Citation: 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 Academic Editor: Calogero Vetro Received: 1 January 2023 Revised: 20 February 2023 Accepted: 21 February 2023 Published: 19 April 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article The n-Point Composite Fractional Formula for Approximating Riemann–Liouville Integrator Iqbal M. Batiha 1,2 , Shameseddin Alshorm 1,* , Abdallah Al-Husban 3 , Rania Saadeh 4 , Gharib Gharib 4 and Shaher Momani 2,5 1 Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan; i.batiha@zuj.edu.jo 2 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman P.O. Box 346, United Arab Emirates 3 Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid P.O. Box 2600, Jordan 4 Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan 5 Department of Mathematics, The University of Jordan, Amman 11942, Jordan * Correspondence: alshormanshams@gmail.com Abstract: In this paper, we aim to present a novel n-point composite fractional formula for approx- imating a Riemann–Liouville fractional integral operator. With the use of the definite fractional integral’s definition coupled with the generalized Taylor’s formula, a novel three-point central frac- tional formula is established for approximating a Riemann–Liouville fractional integrator. Such a new formula, which emerges clearly from the symmetrical aspects of the proposed numerical approach, is then further extended to formulate an n-point composite fractional formula for approximating the same operator. Several numerical examples are introduced to validate our findings. Keywords: Richardson extrapolation; Riemann–Liouville fractional derivative and integral; Lagrange interpolating polynomial; Caputo derivative 1. Introduction Fractional calculus has many uses in the mathematical modeling of chemical phenom- ena, physics, technical, and economics. It has contributed in a significant way in developing many topics and implementations in applied mathematics, although there are different definitions of the fractional-order operators, see [1–6]. In recent times, many evolutions in the theory of fractional calculus have been investigated to be employed in many fields of science and engineering. In particular, the fundamental of fractional calculus has been approved as an illustrious mathematical facility used to describe many actual applica- tions [7,8]. As a result, various fractional-order differentiators and integrators have been established and approved by many researchers. It is important to highlight that there are two main operators for fractional-order differentiators; the first one is Caputo’s derivative operator with a power law function of convolution of a given function related to a local derivative, whereas the other one is the Riemann–Liouville derivative operator with a power law kernel of convolution [9]. In light of the various views of many mathematicians, the Caputo fractional-order differentiator has confirmed that it is more satisfactory for several real applications than that of the Riemann–Liouville derivative operator [10]. This is due to its suitability for using the assumed initial conditions when the fractional deriva- tives are taken [11]. Regardless of the best operator among the two former operators, the Riemann–Liouville integrator represents an inverse operator for both. This is because the Caputo differentiator is just a modification for the Riemann–Liouville differentiator [9,11]. The fractional-order integrator supposes that different constructs are not compatible and not constantly equivalent with each other. Actually, the fractional-order integrator is commonly employed for expressing an indefinite integral. In former research, there were only two endeavors attempted to establish a generalization of the fundamental theorem Symmetry 2023, 15, 938. https://doi.org/10.3390/sym15040938 https://www.mdpi.com/journal/symmetry