In this paper, the G ′ G expansion method with the aid of computer algebraic system Maple, is proposed for seeking the travelling wave solutions for the a class of nonlinear pseudoparabolic equations. The method is straightforward and concise, and it be also applied to other nonlinear pseudoparabolic equations. We studied mostly important four nonlinear pseudoparabolic physical models : the Benjamin-Bona-Mahony-Peregrine-Burger(BBMPB) equation, the Oskolkov-Benjamin-Bona-Mahony-Burgers(OBBMB) equation, the one-dimensional Oskolkov equation and the generalized hyperbolic-elastic-rod wave equation.

Abstract:In this paper, we show that the unified method is not only more general than the family of tanh function methods, also it gives many more general solutions than the new members of the family ofCompared to other methods, the significant contribution of the unified method is firstly to unify the family of tanh function methods and the family of G ′ G -expansion methods. Secondly, it gives many more solutions for NPDEs direct, concise and simple manner than the total of these two families. Also, the unified method gives these abundant solutions without using tedious and complex algorithm on computer programs. Afterwards, we demonstrate the effectiveness of the unified tanh method by seeking more exact solutions of the Lonngren wave equation.

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Over the last twenty years, several "different" hyperbolic tangent function methods have been proposed to search solutions for nonlinear partial differential equations(NPDEs). The most common of these methods were the tanh-function method, the extended tanh-function method, the modified extended tanh-function method, and the complex tanh-function method. Besides the excellent sides of these methods, weaknesses and deficiencies of each method were encountered. The authors realized that they did not actually give "very different and comprehensive results", and some of them are even unnecessary. Therefore, these methods were analysed and significant findings obtained. Firstly, they compared all of these methods with each other and gave the connections between them; and secondly, they proposed a more general method to obtain many more solutions for NPDEs, some of which having never been obtained before, and thus to overcome weaknesses and deficiencies of existing hyperbolic tangent function methods in the literature. This new method, named as the unified method, provides many more solutions in a straightforward, concise and elegant manner without reproducing a lot of different forms of the same solution. Lastly, they demonstrate the effectiveness of the unifed tanh method by seeking more exact solutions of the Rabinovich wave equation which were not obtained before.

In this work, we are interested in finding exact and numerical solutions of well‐known Kolmogorov–Petrovsky–Piskunov (KPP) equation. Here, we introduce modified extended tanh method for finding the exact solutions of KPP equation and an average linear finite difference scheme for numerical investigations. It has been observed that the numerical scheme so employed is stable and accurate enough to produce stable results.