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.
