Lie Symmetry Solution Of Third Order First Degree Nonlinear Wave Equation Of Fourth Degree In Second Derivative

Abstract

In this study the method of Lie symmetry was used to determine the solution to a third order first degree nonlinear ordinary differential equation (ODE) fourth degree in sec ond derivative that arise in waves of systems like water in shallow oceans. Many third order nonlinear ordinary differential equations (ODEs) have been developed using numer ical methods like the finite difference but their solutions are just approximations within known boundary conditions or restrictions. To address such limitations, analytical Lie symmetry method which provides group invariant solutions was applied. This method does not depend on initial boundary values and gives exact solutions to problems. It has been shown what Lie symmetry analysis entails by reviewing some relevant nonlinear or dinary differential equations which have admitted it. The solution to nonlinear ordinary differential equation of the general form: G(x, y, y0 , y00, y000) = 0 that has not been developed by other earlier researchers has been worked out sequentially. A comprehensive Lie symmetry analysis carried out on this nonlinear ordinary differential equation included Lie groups, Lie symmetry generators, prolongations, invariant transfor mations, integrating factors and order reduction. The most significant Lie group theory application used was the order reduction of the nonlinear ODE from a third order to a first order which is easily solvable by other known simple methods. The objectives were to develop and determine both mathematical solution and general solution to a third order first degree nonlinear ODE of fourth degree in the second derivative, a special case of wave equation whose form was y 000 − y 0

y 00 y 4 = 0 using Lie symmetry method. Its solution is the source of knowledge and basis for further future research.