On an improved method for solving evolution equations of higher order importantly in science and technology

Abstract

In this paper we introduce a new algebraic procedure to compute new classes of solutions of (1+1)-nonlinear partial differential equations (nPDEs) both of physical and technical relevance. The basic assumption is that the unknown solution(s) of the nPDE under consideration satisfy an ordinary differential equation (ODE) of the first order that can be integrated completely. This solution manifold of these first-order ODEs play an essential part in solving given nPDEs. A further important aspect however is the fact that we have the freedom in choosing some parameters bearing positively on the algorithm and hence, the solution-manifold of any nPDEs under consideration are therefore augmented naturally. The present algebraic procedure can widely use to study many nPDE and is not only restricted to time-dependent problems. We note that no numerical methods are necessary and so analytical closed-form classes of solutions result. The algorithm works accurately, is clear structured and can be converted in any computer language. On the contrary it is worth to stress the necessity of such sophisticated methods since a general theory of nPDEs does not exist.

Keywords: Nonlinear partial differential equations, evolution equations, special function methods.