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Article

A Novel Approach to Solving Generalised Nonlinear Dynamical Systems Within the Caputo Operator

by
Mashael M. AlBaidani
* and
Rabab Alzahrani
Department of Mathematics, College of Science and Humanities, Prince Sattam Bin Abdulaziz University, Al Kharj 11942, Saudi Arabia
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(8), 503; https://doi.org/10.3390/fractalfract9080503 (registering DOI)
Submission received: 28 June 2025 / Revised: 22 July 2025 / Accepted: 30 July 2025 / Published: 31 July 2025
(This article belongs to the Special Issue Recent Trends in Computational Physics with Fractional Applications)

Abstract

In this study, we focus on solving the nonlinear time-fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations, using the Yang transform iterative method (YTIM). This method combines the Yang transform with a new iterative scheme to construct reliable and efficient solutions. Readers can understand the procedures clearly, since the implementation of Yang transform directly transforms fractional derivative sections into algebraic terms in the given problems. The new iterative scheme is applied to generate series solutions for the provided problems. The fractional derivatives are considered in the Caputo sense. To validate the proposed approach, two numerical examples are analysed and compared with exact solutions, as well as with the results obtained from the fractional reduced differential transform method (FRDTM) and the q-homotopy analysis transform method (q-HATM). The comparisons, presented through both tables and graphical illustrations, confirm the enhanced accuracy and reliability of the proposed method. Moreover, the effect of varying the fractional order is explored, demonstrating convergence of the solution as the order approaches an integer value. Importantly, the time-fractional Hirota–Satsuma coupled KdV and modified Korteweg–de Vries (MKdV) equations investigated in this work are not only of theoretical and computational interest but also possess significant implications for achieving global sustainability goals. Specifically, these equations contribute to the Sustainable Development Goal (SDG) “Life Below Water” by offering advanced modelling capabilities for understanding wave propagation and ocean dynamics, thus supporting marine ecosystem research and management. It is also relevant to SDG “Climate Action” as it aids in the simulation of environmental phenomena crucial to climate change analysis and mitigation. Additionally, the development and application of innovative mathematical modelling techniques align with “Industry, Innovation, and Infrastructure” promoting advanced computational tools for use in ocean engineering, environmental monitoring, and other infrastructure-related domains. Therefore, the proposed method not only advances mathematical and numerical analysis but also fosters interdisciplinary contributions toward sustainable development.
Keywords: time-fractional Hirota–Satsuma coupled KdV; modified KdV; Yang transform; Caputo operator; new iterative method time-fractional Hirota–Satsuma coupled KdV; modified KdV; Yang transform; Caputo operator; new iterative method

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MDPI and ACS Style

AlBaidani, M.M.; Alzahrani, R. A Novel Approach to Solving Generalised Nonlinear Dynamical Systems Within the Caputo Operator. Fractal Fract. 2025, 9, 503. https://doi.org/10.3390/fractalfract9080503

AMA Style

AlBaidani MM, Alzahrani R. A Novel Approach to Solving Generalised Nonlinear Dynamical Systems Within the Caputo Operator. Fractal and Fractional. 2025; 9(8):503. https://doi.org/10.3390/fractalfract9080503

Chicago/Turabian Style

AlBaidani, Mashael M., and Rabab Alzahrani. 2025. "A Novel Approach to Solving Generalised Nonlinear Dynamical Systems Within the Caputo Operator" Fractal and Fractional 9, no. 8: 503. https://doi.org/10.3390/fractalfract9080503

APA Style

AlBaidani, M. M., & Alzahrani, R. (2025). A Novel Approach to Solving Generalised Nonlinear Dynamical Systems Within the Caputo Operator. Fractal and Fractional, 9(8), 503. https://doi.org/10.3390/fractalfract9080503

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