### AIMS Mathematics

2021, Issue 11: 12743-12756. doi: 10.3934/math.2021735
Research article Special Issues

# Numerical approach for approximating the Caputo fractional-order derivative operator

• Received: 09 July 2021 Accepted: 27 August 2021 Published: 06 September 2021
• MSC : 34A08, 65Y99, 65H05, 65L05

• This work aims to propose a new simple robust power series formula with its truncation error to approximate the Caputo fractional-order operator $D_{a}^{\alpha}y(t)$ of order $m-1 < \alpha < m$, where $m\in\mathbb{N}$. The proposed formula, which are derived with the help of the weighted mean value theorem, is expressed ultimately in terms of a fractional-order series and its reminder term. This formula is used successfully to provide approximate solutions of linear and nonlinear fractional-order differential equations in the form of series solution. It can be used to determine the analytic solutions of such equations in some cases. Some illustrative numerical examples, including some linear and nonlinear problems, are provided to validate the established formula.

Citation: Ramzi B. Albadarneh, Iqbal Batiha, A. K. Alomari, Nedal Tahat. Numerical approach for approximating the Caputo fractional-order derivative operator[J]. AIMS Mathematics, 2021, 6(11): 12743-12756. doi: 10.3934/math.2021735

### Related Papers:

• This work aims to propose a new simple robust power series formula with its truncation error to approximate the Caputo fractional-order operator $D_{a}^{\alpha}y(t)$ of order $m-1 < \alpha < m$, where $m\in\mathbb{N}$. The proposed formula, which are derived with the help of the weighted mean value theorem, is expressed ultimately in terms of a fractional-order series and its reminder term. This formula is used successfully to provide approximate solutions of linear and nonlinear fractional-order differential equations in the form of series solution. It can be used to determine the analytic solutions of such equations in some cases. Some illustrative numerical examples, including some linear and nonlinear problems, are provided to validate the established formula.

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沈阳化工大学材料科学与工程学院 沈阳 110142

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