Open Access

ARTICLE

Composite Fractional Trapezoidal Rule with Romberg Integration

by Iqbal M. Batiha^1,2,*, Rania Saadeh³, Iqbal H. Jebril¹, Ahmad Qazza³, Abeer A. Al-Nana⁴, Shaher Momani^2,5

1 Department of Mathematics, Al Zaytoonah University of Jordan, Amman, 11733, Jordan
2 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, United Arab Emirates
3 Department of Mathematics, Faculty of Science, Zarqa University, Zarqa, 13110, Jordan
4 Department of Mathematics, Prince Sattam Bin Abdulaziz University, Alkharj, 11942, Saudi Arabia
5 Department of Mathematics, The University of Jordan, Amman, 11942, Jordan

* Corresponding Author: Iqbal M. Batiha. Email:

(This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)

Computer Modeling in Engineering & Sciences 2024, 140(3), 2729-2745. https://doi.org/10.32604/cmes.2024.051588

Received 09 March 2024; Accepted 23 April 2024; Issue published 08 July 2024

Abstract

The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator. This would be achieved by first establishing a fractional-order version of the -point Trapezoidal rule and then by proposing another fractional-order version of the -composite Trapezoidal rule. In particular, the so-called divided-difference formula is typically employed to derive the -point Trapezoidal rule, which has accordingly been used to derive a more accurate fractional-order formula called the -composite Trapezoidal rule. Additionally, in order to increase the accuracy of the proposed approximations by reducing the true errors, we incorporate the so-called Romberg integration, which is an extrapolation formula of the Trapezoidal rule for integration, into our proposed approaches. Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.

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Keywords

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