A note on improved F -expansion method combined with Riccati equation applied to nonlinear evolution equations

Abstract

The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.

An analytical method for finding exact solutions of modified Korteweg–de Vries equation

“…abstract 25 In this present work, we have studied new extension of the (G 0 /G)-expansion method for finding the sol- 26 itary wave solutions of the modified Korteweg-de Vries (mKdV) equation. It has been shown that the pro- 27 posed method is effective and can be used for many other nonlinear evolution equations (NLEEs) in 28 mathematical physics.…”

New interaction solutions and nonlocal symmetries for the (2 + 1)-dimensional coupled Burgers equation

“…There are many methods to construct the exact solutions for different integrable systems, such as the Boussinesq equation, the unsteady KdV equation, the Benney-Luke equation, etc. (Akbar and Aliz, 2011;Akbar, Ali, and MohyudDin, 2013;Akter and Akbar, 2015;Akbar, 2013, 2015;Alam, Hafez, Akbar, and Roshid, 2015;Alam, Hafez, Belgacem, and Akbar, 2015;Islam, Khan, Akbar, and Mastroberardino, 2014;Akbar, 2013, 2016).…”

Exact solutions of the classical Boussinesq system

“…To seek the exact solutions of the nonlinear partial differential equations, many methods have been proposed, for instance the inverse scattering transformation (IST) (Ablowitz & Segur, 2000), B€ acklund transformation (Rogers & Schief, 2002), Painlev e analysis method (Chowdhury, 1999), Darboux transformation (DT) (Gu, Hu, & Zhou, 2005), Hirota direct method (Hirota, 2004), the tanh-function method (Parkes & Duffy, 1996;Zhang, Xu, & Li, 2002), the improved F-expansion method (Islam, Khan, Akbar, & Mastroberardino, 2014;Wang & Zhang, 2005), the modified simple equation method (Akter & Akbar, 2015;Khan, Akbar, & Alam, 2013), the (G 0 =G)-expansion and extend (G 0 =G)-expansion method (Akbar & Ali, 2011;Akbar, Ali, & Mohyud-Din, 2013;Alam, Hafez, Belgacem, & Akbar, 2015b;Zayed & Shorog, 2010), the Exp-function method (He & Abdou, 2007;, the generalized Kudryashov method (Khan & Akbar, 2016), the expðÀUðgÞÞ-expansion and expðUðgÞÞ method (Alam, Hafez, Akbar, & Roshid, 2015a;Roshid & Rahman, 2014), the extended three-wave method Li, Dai, & Liu, 2011;Singh & Gupta, 2016;Wang, Dai, & Liang, 2010). In this paper, based on the bilinear form, we consider exact solutions including solitary wave solution, periodic solitary solution and rational solution of the classical Boussinesq (CB) system (see Wu & Zhang, 1996) …”