Computational Fractional Dynamical Systems

17(G^′/G)‐Expansion Method

17.1 Introduction

Several mathematical approaches (Diethelm et al. 2002; Erturk and Momani 2008; Behroozifar and Ahmadpour 2017; Jena et al. 2020a, 2020b) have been developed for solving fractional differential equations. However, finding exact solutions to nonlinear fractional differential equations (FDEs) was challenging until (Li and He 2010) introduced a fractional complex transform. Fractional complex transform turns FDEs into ordinary differential equations (ODEs), allowing all analytical methods used to solve ODEs. The (G′/G)‐expansion method (Bin 2012; Gepreel and Omran 2012; Bekir and Guner 2013; Bekir and Guner 2014; Bekir et al. 2015; Khan et al. 2019) is a powerful method among them to solve FDEs. One advantage of this technique is that it allows us to find the exact solution of FDEs without initial or boundary conditions. Exact solutions can be achieved via this method by solving a set of linear or nonlinear algebraic equations.

17.2 Description of the (G′/G)‐Expansion Method

This part will briefly describe the main steps of this method for solving fractional partial differential equations.

Step 1. In order to understand the (G′/G)‐expansion method (Bin 2012; Gepreel and Omran 2012; Bekir et al. 2015), let us consider the following nonlinear fractional partial differential equation in two independent variables x and t of the type

(17.1)

where u = u(x, t) is an unknown function, and Q is a polynomial of u and its partial fractional ...