# A Collocation Method for Numerical Solution of Nonlinear Delay Integro-Differential Equations for Wireless Sensor Network and Internet of Things

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## Abstract

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## 1. Introduction

- To develop efficient technique for the approximate solution of nonlinear FVIDEs arising in WSN and IIoT
- To design algorithm for proposed technique
- To examine the efficiency of the developed technique on some test problems and compare the results of our technique with results [1] available in the literature

## 2. Literature Review

## 3. Haar Wavelet

- To find solutions of different nonlinear FVIDEs arising in WSN and IIoT
- To check the efficiency and accuracy of the developed technique, the proposed method is applied on some test problems

## 4. Numerical Method for Nonlinear Delay IDEs Arising in WSN and IIoT

#### 4.1. Nonlinear Delay Fredholm IDEs

#### 4.2. Nonlinear Delay Volterra IDEs

#### 4.3. Nonlinear Delay Volterra–Fredholm IDEs

## 5. Numerical Assessments

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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J | $\mathit{N}={2}^{\mathit{J}+1}$ | ${\mathit{L}}_{\mathit{a}\mathit{b}\mathit{c}}$ | ${\mathit{M}}_{\mathit{c}}\left(\mathit{N}\right)$ | Results [1] |
---|---|---|---|---|

1 | 4 | 2.1194 $\times {10}^{-5}$ | 1.5933 $\times {10}^{-5}$ | 4.84 $\times {10}^{-3}$ |

2 | 8 | 6.3911 $\times {10}^{-6}$ | 4.0412 $\times {10}^{-6}$ | 1.36 $\times {10}^{-2}$ |

3 | 16 | 1.7632 $\times {10}^{-6}$ | 1.0141 $\times {10}^{-6}$ | 2.01 $\times {10}^{-2}$ |

4 | 32 | 4.6373 $\times {10}^{-7}$ | 2.5447 $\times {10}^{-7}$ | 3.54 $\times {10}^{-2}$ |

5 | 64 | 1.1898 $\times {10}^{-7}$ | 6.4159 $\times {10}^{-8}$ | 3.85 $\times {10}^{-2}$ |

6 | 128 | 3.0129 $\times {10}^{-8}$ | 1.6638 $\times {10}^{-8}$ | — |

7 | 256 | 7.5791 $\times {10}^{-9}$ | 4.9327 $\times {10}^{-9}$ | — |

J | $\mathit{N}={2}^{\mathit{J}+1}$ | ${\mathit{L}}_{\mathit{a}\mathit{b}\mathit{c}}$ | ${\mathit{M}}_{\mathit{c}}\left(\mathit{N}\right)$ | Results [1] |
---|---|---|---|---|

1 | 4 | 5.4262 $\times {10}^{-4}$ | 4.6432 $\times {10}^{-4}$ | 1.1 $\times {10}^{-1}$ |

2 | 8 | 1.5086 $\times {10}^{-4}$ | 3.1776 $\times {10}^{-5}$ | 1.1 $\times {10}^{-2}$ |

3 | 16 | 3.9748 $\times {10}^{-5}$ | 2.9549 $\times {10}^{-5}$ | 1.9 $\times {10}^{-2}$ |

4 | 32 | 1.0203 $\times {10}^{-5}$ | 7.3941 $\times {10}^{-6}$ | 2.8 $\times {10}^{-2}$ |

5 | 64 | 2.5847 $\times {10}^{-6}$ | 1.8489 $\times {10}^{-6}$ | 9.5 $\times {10}^{-4}$ |

6 | 128 | 6.5045 $\times {10}^{-7}$ | 4.6226 $\times {10}^{-7}$ | — |

7 | 256 | 1.6315 $\times {10}^{-7}$ | 1.1556 $\times {10}^{-7}$ | — |

8 | 512 | 4.0855 $\times {10}^{-8}$ | 2.8892 $\times {10}^{-8}$ | — |

9 | 1024 | 1.0222 $\times {10}^{-8}$ | 7.2230 $\times {10}^{-9}$ | — |

10 | 2048 | 2.5566 $\times {10}^{-9}$ | 1.8057 $\times {10}^{-9}$ | — |

J | $\mathit{N}={2}^{\mathit{J}+1}$ | ${\mathit{L}}_{\mathit{a}\mathit{b}\mathit{c}}$ | ${\mathit{M}}_{\mathit{c}}\left(\mathit{N}\right)$ |
---|---|---|---|

1 | 4 | 4.71362 $\times {10}^{-4}$ | 2.89263 $\times {10}^{-4}$ |

2 | 8 | 1.02141 $\times {10}^{-4}$ | 1.00189 $\times {10}^{-4}$ |

3 | 16 | 3.72674 $\times {10}^{-5}$ | 3.60812 $\times {10}^{-5}$ |

4 | 32 | 2.61325 $\times {10}^{-5}$ | 1.46973 $\times {10}^{-5}$ |

5 | 64 | 4.58470 $\times {10}^{-6}$ | 2.25831 $\times {10}^{-6}$ |

6 | 128 | 1.84356 $\times {10}^{-6}$ | 3.96374 $\times {10}^{-7}$ |

7 | 256 | 4.01407 $\times {10}^{-7}$ | 1.32281 $\times {10}^{-7}$ |

8 | 512 | 1.37851 $\times {10}^{-7}$ | 2.13459 $\times {10}^{-8}$ |

9 | 1024 | 3.93057 $\times {10}^{-8}$ | 4.53722 $\times {10}^{-9}$ |

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**MDPI and ACS Style**

Amin, R.; Nazir, S.; García-Magariño, I. A Collocation Method for Numerical Solution of Nonlinear Delay Integro-Differential Equations for Wireless Sensor Network and Internet of Things. *Sensors* **2020**, *20*, 1962.
https://doi.org/10.3390/s20071962

**AMA Style**

Amin R, Nazir S, García-Magariño I. A Collocation Method for Numerical Solution of Nonlinear Delay Integro-Differential Equations for Wireless Sensor Network and Internet of Things. *Sensors*. 2020; 20(7):1962.
https://doi.org/10.3390/s20071962

**Chicago/Turabian Style**

Amin, Rohul, Shah Nazir, and Iván García-Magariño. 2020. "A Collocation Method for Numerical Solution of Nonlinear Delay Integro-Differential Equations for Wireless Sensor Network and Internet of Things" *Sensors* 20, no. 7: 1962.
https://doi.org/10.3390/s20071962