Citation: Naeem, M.; Yasmin, H.; Shah, R.; Shah, N.A.; Nonlaopon, K. Investigation of Fractional Nonlinear Regularized Long-Wave Models via Novel Techniques. Symmetry 2023, 15, 220. https://doi.org/10.3390/ sym15010220 Academic Editors: Dongfang Li, Hongyu Qin, Xiaoli Chen and Calogero Vetro Received: 16 November 2022 Revised: 16 December 2022 Accepted: 3 January 2023 Published: 12 January 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 Investigation of Fractional Nonlinear Regularized Long-Wave Models via Novel Techniques Muhammad Naeem 1,† , Humaira Yasmin 2 , Rasool Shah 3 , Nehad Ali Shah 4,† and Kamsing Nonlaopon 5, * 1 Department of Mathematics, Umm Al-Qura University, Makkah 517, Saudi Arabia 2 Department of Basic Sciences, King Faisal University, Al-Ahsa 31982, Saudi Arabia 3 Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan 4 Department of Mechanical Engineering, Sejong University, Seoul 05006, Republic of Korea 5 Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand * Correspondence: nkamsi@kku.ac.th or kamsingn@yahoo.com † These authors contributed equally to this work and are co-first authors. Abstract: The main goal of the current work is to develop numerical approaches that use the Yang transform, the homotopy perturbation method (HPM), and the Adomian decomposition method to analyze the fractional model of the regularized long-wave equation. The shallow-water waves and ion-acoustic waves in plasma are both explained by the regularized long-wave equation. The first method combines the Yang transform with the homotopy perturbation method and He’s polynomials. In contrast, the second method combines the Yang transform with the Adomian polynomials and the decomposition method. The Caputo sense is applied to the fractional derivatives. The strategy’s effectiveness is shown by providing a variety of fractional and integer-order graphs and tables. To confirm the validity of each result, the technique was substituted into the equation. The described methods can be used to find the solutions to these kinds of equations as infinite series, and when these series are in closed form, they give the precise solution. The results support the claim that this approach is simple, strong, and efficient for obtaining exact solutions for nonlinear fractional differential equations. The method is a strong contender to contribute to the existing literature. Keywords: nonlinear regularized long-wave model; Adomian decomposition method; homotopy perturbation method; Caputo operator; Yang transform 1. Introduction As far back as the classical integer order analysis goes, fractional-order calculus studies have a long history. However, have not been utilized in the physical sciences for a long time. Furthermore, over the past few decades, applications of fractional calculus in applied mathematics, control [1], viscoelasticity [2], electromagnetic [3], and electrochemistry [4] have grown in popularity. The advancement of symbolic computation software has further aided this growth. Fractional derivatives and integrals can be used to represent a variety of multidisciplinary applications. Some basic explanations and applications of fractional calculus are provided in [5,6]. In [7], the existence and distinctiveness of the solutions are also explored. Fractional derivatives and integrals have recently received new definitions from several scientists and engineers, who have utilized them to describe a variety of physical phenomena. In a study, Vazquez [8] provided a brief, non-exhaustive, compre- hensive overview of the mathematical tool connected to fractional-order derivatives and integrals, along with an interpretation of various domains where they are either are being used or may one day be used. The existence, uniqueness, and regularity of solutions to the heat equation of the arbitrary order were investigated by Bonfortea et al. in a research study [9]. Excellent research on differential equations of fractional-order derivatives and their uses in bioengineering may be found in a monograph by Magin [10]. Magin [11] Symmetry 2023, 15, 220. https://doi.org/10.3390/sym15010220 https://www.mdpi.com/journal/symmetry