# Hermite Wavelet Method for Nonlinear Fractional Differential Equations

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Foundations

**Definition**

**1.**

**Definition**

**2.**

## 3. Hermite Wavelets (HWs)

## 4. Function Approximation of HWs

## 5. Convergence Analysis

**Theorem**

**1:**

**Proof**

## 6. Operational Matrices of HWs

**Definition**:

## 7. Numeric Solution Examples

- Example 1:

- Example 2:

- Example 3:

- Example 4:

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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t | ${\mathit{m}}^{\prime}=24$ | ${\mathit{m}}^{\prime}=48$ | ${\mathit{m}}^{\prime}=96$ | ${\mathit{m}}^{\prime}=192$ |
---|---|---|---|---|

0 | 5.27 × 10^{−4} | 1.40 × 10^{−4} | 3.58 × 10^{−5} | 9.01 × 10^{−6} |

0.1 | 5.23 × 10^{−4} | 1.31 × 10^{−4} | 3.32 × 10^{−5} | 8.32 × 10^{−6} |

0.2 | 4.15 × 10^{−4} | 1.09 × 10^{−4} | 2.71 × 10^{−5} | 6.56 × 10^{−6} |

0.3 | 3.29 × 10^{−4} | 7.49 × 10^{−5} | 1.88 × 10^{−5} | 4.80 × 10^{−6} |

0.4 | 1.77 × 10^{−4} | 4.36 × 10^{−5} | 9.96 × 10^{−6} | 2.62 × 10^{−6} |

0.5 | 1.94 × 10^{−4} | 2.37 × 10^{−5} | 3.11 × 10^{−6} | 6.66 × 10^{−7} |

0.6 | 1.74 × 10^{−4} | 4.29 × 10^{−5} | 9.86 × 10^{−6} | 2.81 × 10^{−6} |

0.7 | 3.41 × 10^{−4} | 7.54 × 10^{−5} | 1.90 × 10^{−5} | 4.93 × 10^{−6} |

0.8 | 4.26 × 10^{−4} | 1.17 × 10^{−4} | 2.91 × 10^{−5} | 7.17 × 10^{−6} |

0.9 | 5.68 × 10^{−4} | 1.43 × 10^{−4} | 3.70 × 10^{−5} | 9.22 × 10^{−6} |

t | ${\mathit{y}}_{\mathit{e}\mathit{x}\mathit{a}\mathit{c}\mathit{t}}$ | ${\mathit{y}}_{\mathit{H}\mathit{W}\mathit{M}}$ $(\mathit{k}=8,\mathit{M}=3)$ | ${\mathit{y}}_{\mathit{O}\mathit{F}\mathit{M}}$ [32] | ${\mathit{y}}_{\mathit{V}\mathit{I}\mathit{M}}$ [4] | ${\mathit{y}}_{\mathit{A}\mathit{D}\mathit{M}}$ [4] | ${\mathit{y}}_{\mathit{F}\mathit{D}\mathit{M}}$ [1] |
---|---|---|---|---|---|---|

0.1 | 0.039750 | 0.039752 | 0.039754 | 0.039874 | 0.039874 | 0.039473 |

0.2 | 0.157036 | 0.157038 | 0.157043 | 0.158512 | 0.158512 | 0.157703 |

0.3 | 0.347370 | 0.347371 | 0.347373 | 0.353625 | 0.353625 | 0.352402 |

0.4 | 0.604695 | 0.604696 | 0.604699 | 0.622083 | 0.622083 | 0.620435 |

0.5 | 0.921768 | 0.921768 | 0.921768 | 0.960047 | 0.960047 | 0.957963 |

0.6 | 1.290457 | 1.290456 | 1.290458 | 1.363093 | 1.363093 | 1.360551 |

0.7 | 1.702008 | 1.702007 | 1.702007 | 1.826257 | 1.826257 | 1.823267 |

0.8 | 2.147287 | 2.147285 | 2.147286 | 2.344224 | 2.344224 | 2.340749 |

0.9 | 2.617001 | 2.616999 | 2.616998 | 2.911278 | 2.911278 | 2.907324 |

t | ${\mathit{m}}^{\prime}=12$ | ${\mathit{m}}^{\prime}=24$ | ${\mathit{m}}^{\prime}=48$ | ${\mathit{m}}^{\prime}=96$ | ${\mathit{m}}^{\prime}=192$ |
---|---|---|---|---|---|

