Lie Symmetry Solution of Fourth Order Nonlinear Ordinary Differential Equation: (yy'(y(y') -1)'')'=0

Abstract

The equation F(x, y, y, y, y, y (4))  0 is a one-space dimension version of wave equation. Its solutions can be classified either as analytic or numerical using finite difference approach, where the convergence of the numerical schemes depends entirely on the initial and boundary values given. In this paper, we have used Lie symmetry analysis approach to solve the wave equation given since the solution does not depend on either boundary or initial values. Thus in our search for the solution we exploited a systematic procedure of developing infinitesimal transformations, generators, prolongations (extended transformations), variational symmetries, adjoint-symmetries, integrating factors and the invariant transformations of the problem. The procedure is aimed at lowering the order of the equation from fourth to first order, which is then solved to provide its Lie symmetry solution.