# The n-Point Composite Fractional Formula for Approximating Riemann–Liouville Integrator

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**The fractional Riemann–Liouville integrator of a function $h\left(t\right)$ of order $\mu >0$ can be expressed as:

**Definition**

**2**

**.**The Caputo fractional differentiator of a function $h\left(t\right)$ of order $\mu >0$ can be expressed as:

**Definition**

**3**

**.**The Riemann–Liouville fractional differentiator of a function $h\left(t\right)$ of order $\mu >0$ might be outlined in terms of the Riemann–Liouville fractional integrator as:

**Definition**

**4**

**.**The definite fractional integral of the function f of order α is given by:

**Theorem**

**1**

**.**(Generalized Taylor’s Theorem) Suppose that ${D}_{t}^{k\alpha}f\left(x\right)\in {C}^{n+1}(a,b]$ for $k=0,1,\cdots ,n+1,$ where $0<\alpha \le 1$. Then, the function f can be expanded about $x={x}_{0}$ as follows:

## 3. Main Results

**Theorem**

**2**

**.**Suppose that $f\in {C}^{3}[a,b]$ and ${x}_{0},{x}_{1},{x}_{2}$ are distinct points in the interval $[a,b]$ such that $a={x}_{0}<{x}_{1}={x}_{0}+h<{x}_{2}={x}_{0}+2h=b$ with $h>0$. Then, the modified three-point fractional formula for approximating the Caputo fractional differentiator can be given by:

**Corollary**

**1.**

**Proof.**

**Remark**

**1.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

**Proof.**

## 4. Numerical Examples

**Example**

**1.**

- Case 1: When $\alpha =1$, we obtain:$$\begin{array}{cc}\hfill {J}^{1}f\left(x\right)& =0.5{x}^{4}+4{x}^{2}.\hfill \end{array}$$
- Case 2: When $\alpha =0.75$, we obtain:$$\begin{array}{cc}\hfill {J}^{0.75}f\left(x\right)& =0.7234{x}^{3.75}+6.0182{x}^{1.75}.\hfill \end{array}$$
- Case 3: When $\alpha =0.5$, we obtain:$$\begin{array}{cc}\hfill {J}^{0.5}f\left(x\right)& =1.0316{x}^{3.5}+7.7549{x}^{1.5}.\hfill \end{array}$$

**Example**

**2.**

- Case 1: When $\alpha =1$, we have:$$\begin{array}{cc}\hfill {J}^{1}f\left(x\right)& =\sum _{k=0}^{\infty}\frac{{(-1)}^{k}}{(2k+1)!}\frac{\Gamma (2k+2)}{\Gamma (2k+3)}{x}^{2k+2}.\hfill \end{array}$$
- Case 2: When $\alpha =0.65$, we have:$$\begin{array}{cc}\hfill {J}^{0.65}f\left(x\right)& =\sum _{k=0}^{\infty}\frac{{(-1)}^{k}}{(2k+1)!}\frac{\Gamma (2k+2)}{\Gamma (2k+2.65)}{x}^{2k+1.65}.\hfill \end{array}$$
- Case 3: When $\alpha =0.45$, we have:$$\begin{array}{cc}\hfill {J}^{0.45}f\left(x\right)& =\sum _{k=0}^{\infty}\frac{{(-1)}^{k}}{(2k+1)!}\frac{\Gamma (2k+2)}{\Gamma (2k+2.45)}{x}^{2k+1.45}.\hfill \end{array}$$

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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$\mathit{n}\setminus \mathit{\alpha}$ | 0.5 | 0.75 | 1 |
---|---|---|---|

2 | 22.27105816166 | 26.16734499344 | 23.68537600000 |

4 | 22.69349110102 | 26.26331428545 | 23.69932800000 |

6 | 24.03916177943 | 26.50011822893 | 23.72599466666 |

8 | 27.18307909882 | 26.96769964223 | 23.77049600000 |

10 | 28.49632000000 | 26.79576811488 | 23.84000000000 |

Exact: | 28.69344858658 | 26.46463660779 | 24 |

$\mathit{n}\setminus \mathit{\alpha}$ | 0.45 | 0.65 | 1 |
---|---|---|---|

2 | 0.988434759968 | 1.109574041883 | 1.416657631095 |

4 | 1.046664086456 | 1.129294584780 | 1.418163344357 |

6 | 1.098873318034 | 1.271316766203 | 1.420407266983 |

8 | 1.156966874475 | 1.233598909270 | 1.423035132930 |

10 | 1.293240352453 | 1.306986260607 | 1.425632059945 |

Exact: | 1.273563291144 | 1.362300353262 | 1.416146836547 |

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**MDPI and ACS Style**

Batiha, I.M.; Alshorm, S.; Al-Husban, A.; Saadeh, R.; Gharib, G.; Momani, S.
The n-Point Composite Fractional Formula for Approximating Riemann–Liouville Integrator. *Symmetry* **2023**, *15*, 938.
https://doi.org/10.3390/sym15040938

**AMA Style**

Batiha IM, Alshorm S, Al-Husban A, Saadeh R, Gharib G, Momani S.
The n-Point Composite Fractional Formula for Approximating Riemann–Liouville Integrator. *Symmetry*. 2023; 15(4):938.
https://doi.org/10.3390/sym15040938

**Chicago/Turabian Style**

Batiha, Iqbal M., Shameseddin Alshorm, Abdallah Al-Husban, Rania Saadeh, Gharib Gharib, and Shaher Momani.
2023. "The n-Point Composite Fractional Formula for Approximating Riemann–Liouville Integrator" *Symmetry* 15, no. 4: 938.
https://doi.org/10.3390/sym15040938