0 | 3.79 × 10^{−4} | 4.57 × 10^{−5} | 5.67 × 10^{−6} | 7.07 × 10^{−7} | 8.83 × 10^{−8} |

0.1 | 6.20 × 10^{−5} | 3.38 × 10^{−5} | 8.23 × 10^{−6} | 1.78 × 10^{−6} | 4.48 × 10^{−7} |

0.2 | 2.87 × 10^{−4} | 7.03 × 10^{−5} | 1.51 × 10^{−5} | 3.80 × 10^{−6} | 9.89 × 10^{−7} |

0.3 | 3.63 × 10^{−4} | 9.45 × 10^{−5} | 2.67 × 10^{−5} | 6.64 × 10^{−6} | 1.61 × 10^{−6} |

0.4 | 7.33 × 10^{−4} | 1.52 × 10^{−4} | 3.84 × 10^{−5} | 1.01 × 10^{−5} | 2.52 × 10^{−6} |

0.5 | 2.26 × 10^{−3} | 3.72 × 10^{−4} | 7.51 × 10^{−5} | 1.68 × 10^{−5} | 3.97 × 10^{−6} |

0.6 | 1.25 × 10^{−3} | 3.76 × 10^{−4} | 9.34 × 10^{−5} | 2.24 × 10^{−5} | 5.61 × 10^{−6} |

0.7 | 2.39 × 10^{−3} | 5.92 × 10^{−4} | 1.36 × 10^{−4} | 3.43 × 10^{−5} | 8.75 × 10^{−6} |

0.8 | 3.17 × 10^{−3} | 8.29 × 10^{−4} | 2.26 × 10^{−4} | 5.64 × 10^{−5} | 1.38 × 10^{−5} |

0.9 | 6.76 × 10^{−3} | 1.41 × 10^{−3} | 3.58 × 10^{−4} | 9.36 × 10^{−5} | 2.34 × 10^{−5} |

t | ${\mathit{m}}^{\prime}=12$ | ${\mathit{m}}^{\prime}=24$ | ${\mathit{m}}^{\prime}=48$ | ${\mathit{m}}^{\prime}=96$ | ${\mathit{m}}^{\prime}=192$ |
---|---|---|---|---|---|

0 | 1.00 × 10^{−3} | 4.47 × 10^{−4} | 1.54 × 10^{−4} | 4.54 × 10^{−5} | 1.24 × 10^{−5} |

0.1 | 1.90 × 10^{−3} | 3.63 × 10^{−4} | 9.31 × 10^{−5} | 2.49 × 10^{−5} | 6.20 × 10^{−6} |

0.2 | 6.26 × 10^{−4} | 1.73 × 10^{−4} | 4.97 × 10^{−5} | 1.24 × 10^{−5} | 3.00 × 10^{−6} |

0.3 | 4.34 × 10^{−4} | 1.08 × 10^{−4} | 2.34 × 10^{−5} | 5.90 × 10^{−6} | 1.53 × 10^{−6} |

0.4 | 1.24 × 10^{−4} | 4.97 × 10^{−5} | 1.23 × 10^{−5} | 2.81 × 10^{−6} | 7.05 × 10^{−7} |

0.5 | 1.04 × 10^{−4} | 7.99 × 10^{−6} | 8.56 × 10^{−7} | 6.21 × 10^{−7} | 2.10 × 10^{−7} |

0.6 | 2.26 × 10^{−5} | 2.09 × 10^{−6} | 3.80 × 10^{−7} | 2.84 × 10^{−8} | 5.79 × 10^{−9} |

0.7 | 5.47 × 10^{−5} | 1.25 × 10^{−5} | 2.42 × 10^{−6} | 6.12 × 10^{−7} | 1.64 × 10^{−7} |

0.8 | 6.02 × 10^{−5} | 1.51 × 10^{−5} | 4.29 × 10^{−6} | 1.06 × 10^{−6} | 2.58 × 10^{−7} |

