On non-linear evolution equations of higher order – the introduction and application of a novel computational approach

Abstract

The purpose of the present paper is to introduce a new computational algebraic procedure that can easily be applied to derive class of solutions of non-linear partial differential equations (nPDE) especially of higher order. The crucial step needs an auxiliary variable satisfying some ordinary differential equations (ODE) of first order containing sine, cosine and their hyperbolic varieties introducing to the first time. General transformations are given to determine class of solutions explicitly. The validity and reliability of the method is tested by its application to some important non-linear evolution equations leading to new class of solutions with physical significance. Nevertheless it should be emphasised that this techniques do not need the solution of complicate nODEs as in the case of similarity reduction. Further, the algorithm works efficiently, is clear structured and can be used in any applications independent of the order of the nPDE. For computational purposes the method is appropriate to rewrite it in any computer languages. Therefore, the given novel algebraic approach is suitable for a wider class of nPDE in order to augment the solution manifold by a straightforward alternative approach.