Computational Fractional Dynamical Systems

19(G′/G, 1/G)‐Expansion Method

19.1 Introduction

In recent decades, exact traveling wave solutions to nonlinear fractional differential equations have become more significant in studying complex physical and mechanical phenomena. The (G′/G,1/G)‐expansion method (Zayed et al. 2012; Zayed and Alurrfi 2014; Yasar and Giresunlu 2016; Al‐Shawba et al. 2018; Sirisubtawee et al. 2019; Duran 2021) has been utilized to obtain the solutions of time and space fractional differential equations. Recently, Jumarie (2006) proposed the modified Riemann–Liouville derivative. Using fractional complex transformation, one may convert the fractional differential equation into integer‐order differential equations (Li and He 2010). The original (G′/G)‐expansion approach assumes that a polynomial may describe the exact solutions of nonlinear PDEs in one variable (G′/G) satisfying the second‐order ordinary differential equation G^″(ξ) + λG^′(ξ) + μ = 0, where λ and μ are constants. On the other hand, the two variables (G′/G,1/G)‐expansion approach is an extension of the original (G′/G)‐expansion method. The two variables (G′/G,1/G)‐expansion approach is based on the assumption that a polynomial may describe exact traveling wave solutions to nonlinear PDEs in the two variables (G′/G) and (1/G), in which G = G(ξ) satisfies second‐order linear ODE, namely, G^″(ξ) + λG(ξ) = μ, where λ and μ are constants. The degree of this polynomial can be found by evaluating the homogeneous balance between the highest ...