On the connection of the solvability of a nonlinear evolution equation of the fourth order and elliptic integrals

Abstract

The purpose of the present paper is to cover two intentions: Firstly we introduce a new computational algebraic procedure that can be applied to derive classes of solutions of nonlinear partial differential equations especially of higher order important in scientific and technical applications. The crucial step needs an auxiliary variable satisfying some ordinary differential equations of the first order containing sine, cosine and their hyperbolic varieties introducing for the first time connected by a special functionally dependence. Stated in most general form the solution manifold of these ordinary differential equations admits elliptic integrals and functions, respectively, as analytical solutions. Secondly the validity and reliability of the method is tested by its application to a rarely studied nonlinear evolution equation of the fourth order and leads to new classes of solutions. Nevertheless it should be emphasised that this technique does not need the solution of complicated nonlinear ordinary differential equations as caused the case of similarity reduction techniques. Further the algorithm works efficiently, is clearly structured and can be used in applications independently of the order of the equation. For computational purposes the method is appropriate to be written in any computer language. Therefore the given novel algebraic approach is suitable for wider classes of nonlinear partial differential equations in order to augment the solution manifold by a straightforward alternative approach.