0.9 | 9.23 × 10^{−5} | 1.98 × 10^{−5} | 4.98 × 10^{−6} | 1.29 × 10^{−6} | 3.23 × 10^{−7} |

t | ${\mathit{m}}^{\prime}=12$ | ${\mathit{m}}^{\prime}=24$ | ${\mathit{m}}^{\prime}=48$ | ${\mathit{m}}^{\prime}=96$ | ${\mathit{m}}^{\prime}=192$ | ${\mathit{m}}^{\prime}=384$ | MHPM [34] | IRKHSM [35] |
---|---|---|---|---|---|---|---|---|

0.1 | 6.40 × 10^{−5} | 3.24 × 10^{−5} | 7.88 × 10^{−6} | 1.71 × 10^{−6} | 4.31 × 10^{−7} | 1.10 × 10^{−8} | 0 | 9.05 × 10^{−6} |

0.2 | 2.44 × 10^{−4} | 5.91 × 10^{−5} | 1.29 × 10^{−5} | 3.25 × 10^{−6} | 8.42 × 10^{−7} | 1.12 × 10^{−7} | 0 | 1.72 × 10^{−5} |

0.3 | 2.71 × 10^{−4} | 6.82 × 10^{−5} | 1.85 × 10^{−5} | 4.60 × 10^{−6} | 1.13 × 10^{−6} | 2.10 × 10^{−7} | 1.00 × 10^{−6} | 2.38 × 10^{−5} |

0.4 | 3.64 × 10^{−4} | 8.24 × 10^{−5} | 2.06 × 10^{−5} | 5.29 × 10^{−6} | 1.32 × 10^{−6} | 2.82 × 10^{−7} | 5.00 × 10^{−6} | 2.85 × 10^{−5} |

0.5 | 3.92 × 10^{−4} | 9.67 × 10^{−5} | 2.33 × 10^{−5} | 5.68 × 10^{−6} | 1.40 × 10^{−6} | 3.28 × 10^{−7} | 3.90 × 10^{−5} | 3.11 × 10^{−5} |

0.6 | 3.35 × 10^{−4} | 8.53 × 10^{−5} | 2.13 × 10^{−5} | 5.29 × 10^{−6} | 1.32 × 10^{−6} | 3.46 × 10^{−7} | 1.93 × 10^{−4} | 3.17 × 10^{−5} |

0.7 | 3.05 × 10^{−4} | 7.56 × 10^{−5} | 1.90 × 10^{−5} | 4.75 × 10^{−6} | 1.19 × 10^{−6} | 3.31 × 10^{−7} | 7.37 × 10^{−4} | 3.07 × 10^{−5} |

0.8 | 2.62 × 10^{−4} | 6.47 × 10^{−5} | 1.58 × 10^{−5} | 3.95 × 10^{−6} | 9.93 × 10^{−7} | 2.96 × 10^{−7} | 2.33 × 10^{−3} | 2.81 × 10^{−5} |

0.9 | 1.84 × 10^{−4} | 5.03 × 10^{−5} | 1.25 × 10^{−5} | 3.06 × 10^{−6} | 7.65 × 10^{−7} | 2.48 × 10^{−7} | 6.37 × 10^{−3} | 2.32 × 10^{−5} |

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**MDPI and ACS Style**

Turan Dincel, A.; Tural Polat, S.N.; Sahin, P.
Hermite Wavelet Method for Nonlinear Fractional Differential Equations. *Fractal Fract.* **2023**, *7*, 346.
https://doi.org/10.3390/fractalfract7050346

**AMA Style**

Turan Dincel A, Tural Polat SN, Sahin P.
Hermite Wavelet Method for Nonlinear Fractional Differential Equations. *Fractal and Fractional*. 2023; 7(5):346.
https://doi.org/10.3390/fractalfract7050346

**Chicago/Turabian Style**

Turan Dincel, Arzu, Sadiye Nergis Tural Polat, and Pelin Sahin.
2023. "Hermite Wavelet Method for Nonlinear Fractional Differential Equations" *Fractal and Fractional* 7, no. 5: 346.
https://doi.org/10.3390/fractalfract7050346