Comparison between the (G’/G) - expansion method and the modified extended tanh method

Abstract

Abstract In this paper, firstly, we give a connection between well known and commonly used methods called the ( G ' G ) $\left( {{{G'} \over G}} \right)$ -expansion method and the modified extended tanh method which are often used for finding exact solutions of nonlinear partial differential equations (NPDEs). We demonstrate that giving a convenient transformation and formula, all of the solutions obtained by using the ( G ' G ) $\left( {{{G'} \over G}} \right)$ - expansion method can be converted the solutions obtained by using the modified extended tanh method. Secondly, contrary to the assertion in some papers, the ( G ' G ) $\left( {{{G'} \over G}} \right)$ -expansion method gives neither all of the solutions obtained by using the other method nor new solutions for NPDEs. Namely, while the modified extended tanh method gives more solutions in a straightforward, concise and elegant manner without reproducing a lot of different forms of the same solution. On the other hand, the ( G ' G ) $\left( {{{G'} \over G}} \right)$ -expansion method provides less solutions in a rather cumbersome form. Lastly, we obtain new exact solutions for the Lonngren wave equation as an illustrative example by using these methods.