# Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

- -
- Good accuracy and simple implementation.
- -
- Exponential convergence.
- -
- Provide a global approach in $(0,\infty )$
- -
- Provide the possibility of error analysis and convergence of method due to the clear and strong theoretical structure.
- -
- It can be a basis for solving problems with higher complexity.

## 2. Laplace Transform Method and Quadrature Rule

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 3. Special Cases and Their Examples

#### 3.1. First Order FDE

#### 3.2. FDE with Delay Term

#### 3.3. Second Order FDE

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Comparing the exact and the approximate solutions with $\alpha =0.8$ and $N=1000$ for Equation (23).

**Figure 3.**Comparing the exact and the approximate solutions for Example Section 3.2.

**Figure 4.**Comparing the exact and the approximate solutions for Equation (24).

**Figure 5.**Comparing the exact and the approximate solutions for Equation (25).

**Table 1.**Exact and approximate values of the proposed method for Example Section 3.1.

t | $\mathbf{Exact}\phantom{\rule{4.pt}{0ex}}\mathbf{Solution}$ | $\mathbf{Approximate}\phantom{\rule{4.pt}{0ex}}\mathbf{Solution}$ | Error |
---|---|---|---|

$0.0$ | $0.0000$ | $0.0005$ | $0.0005$ |

$0.3$ | $0.1619$ | $0.1625$ | $0.0006$ |

$0.6$ | $0.4374$ | $0.4376$ | $0.0002$ |

$0.9$ | $0.7431$ | $0.7432$ | $0.0001$ |

$1.2$ | $1.0210$ | $1.0213$ | $0.0003$ |

$1.5$ | $1.2217$ | $1.2224$ | $0.0007$ |

$1.8$ | $1.3066$ | $1.3067$ | $0.0002$ |

$2.1$ | $1.2509$ | $1.2510$ | $0.0001$ |

$2.4$ | $1.0464$ | $1.0472$ | $0.0008$ |

$2.7$ | $0.7023$ | $0.7029$ | $0.0006$ |

$3.0$ | $0.2444$ | $0.2447$ | $0.0003$ |

**Table 2.**Exact and approximate values of the proposed method with $\alpha =0.8$ and $N=1000$ for Equation (23).

t | $\mathbf{Exact}\phantom{\rule{4.pt}{0ex}}\mathbf{Solution}$ | $\mathbf{Approximate}\phantom{\rule{4.pt}{0ex}}\mathbf{Solution}$ |
---|---|---|

$0.0$ | $0.0000$ | $0.0000$ |

$0.5$ | $0.0000$ | $0.0000$ |

$1.0$ | $0.5000$ | $0.5000$ |

$1.5$ | $3.3750$ | $3.3750$ |

$2.0$ | $12.0000$ | $12.0000$ |

$2.5$ | $31.2500$ | $31.2500$ |

$3.0$ | $67.5000$ | $67.5000$ |

$3.5$ | $128.6250$ | $128.6250$ |

$4.0$ | $224.0000$ | $224.0000$ |

$4.5$ | $364.5000$ | $364.5000$ |

$5.0$ | $562.5000$ | $562.5000$ |

**Table 3.**Maximum error of the proposed method with different values of $\alpha $ and N for Equation (23).

$\mathit{\alpha}=0.8$ | $\mathit{\alpha}=0.4$ | |||||
---|---|---|---|---|---|---|

$\mathit{t}$ | $\mathit{N}\mathbf{=}\mathit{1000}$ | $\mathit{N}\mathbf{=}\mathit{2000}$ | $\mathit{N}\mathbf{=}\mathit{4000}$ | $\mathit{N}\mathbf{=}\mathit{1000}$ | $\mathit{N}\mathbf{=}\mathit{2000}$ | $\mathit{N}\mathbf{=}\mathit{4000}$ |

$0.0$ | $7.5582e-8$ | $1.197e-10$ | $1.310e-13$ | $2.011e-8$ | $1.187e-10$ | $2.301e-13$ |

$0.5$ | $2.9539e-8$ | $1.094e-10$ | $1.037e-13$ | $2.000e-8$ | $1.194e-09$ | $2.037e-13$ |

$1.0$ | $3.2312e-8$ | $1.006e-10$ | $1.124e-13$ | $1.908e-8$ | $1.105e-09$ | $2.325e-13$ |

$1.5$ | $4.6016e-8$ | $0.923e-10$ | $1.130e-13$ | $1.901e-8$ | $0.921e-10$ | $2.159e-13$ |

$2.0$ | $1.2322e-8$ | $0.845e-10$ | $1.167e-13$ | $1.852e-8$ | $0.841e-10$ | $2.167e-13$ |

