The new types of wave solutions of the Burger's equation and the Benjamin–Bona–Mahony equation

Abstract

In this article, we suggest the two variable (G′/G, 1/G)-expansion method for extracting further general closed form wave solutions of two important nonlinear evolution equations (NLEEs) that model one-dimensional internal waves in deep water and the long surface gravity waves of small amplitude propagating uni-directionally. The method can be regarded as an extension of the(G′/G)-expansion method. The ansatz of this extension method to obtain the solution is based on homogeneous balance between the highest order dispersion terms and nonlinearity which is similar to the (G′/G) method whereas the auxiliary linear ordinary differential equation (LODE) and polynomial solution differs. We applied this method to find explicit form solutions to the Burger's and Benjamin–Bona–Mahony (BBM) equations to examine the effectiveness of the method and tested through mathematical computational software Maple. Some new exact travelling wave solutions in more general form of these two nonlinear equations are derived by this extended method. The method introduced here appears to be easier and faster comparatively by means of symbolic computation system.