Exact Solutions to Some Nonlinear Time-Fractional Evolution Equations Using the Generalized Kudryashov Method in Mathematical Physics

Abstract

In this study, we utilize the potent generalized Kudryashov method to address the intricate obstacles presented by fractional differential equations in the field of mathematical physics. Specifically, our focus centers on obtaining novel exact solutions for three pivotal equations: the time-fractional seventh-order Sawada-Kotera-Ito equation, the time-fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation, and the time-fractional seventh-order Kaup–Kupershmidt equation. The generalized Kudryashov method, celebrated for its versatility and efficacy in addressing intricate nonlinear problems, plays a central role in our research. This method not only simplifies the equations but also unveils their inner dynamics, rendering them amenable to meticulous analysis. It is worth noting that our fractional derivatives are defined in the context of the conformable fractional derivative, providing a solid foundation for our mathematical investigations. One notable aspect of our study is the visual representation of our findings. Graphical representations of the yielded solutions enliven intricate mathematical structures, providing a concrete insight into the dynamics and behaviors of said equations. This paper highlights the proficiency of the generalized Kudryashov method in resolving complex issues presented by fractional differential equations. Our study not only broadens the range of mathematical methods but also enhances our comprehension of the intriguing realm of nonlinear physical phenomena.

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Chaotic Phenomena, Sensitivity Analysis, Bifurcation Analysis, and New Abundant Solitary Wave Structures of The Two Nonlinear Dynamical Models in Industrial Optimization

“…It is highly important to investigate the progressive wave-like solution for the best perception of NLPDEs and their application in real life. Lately, different kinds of techniques have been exhibited for generating numerical and analytical demonstrations by many experts, such as the Riccati equation method [1], the F-expansion technique [2,3], the auxiliary equation method [4,5], the Jacobi elliptic function method [6,7], the direct algebraic function technique [8,9], the Cole-Hopf conversion technique [10,11], the tanhfunction method [12,13], the Backlund transform technique [14,15], the Hirota's bilinear technique [16][17][18], the exp(−ϕ(ξ))-expansion method [19,20], the generalized Kudryashov method [21,22], the homotopy exploration technique [23], the homogeneous balance technique [24][25][26], the variational iteration method [27], the sine cosine algorithm [28], and the G ′ G ′ +G+A -expansion technique [29][30][31]. In addition, the (G ′ /G)-expansion technique was introduced by Wang et al for describing the outcomes of NLPDEs [32].…”

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