$2.5$ | $5.1819e-8$ | $0.772e-10$ | $1.248e-13$ | $1.798e-8$ | $0.767e-10$ | $1.769e-13$ |

$3.0$ | $1.5116e-8$ | $0.705e-10$ | $1.216e-13$ | $1.775e-8$ | $0.705e-10$ | $1.726e-13$ |

$3.5$ | $6.2418e-8$ | $0.641e-10$ | $1.049e-13$ | $1.687e-8$ | $0.631e-10$ | $1.649e-13$ |

$4.0$ | $3.4868e-8$ | $0.581e-10$ | $1.140e-13$ | $1.625e-8$ | $0.538e-10$ | $1.591e-13$ |

$4.5$ | $6.5410e-8$ | $0.526e-10$ | $1.154e-13$ | $1.527e-8$ | $0.502e-10$ | $1.514e-13$ |

$5.0$ | $4.2626e-8$ | $0.475e-10$ | $1.103e-13$ | $1.499e-8$ | $0.397e-10$ | $1.445e-13$ |

Error of the Approximate Solution | ||||
---|---|---|---|---|

$\mathit{t}$ | $\mathbf{Exact}\phantom{\rule{4.pt}{0ex}}\mathbf{Solution}$ | $\mathit{N}\mathbf{=}\mathit{1000}$ | $\mathit{N}\mathbf{=}\mathit{2000}$ | $\mathit{N}\mathbf{=}\mathit{4000}$ |

$0.0$ | $0.0000$ | $0.0065$ | $7.227e-6$ | $1.301e-10$ |

$0.5$ | $0.8243$ | $0.0007$ | $6.894e-6$ | $1.247e-10$ |

$1.0$ | $2.7182$ | $0.0012$ | $5.054e-6$ | $1.191e-10$ |

$1.5$ | $6.7225$ | $0.0011$ | $5.203e-6$ | $1.112e-10$ |

$2.0$ | $14.7781$ | $0.0052$ | $5.111e-6$ | $1.097e-10$ |

$2.5$ | $30.4562$ | $0.0060$ | $5.072e-6$ | $1.048e-10$ |

$3.0$ | $60.2566$ | $0.0056$ | $4.774e-6$ | $1.018e-10$ |

$3.5$ | $115.9040$ | $0.0065$ | $4.002e-6$ | $9.049e-11$ |

$4.0$ | $218.3926$ | $0.0035$ | $3.332e-6$ | $8.141e-11$ |

$4.5$ | $405.0770$ | $0.0039$ | $2.657e-6$ | $7.754e-11$ |

$5.0$ | $742.0657$ | $0.0022$ | $1.011e-6$ | $5.914e-11$ |

**Table 5.**Exact and approximate values of the proposed method for Equation (24).

t | $\mathbf{Exact}\phantom{\rule{4.pt}{0ex}}\mathbf{Solution}$ | $\mathbf{Approximate}\phantom{\rule{4.pt}{0ex}}\mathbf{Solution}$ | $\mathbf{Error}$ |
---|---|---|---|

$0.0$ | $0.0000$ | $0.0065$ | $0.0065$ |

$0.5$ | $0.8243$ | $0.8251$ | $0.0007$ |

$1.0$ | $2.7182$ | $2.7195$ | $0.0012$ |

$1.5$ | $6.7225$ | $6.7236$ | $0.0011$ |

$2.0$ | $14.7781$ | $14.7833$ | $0.0052$ |

$2.5$ | $30.4562$ | $30.4622$ | $0.0060$ |

$3.0$ | $60.2566$ | $60.2622$ | $0.0056$ |

$3.5$ | $115.9040$ | $115.9106$ | $0.0065$ |

$4.0$ | $218.3926$ | $218.3961$ | $0.0035$ |

$4.5$ | $405.0770$ | $405.0810$ | $0.0039$ |

$5.0$ | $742.0657$ | $742.0680$ | $0.0022$ |

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**MDPI and ACS Style**

Soradi-Zeid, S.; Mesrizadeh, M.; Cattani, C.
Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule. *Fractal Fract.* **2021**, *5*, 111.
https://doi.org/10.3390/fractalfract5030111

**AMA Style**

Soradi-Zeid S, Mesrizadeh M, Cattani C.
Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule. *Fractal and Fractional*. 2021; 5(3):111.
https://doi.org/10.3390/fractalfract5030111

**Chicago/Turabian Style**

Soradi-Zeid, Samaneh, Mehdi Mesrizadeh, and Carlo Cattani.
2021. "Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule" *Fractal and Fractional* 5, no. 3: 111.
https://doi.org/10.3390/fractalfract5030111