A New Version of the Generalized F-Expansion Method for the Fractional Biswas-Arshed Equation and Boussinesq Equation with the Beta-Derivative

Abstract

In this article, a new version of the generalized F-expansion method is proposed enabling to obtain the exact solutions of the Biswas-Arshed equation and Boussinesq equation defined by Atangana’s beta-derivative. First, the new version generalized F-expansion method is introduced, and then, the exact solutions of the nonlinear fractional differential equations expressed with Atangana’s beta-derivative are given. When the results are examined, it is seen that single, combined, and mixed Jacobi elliptic function solutions are obtained. From the point of view, it is understood that the new version generalized F-expansion method can give significant results in finding the exact solutions of equations containing beta-derivatives.

New exact solitary waves for the Sasa-Satsuma model with variable coefficients

“…In recent years, a number of significant approaches have been established, such as the Riemann-Hilbert approach [26,27], the tanh-coth technology [28,29], the Darboux transformation technology [30,31], the Fexpansion technology [32][33][34], the sine-Gordon expansion technology [35,36] and so on. The Weierstrass elliptic function expansion technology and the Jacobi elliptic function expansion technology are very effective and straightforward methods for constructing traveling wave solutions of the nonlinear evolutionary equations [37][38][39].…”

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].…”