Analytical Insight into Some Fractional Nonlinear Dynamical Systems Involving the Caputo Fractional Derivative Operator

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Detalles Bibliográficos
Publicado en:	Fractal and Fractional vol. 9, no. 5 (2025), p. 320
Autor principal:	AlBaidani, Mashael M
Publicado:	MDPI AG
Materias:	United States > US Calculus Applied mathematics Physics Partial differential equations Convergence Mathematical analysis Shallow water equations Iterative methods Operators (mathematics) Water waves Signal processing Effectiveness Decomposition Algorithms Methods Fractional calculus Nonlinear systems Differential equations Graphical representations Wave equations Dynamical systems Perturbation methods Ordinary differential equations Approximation
Acceso en línea:	Citation/Abstract Full Text + Graphics Full Text - PDF
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Descripción
Resumen:	This work explores modern mathematical avenues as part of fractional calculus research. We apply fractional dispersion relations to the fractional wave equation to numerically examine various formulations of the generalized fractional wave equation. The research explores Drinfeld–Sokolov–Wilson and shallow water equations as fundamental differential equations forming the basis of wave theory studies. This work presents effective methods to obtain the numerical solution of the fractional-order FDSW and FSW coupled system equations. The analysis employs Caputo fractional derivatives during studies of fractional orders. This study develops the new iterative transform technique (NITM) and homotopy perturbation transform method (HPTM) using Elzaki transform (ET) with a new iteration method and a homotopy perturbation method. The proposed techniques generate approximation solutions that adopt an infinite fractional series with fractional order solutions converging towards analytic integer solutions. The proposed method demonstrates its precision through tabular simulations of computed approximations and their absolute error values while representing results with 2D and 3D graphics. The paper presents the physical analysis of solution dynamics across diverse <inline-formula>ϵ</inline-formula> ranges during a suitable time frame. The developed computational techniques yield numerical and graphical output, which are compared to analytic results to verify the solution convergence. The computational algorithms have proven their high accuracy, flexibility, effectiveness, and simplicity in evaluating fractional-order mathematical models.
ISSN:	2504-3110
DOI:	10.3390/fractalfract9050320
Fuente:	Engineering Database