Traveling wave solutions to the nonlinear space–time fractional extended KdV equation via efficient analytical approaches

Abstract

In this paper, the nonlinear space-time fractional extended Korteweg-de Vries (F-eKdV) equation involving the conformable fractional derivative (CFD) is considered. Bernoulli and Riccati equations as a simplest equation method together with applying modified traveling wave transformation are used to obtain the exact solution of F-eKdV equation. Also, by exp⁡(−Φ(ζ))-expansion method, different kind of exact traveling wave solutions are gained, including periodic-singular, dark-singular and plane-wave solutions. As a result, several soliton solutions of the proposed equation are revealed. These novel soliton waves are constructed using the software package Maple with some given parametric values and displayed graphically in 2-dimensional (2D) and 3-dimensional (3D) plots by using MATLAB. To illustrate how the fractional operator affects the results, some of the acquired solutions are presented for various values of the fractional order α and also compared to their exact solutions in classical case (i.e. α=1). In the given graphical representations, the physical meaning of the reported solutions is illustrated. From the obtained results, it is observed that the methods are effective and dependable to solve nonlinear fractional partial differential equations (NFPDEs).