Fractional View Analysis of Kuramoto–Sivashinsky Equations with Non-Singular Kernel Operators

Abstract

In this article, we investigate the nonlinear model describing the various physical and chemical phenomena named the Kuramoto–Sivashinsky equation. We implemented the natural decomposition method, a novel technique, mixed with the Caputo–Fabrizio (CF) and Atangana–Baleanu deriavatives in Caputo manner (ABC) fractional derivatives for obtaining the approximate analytical solution of the fractional Kuramoto–Sivashinsky equation (FKS). The proposed method gives a series form solution which converges quickly towards the exact solution. To show the accuracy of the proposed method, we examine three different cases. We presented proposed method results by means of graphs and tables to ensure proposed method validity. Further, the behavior of the achieved results for the fractional order is also presented. The results we obtain by implementing the proposed method shows that our technique is extremely efficient and simple to investigate the behaviour of nonlinear models found in science and technology.

Solving the fractional Fornberg-Whitham equation within Caputo framework using the optimal auxiliary function method

“…The primary objective of this research paper is to address the fractional order Fornberg-Whitham (FW) equation by employing the optimal auxiliary function method, with a specific focus on the fractional derivative in the Caputo sense. Additionally, this investigation explores the physical characteristics of the solution it provides using 3D and contour plots, while examining various fractional order values across three distinct scenarios [34][35][36][37].…”

Fractional Fokas-Lenells equation: analyzing travelling waves via advanced analytical method

“…Lastly, Kai and Yin explore Gaussian traveling wave solutions to Schrdinger equations with logarithmic nonlinearity [14] and investigate the linear structure and soliton molecules of the Sharma-Tasso-Olver-Burgers equation [15]. These studies collectively highlight the interdisciplinary nature of contemporary research, emphasizing the fusion of mathematical modeling, engineering innovation, and physical phenomena analysis [16][17][18][19][20].…”

Unraveling the Dynamics of Singular Stochastic Solitons in Stochastic Fractional Kuramoto–Sivashinsky Equation

“…SFDEs are useful in finance for asset price modeling and risk assessment and in physics, biology, geophysics, and environmental science for understanding various natural phenomena. They are also useful in control theory, signal processing, and image analysis, providing novel solutions to difficult system and data analysis issues [4][5][6][7][8].…”

“…The soliton solutions to (6) are then explored by determining the unidentified coefficients and additional parameters and placing them in (8) together with the χ(Ω) (general solution of ( 9)). The families of soliton solutions shown below may be produced using this generic solution of (9).…”