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13 pages, 9670 KiB

Open AccessArticle

Exact Solitary Wave Solutions and Sensitivity Analysis of the Fractional (3+1)D KdV–ZK Equation

by Asif Khan, Fehaid Salem Alshammari, Sadia Yasin and Beenish

Fractal Fract. 2025, 9(7), 476; https://doi.org/10.3390/fractalfract9070476 - 21 Jul 2025

Viewed by 293

The present paper examines a novel exact solution to nonlinear fractional partial differential equations (FDEs) through the Sardar sub-equation method (SSEM) coupled with Jumarie’s Modified Riemann–Liouville derivative (JMRLD). We take the (3+1)-dimensional space–time fractional modified Korteweg-de Vries (KdV) -Zakharov-Kuznetsov (ZK) equation as a case study, which describes some intricate phenomena of wave behavior in plasma physics and fluid dynamics. With the implementation of SSEM, we yield new solitary wave solutions and explicitly examine the role of the fractional-order parameter in the dynamics of the solutions. In addition, the sensitivity analysis of the results is conducted in the Galilean transformation in order to ensure that the obtained results are valid and have physical significance. Besides expanding the toolbox of analytical methods to address high-dimensional nonlinear FDEs, the proposed method helps to better understand how fractional-order dynamics affect the nonlinear wave phenomenon. The results are compared to known methods and a discussion about their possible applications and limitations is given. The results show the effectiveness and flexibility of SSEM along with JMRLD in forming new categories of exact solutions to nonlinear fractional models. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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21 pages, 1723 KiB

Open AccessArticle

Exploring Solitons Solutions of a (3+1)-Dimensional Fractional mKdV-ZK Equation

by Amjad E. Hamza, Osman Osman, Muhammad Umair Sarwar, Khaled Aldwoah, Hicham Saber and Manel Hleili

Fractal Fract. 2024, 8(9), 498; https://doi.org/10.3390/fractalfract8090498 - 24 Aug 2024

Cited by 1 | Viewed by 1213

Abstract

This study presents the application of the

ϕ^{6}

This study presents the application of the

ϕ^{6}

model expansion technique to find exact solutions for the (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation under Jumarie’s modified Riemann–Liouville derivative (JMRLD). The suggested method captures dark, periodic, traveling, and singular soliton solutions, providing deep insights into wave behavior. Clear graphics demonstrate that the solutions are greatly affected by changes in the fractional order, deepening our understanding and revealing the hidden dynamics of wave propagation. The considered equation has several applications in fluid dynamics, plasma physics, and nonlinear optics. Full article

(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)

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15 pages, 78781 KiB

Open AccessArticle

New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method

by Mariam Sultana, Uroosa Arshad, Md. Nur Alam, Omar Bazighifan, Sameh Askar and Jan Awrejcewicz

Symmetry 2021, 13(12), 2296; https://doi.org/10.3390/sym13122296 - 2 Dec 2021

Cited by 7 | Viewed by 2108

Abstract

Symmetry performs an essential function in finding the correct techniques for solutions to time space fractional differential equations (TSFDEs). In this article, we present the Novel Analytic Method (NAM) for approximate solutions of the linear and non-linear KdV equation for TSFDs. To enunciate the non-integer derivative for the aforementioned equation, the Caputo operator is manipulated. Furthermore, the formula implemented is a numerical way that is postulated from Taylor’s series, which confirms an analytical answer in the form of a convergent series. For delineation of the efficiency and functionality of the method in question, four applications are exemplified along with graphical interpretation and numerical solutions to finitely illustrate the behavior of the solution to this equation. Moreover, the 3D graphs of some of these numerical examples are plotted with specific values. Observing the effectiveness of this process, we can easily decide that this process can be implemented to other TSFDEs applied in the mathematical modeling of a real-world aspect. Full article

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29 pages, 19412 KiB

Open AccessArticle

Some Applications of the (G′/G,1/G)-Expansion Method for Finding Exact Traveling Wave Solutions of Nonlinear Fractional Evolution Equations

by Sekson Sirisubtawee, Sanoe Koonprasert and Surattana Sungnul

Symmetry 2019, 11(8), 952; https://doi.org/10.3390/sym11080952 - 26 Jul 2019

Cited by 28 | Viewed by 4544

Abstract

In this paper, the

(G^{'} / G, 1 / G)

In this paper, the

(G^{'} / G, 1 / G)

-expansion method is applied to acquire some new, exact solutions of certain interesting, nonlinear, fractional-order partial differential equations arising in mathematical physics. The considered equations comprise the time-fractional, (2+1)-dimensional extended quantum Zakharov-Kuznetsov equation, and the space-time-fractional generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) system in the sense of the conformable fractional derivative. Applying traveling wave transformations to the equations, we obtain the corresponding ordinary differential equations in which each of them provides a system of nonlinear algebraic equations when the method is used. As a result, many analytical exact solutions obtained of these equations are expressed in terms of hyperbolic function solutions, trigonometric function solutions, and rational function solutions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, solitary wave solutions of kink type, and so on. The method is very efficient, powerful, and reliable for solving the proposed equations and other nonlinear fractional partial differential equations with the aid of a symbolic software package. Full article

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15 pages, 842 KiB

Open AccessArticle

Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation

by Wen Cao, Yufeng Xu and Zhoushun Zheng

Appl. Sci. 2018, 8(1), 42; https://doi.org/10.3390/app8010042 - 1 Jan 2018

Cited by 18 | Viewed by 4929

Abstract

In this paper, we studied the numerical solution of a time-fractional Korteweg–de Vries (KdV) equation with new generalized fractional derivative proposed recently. The fractional derivative employed in this paper was defined in Caputo sense and contained a scale function and a weight function. A finite difference/collocation scheme based on Jacobi–Gauss–Lobatto (JGL) nodes was applied to solve this equation and the corresponding stability was analyzed theoretically, while the convergence was verified numerically. Furthermore, we investigated the behavior of solution of the generalized KdV equation depending on its parameter

δ

, scale function

z (t)

in fractional derivative. We found that the full discrete scheme was effective to obtain a numerical solution of the new KdV equation with different conditions. The wave number

δ

in front of the third order space derivative term played a significant role in splitting a soliton wave into multiple small pieces. Full article

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