathit{\eta}$ = 0.9 | $\mathit{\eta}$ = 1.0 | |||

$\rho $ | 1 | MIN | 6.710 × $10{}^{-9}$ | 3.990 × $10{}^{-8}$ | 1.180 × $10{}^{-9}$ | 2.350 × $10{}^{-8}$ | 1.360 × $10{}^{-8}$ | 1.040 × $10{}^{-8}$ | 3.080 × $10{}^{-9}$ | 3.450 × $10{}^{-9}$ | 1.860 × $10{}^{-8}$ | 2.720 × $10{}^{-8}$ | 3.100 × $10{}^{-8}$ |

MAX | 1.93 × $10{}^{-6}$ | 6.98 × $10{}^{-6}$ | 1.84 × $10{}^{-6}$ | 3.28 × $10{}^{-6}$ | 2.22 × $10{}^{-6}$ | 1.94 × $10{}^{-6}$ | 1.35 × $10{}^{-6}$ | 2.17 × $10{}^{-6}$ | 2.72 × $10{}^{-6}$ | 9.43 × $10{}^{-7}$ | 4.26 × $10{}^{-6}$ | ||

MEAN | 3.84 × $10{}^{-7}$ | 1.39 × $10{}^{-6}$ | 3.16 × $10{}^{-7}$ | 4.10 × $10{}^{-7}$ | 5.05 × $10{}^{-7}$ | 3.46 × $10{}^{-7}$ | 1.75 × $10{}^{-7}$ | 3.13 × $10{}^{-7}$ | 5.56 × $10{}^{-7}$ | 2.90 × $10{}^{-7}$ | 7.02 × $10{}^{-7}$ | ||

STD | 3.52 × $10{}^{-7}$ | 1.06 × $10{}^{-6}$ | 3.50 × $10{}^{-7}$ | 3.66 × $10{}^{-7}$ | 4.30 × $10{}^{-7}$ | 3.31 × $10{}^{-7}$ | 1.63 × $10{}^{-7}$ | 2.88 × $10{}^{-7}$ | 4.81 × $10{}^{-7}$ | 2.14 × $10{}^{-7}$ | 6.13 × $10{}^{-7}$ | ||

2 | MIN | 1.73 × $10{}^{-9}$ | 1.97 × $10{}^{-8}$ | 2.69 × $10{}^{-10}$ | 1.16 × $10{}^{-8}$ | 1.65 × $10{}^{-9}$ | 9.28 × $10{}^{-10}$ | 4.86 × $10{}^{-9}$ | 1.21 × $10{}^{-9}$ | 7.76 × $10{}^{-9}$ | 5.11 × $10{}^{-9}$ | 9.06 × $10{}^{-9}$ | |

MAX | 2.28 × $10{}^{-6}$ | 6.20 × $10{}^{-6}$ | 1.05 × $10{}^{-6}$ | 1.25 × $10{}^{-6}$ | 2.97 × $10{}^{-6}$ | 1.23 × $10{}^{-6}$ | 3.14 × $10{}^{-7}$ | 1.81 × $10{}^{-6}$ | 2.73 × $10{}^{-6}$ | 8.44 × $10{}^{-7}$ | 3.57 × $10{}^{-6}$ | ||

MEAN | 2.53 × $10{}^{-7}$ | 6.61 × $10{}^{-7}$ | 1.48 × $10{}^{-7}$ | 1.37 × $10{}^{-7}$ | 2.76 × $10{}^{-7}$ | 1.84 × $10{}^{-7}$ | 5.56 × $10{}^{-8}$ | 1.44 × $10{}^{-7}$ | 2.97 × $10{}^{-7}$ | 1.28 × $10{}^{-7}$ | 3.53 × $10{}^{-7}$ | ||

STD | 3.65 × $10{}^{-7}$ | 8.94 × $10{}^{-7}$ | 1.70 × $10{}^{-7}$ | 1.74 × $10{}^{-7}$ | 4.15 × $10{}^{-7}$ | 2.04 × $10{}^{-7}$ | 5.35 × $10{}^{-8}$ | 2.41 × $10{}^{-7}$ | 3.82 × $10{}^{-7}$ | 1.21 × $10{}^{-7}$ | 4.87 × $10{}^{-7}$ | ||

3 | MIN | 7.61 × $10{}^{-9}$ | 1.82 × $10{}^{-8}$ | 5.97 × $10{}^{-9}$ | 2.16 × $10{}^{-9}$ | 5.27 × $10{}^{-9}$ | 8.92 × $10{}^{-9}$ | 4.97 × $10{}^{-10}$ | 1.68 × $10{}^{-9}$ | 3.02 × $10{}^{-9}$ | 3.45 × $10{}^{-9}$ | 1.19 × $10{}^{-8}$ | |

MAX | 1.13 × $10{}^{-5}$ | 2.22 × $10{}^{-6}$ | 7.58 × $10{}^{-6}$ | 6.43 × $10{}^{-6}$ | 1.84 × $10{}^{-6}$ | 8.41 × $10{}^{-7}$ | 4.68 × $10{}^{-6}$ | 8.03 × $10{}^{-6}$ | 5.34 × $10{}^{-6}$ | 3.54 × $10{}^{-7}$ | 1.51 × $10{}^{-5}$ | ||

