# Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Basic Concepts

**Definition**

**1**

**Theorem**

**1**

**Definition**

**2**

**Lemma**

**1.**

- i.
- $\mathcal{L}\left\{{c}_{1}{u}_{1}\left(t\right)+{c}_{2}{u}_{2}\left(t\right)\right\}={c}_{1}{U}_{1}\left(\xi \right)+{c}_{2}{U}_{2}\left(\xi \right)$.
- ii.
- ${\mathcal{L}}^{-1}\left\{{c}_{1}{U}_{1}\left(\xi \right)+{c}_{2}{U}_{2}\left(\xi \right)\right\}={c}_{1}{u}_{1}\left(t\right)+{c}_{2}{u}_{2}\left(t\right).$
- iii.
- $\underset{\xi \to \infty}{\mathit{lim}}\xi U\left(\xi \right)=u\left(0\right)$.
- iv.
- $\mathcal{L}\left\{{\mathfrak{D}}_{t}^{\alpha}u\left(t\right)\right\}={\xi}^{\alpha}U\left(\xi \right)-{\displaystyle \sum}_{k=0}^{n-1}{\xi}^{\alpha -k-1}{u}^{\left(k\right)}\left(0\right),\text{}\alpha \in \left(n-1,n\right],\text{}n\in \mathbb{N}.$

**Theorem**

**2**

## 3. Principle of the LFPS Algorithm

**Theorem**

**3.**

**Proof.**

## 4. Illustrated Examples

**Example**

**1.**

**Example**

**2.**

- If $qi$ is odd, then

- If $qi$ is even, then

- The nonlinear term

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**2D plots of fractional curves of the LFPS approximate solution for Example 1, at various values of $\alpha $, versus the exact solution .

**Figure 2.**2D plots of fractional curves of the LFPS approximate solution for Example 2 at various values of $\alpha $, versus the exact solution .

t_{i} | α = 0.9 | α = 0.8 | α = 0.7 | α = 0.6 |
---|---|---|---|---|

0.1 | $3.05125\times {10}^{-8}$ | $2.76818\times {10}^{-8}$ | $1.25598\times {10}^{-7}$ | $3.32936\times {10}^{-7}$ |

0.2 | $5.44445\times {10}^{-6}$ | $2.4180\times {10}^{-6}$ | $2.70504\times {10}^{-5}$ | $7.78659\times {10}^{-5}$ |

0.3 | $1.15718\times {10}^{-4}$ | $1.38234\times {10}^{-6}$ | $6.74897\times {10}^{-4}$ | $2.07896\times {10}^{-3}$ |

0.4 | $1.03751\times {10}^{-3}$ | $4.43181\times {10}^{-4}$ | $7.01985\times {10}^{-3}$ | $2.28483\times {10}^{-2}$ |

**Table 2.**The residual error of the LFPS solutions at different terms and times with $\mathsf{\alpha}=0.8$ for Example 1.

J | 20 | 40 | 60 | 80 | |
---|---|---|---|---|---|

t_{i} | |||||

0.1 | $2.23231\times {10}^{-2}$ | $6.03273\times {10}^{-4}$ | $5.48087\times {10}^{-6}$ | $2.76818\times {10}^{-8}$ | |

0.2 | $8.31149\times {10}^{-2}$ | $7.32976\times {10}^{-3}$ | $2.58114\times {10}^{-4}$ | $2.4180\times {10}^{-6}$ | |

0.3 | $1.85919\times {10}^{-1}$ | $3.28381\times {10}^{-2}$ | $2.64048\times {10}^{-3}$ | $1.38234\times {10}^{-6}$ | |

0.4 | $3.35873\times {10}^{-1}$ | $9.8273\times {10}^{-2}$ | $1.45347\times {10}^{-2}$ | $4.43181\times {10}^{-4}$ |

t_{i} | α = 0.9 | α = 0.8 | α = 0.7 | α = 0.6 | α = 0.5 |
---|---|---|---|---|---|

0.1 | $4.01872\times {10}^{-12}$ | $1.14969\times {10}^{-10}$ | $3.51448\times {10}^{-11}$ | $1.2662\times {10}^{-10}$ | $7.2709\times {10}^{-11}$ |

0.2 | $8.73393\times {10}^{-10}$ | $3.65742\times {10}^{-8}$ | $1.19959\times {10}^{-8}$ | $3.75249\times {10}^{-8}$ | $2.55736\times {10}^{-8}$ |

0.3 | $1.60784\times {10}^{-8}$ | $1.06161\times {10}^{-6}$ | $3.56381\times {10}^{-7}$ | $9.78584\times {10}^{-7}$ | $6.88552\times {10}^{-7}$ |

0.4 | $8.94261\times {10}^{-8}$ | $1.15723\times {10}^{-5}$ | $3.91616\times {10}^{-6}$ | $9.47861\times {10}^{-6}$ | $6.18322\times {10}^{-6}$ |

0.5 | $9.02043\times {10}^{-8}$ | $7.37875\times {10}^{-5}$ | $2.49509\times {10}^{-5}$ | $5.35762\times {10}^{-5}$ | $2.78115\times {10}^{-5}$ |

**Table 4.**The residual error of the LFPS solutions at different terms and times with $\mathsf{\alpha}=0.8$ for Example 2.

J | 20 | 40 | 60 | 80 | |
---|---|---|---|---|---|

t_{i} | |||||

0.1 | $3.23928\times {10}^{-3}$ | $4.56363\times {10}^{-5}$ | $3.98756\times {10}^{-7}$ | $2.06271\times {10}^{-9}$ | |

0.2 | $1.81809\times {10}^{-3}$ | $5.3666\times {10}^{-4}$ | $1.43983\times {10}^{-5}$ | $3.32608\times {10}^{-7}$ | |

0.3 | $1.00238\times {10}^{-2}$ | $2.20152\times {10}^{-3}$ | $1.13732\times {10}^{-4}$ | $6.45179\times {10}^{-6}$ | |

0.4 | $3.55286\times {10}^{-2}$ | $5.84874\times {10}^{-3}$ | $4.83321\times {10}^{-4}$ | $5.2582\times {10}^{-5}$ |

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**MDPI and ACS Style**

Alomari, A.-K.; Alaroud, M.; Tahat, N.; Almalki, A.
Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations. *Symmetry* **2023**, *15*, 1296.
https://doi.org/10.3390/sym15071296

**AMA Style**

Alomari A-K, Alaroud M, Tahat N, Almalki A.
Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations. *Symmetry*. 2023; 15(7):1296.
https://doi.org/10.3390/sym15071296

**Chicago/Turabian Style**

Alomari, Abedel-Karrem, Mohammad Alaroud, Nedal Tahat, and Adel Almalki.
2023. "Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations" *Symmetry* 15, no. 7: 1296.
https://doi.org/10.3390/sym15071296