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Fractal Fract
Volume 9
Issue 12
10.3390/fractalfract9120770
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Article

A Note on Solutions of Fractional Third-Order Dispersive Partial Differential Equations Using the Natural Generalized Laplace Transform Decomposition Method

by
Hassan Eltayeb
*,
Shayea Aldossari
and
Said Mesloub
Mathematics Department, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(12), 770; https://doi.org/10.3390/fractalfract9120770
Submission received: 22 October 2025 / Revised: 20 November 2025 / Accepted: 21 November 2025 / Published: 25 November 2025
(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus in Computational Physics)
Versions Notes

Abstract

The present research offers reliable analytical solutions for time-fractional linear and nonlinear dispersive Korteweg–de Vries (dKdV)-type equations by employing the Natural Generalized Laplace Transform Decomposition Method (NGLTDM). The nonlinear differential dispersive Korteweg–de Vries (dKdV) equation involves a nonlinear derivative term that depends on ϕ and its partial derivative with respect to x. We employ Adomian polynomials to deal with this nonlinear part, and we utilize the Caputo derivative to illustrate the fractional part of the equation. The work provides exact theorems regarding the stability, convergence, and accuracy of the generated solutions. Illustrative examples demonstrate the effectiveness and precision of the method by delivering solutions for quickly converging series with easily calculable coefficients. We use Maple software to show graphical comparisons between the approximate and exact solutions to show how rapidly the method converges.
Keywords: fractional dKdV equation; one-dimensional time-fractional equation; NGLT; DM; DNGLTDM fractional dKdV equation; one-dimensional time-fractional equation; NGLT; DM; DNGLTDM

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MDPI and ACS Style

Eltayeb, H.; Aldossari, S.; Mesloub, S. A Note on Solutions of Fractional Third-Order Dispersive Partial Differential Equations Using the Natural Generalized Laplace Transform Decomposition Method. Fractal Fract. 2025, 9, 770. https://doi.org/10.3390/fractalfract9120770

AMA Style

Eltayeb H, Aldossari S, Mesloub S. A Note on Solutions of Fractional Third-Order Dispersive Partial Differential Equations Using the Natural Generalized Laplace Transform Decomposition Method. Fractal and Fractional. 2025; 9(12):770. https://doi.org/10.3390/fractalfract9120770

Chicago/Turabian Style

Eltayeb, Hassan, Shayea Aldossari, and Said Mesloub. 2025. "A Note on Solutions of Fractional Third-Order Dispersive Partial Differential Equations Using the Natural Generalized Laplace Transform Decomposition Method" Fractal and Fractional 9, no. 12: 770. https://doi.org/10.3390/fractalfract9120770

APA Style

Eltayeb, H., Aldossari, S., & Mesloub, S. (2025). A Note on Solutions of Fractional Third-Order Dispersive Partial Differential Equations Using the Natural Generalized Laplace Transform Decomposition Method. Fractal and Fractional, 9(12), 770. https://doi.org/10.3390/fractalfract9120770

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