Numerical approach for approximating the Caputo fractional-order derivative operator

Ramzi B. Albadarneh; Iqbal Batiha; A. K. Alomari; Nedal Tahat; Ramzi B. Albadarneh; Iqbal Batiha; A. K. Alomari; Nedal Tahat

doi:10.3934/math.2021735

AIMS Mathematics

2021, Volume 6, Issue 11: 12743-12756. doi: 10.3934/math.2021735

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Numerical approach for approximating the Caputo fractional-order derivative operator

1.
Department of Mathematics, Faculty of Science, The Hashemite University, P.O Box 330127, Zarqa 13133, Jordan
2.
Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 2600, Jordan
3.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, UAE
4.
Department of Mathematics, Yarmouk University, Irbid 211-63, Jordan

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.
- Caputo fractional-order operator,
- fractional-order differential equation,
- weighted mean value theorem,
- power series,
- variable-order fractional operator
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

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Abstract

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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