MEAN | 3.20 × $10{}^{-7}$ | 3.45 × $10{}^{-7}$ | 2.03 × $10{}^{-7}$ | 1.34 × $10{}^{-7}$ | 1.70 × $10{}^{-7}$ | 1.11 × $10{}^{-7}$ | 8.67 × $10{}^{-8}$ | 1.81 × $10{}^{-7}$ | 2.20 × $10{}^{-7}$ | 6.19 × $10{}^{-8}$ | 3.76 × $10{}^{-7}$ | ||

STD | 1.15 × $10{}^{-6}$ | 3.54 × $10{}^{-7}$ | 7.67 × $10{}^{-7}$ | 6.53 × $10{}^{-7}$ | 2.22 × $10{}^{-7}$ | 1.22 × $10{}^{-7}$ | 4.71 × $10{}^{-7}$ | 8.09 × $10{}^{-7}$ | 5.45 × $10{}^{-7}$ | 5.98 × $10{}^{-8}$ | 1.52 × $10{}^{-6}$ | ||

4 | MIN | 3.00 × $10{}^{-10}$ | 1.25 × $10{}^{-10}$ | 4.10 × $10{}^{-10}$ | 1.95 × $10{}^{-11}$ | 1.04 × $10{}^{-10}$ | 9.47 × $10{}^{-10}$ | 1.75 × $10{}^{-10}$ | 2.04 × $10{}^{-10}$ | 1.25 × $10{}^{-9}$ | 2.17 × $10{}^{-10}$ | 1.07 × $10{}^{-9}$ | |

MAX | 3.95 × $10{}^{-6}$ | 3.65 × $10{}^{-7}$ | 1.41 × $10{}^{-6}$ | 1.46 × $10{}^{-6}$ | 1.49 × $10{}^{-6}$ | 1.49 × $10{}^{-6}$ | 9.68 × $10{}^{-7}$ | 1.28 × $10{}^{-6}$ | 1.08 × $10{}^{-6}$ | 7.81 × $10{}^{-7}$ | 3.27 × $10{}^{-6}$ | ||

MEAN | 2.31 × $10{}^{-7}$ | 4.36 × $10{}^{-8}$ | 1.16 × $10{}^{-7}$ | 9.53 × $10{}^{-8}$ | 4.89 × $10{}^{-8}$ | 3.51 × $10{}^{-8}$ | 6.13 × $10{}^{-8}$ | 9.36 × $10{}^{-8}$ | 7.80 × $10{}^{-8}$ | 2.08 × $10{}^{-8}$ | 1.99 × $10{}^{-7}$ | ||

STD | 5.97 × $10{}^{-7}$ | 5.45 × $10{}^{-8}$ | 2.38 × $10{}^{-7}$ | 2.50 × $10{}^{-7}$ | 1.66 × $10{}^{-7}$ | 1.49 × $10{}^{-7}$ | 1.41 × $10{}^{-7}$ | 2.27 × $10{}^{-7}$ | 1.77 × $10{}^{-7}$ | 7.93 × $10{}^{-8}$ | 4.97 × $10{}^{-7}$ | ||

E | 1 | MIN | 6.71 × $10{}^{-9}$ | 3.99 × $10{}^{-8}$ | 1.18 × $10{}^{-9}$ | 2.35 × $10{}^{-8}$ | 1.36 × $10{}^{-8}$ | 1.04 × $10{}^{-8}$ | 3.08 × $10{}^{-9}$ | 3.45 × $10{}^{-9}$ | 1.86 × $10{}^{-8}$ | 2.72 × $10{}^{-8}$ | 3.10 × $10{}^{-8}$ |

MAX | 1.93 × $10{}^{-6}$ | 6.98 × $10{}^{-6}$ | 1.84 × $10{}^{-6}$ | 3.28 × $10{}^{-6}$ | 2.22 × $10{}^{-6}$ | 1.94 × $10{}^{-6}$ | 1.35 × $10{}^{-6}$ | 2.17 × $10{}^{-6}$ | 2.72 × $10{}^{-6}$ | 9.43 × $10{}^{-7}$ | 4.26 × $10{}^{-6}$ | ||

MEAN | 3.84 × $10{}^{-7}$ | 1.39 × $10{}^{-6}$ | 3.16 × $10{}^{-7}$ | 4.10 × $10{}^{-7}$ | 5.05 × $10{}^{-7}$ | 3.46 × $10{}^{-7}$ | 1.75 × $10{}^{-7}$ | 3.13 × $10{}^{-7}$ | 5.56 × $10{}^{-7}$ | 2.90 × $10{}^{-7}$ | 7.02 × $10{}^{-7}$ | ||

STD | 3.84 × $10{}^{-7}$ | 1.39 × $10{}^{-6}$ | 3.16 × $10{}^{-7}$ | 4.10 × $10{}^{-7}$ | 5.05 × $10{}^{-7}$ | 3.46 × $10{}^{-7}$ | 1.75 × $10{}^{-7}$ | 3.13 × $10{}^{-7}$ | 5.56 × $10{}^{-7}$ | 2.90 × $10{}^{-7}$ | 7.02 × $10{}^{-7}$ | ||