Symbolic Implementation of Preliminary Group Classiffication for Ordinary Differential Equations

K S Govinder

doi:10.1155/2013/976271

Abstract

We apply the method of preliminary group classification to ordinary differential equations of the formy(n)=E(x,y,y',…,y(n-1),fi(x)), i=0,…,m.We show how this method can be implemented via the popular symmetry analysis packagePROGRAM LIEprovided that the package is supplemented with a small file. We then apply this method to an extension of the generalised Emden-Fowler equation given byy′′=∑i=25fixyi.

Highlights

We apply the method of preliminary group classification to ordinary differential equations of the form y(n) = E(x, y, y󸀠, . . . , y(n−1), fi(x)), i = 0, . . . , m. We show how this method can be implemented via the popular symmetry analysis package PROGRAM LIE provided that the package is supplemented with a small file
Lie symmetry analysis [1,2,3,4] is a useful tool in the analysis of differential equations
A major disadvantage of the Lie approach was that it was quite cumbersome—one had to wade through a system of linear partial differential equations in order to find the symmetries before being able to use the symmetries themselves

Summary

Introduction

Lie symmetry analysis [1,2,3,4] is a useful tool in the analysis of differential equations. One ends up with a differential equation for the arbitrary function, and one has to analyse this equation to determine the possible forms of the function which will lead to additional symmetries. This approach is largely dependent on the perspicacity and persistence of the practitioner; it is easy to omit relevant cases [10]. The method of preliminary group classification [11, 12] can be used to good effect in such situations Again, this method must largely be completed manually. We apply the method to an important equation in mathematical physics

Preliminary Group Classification

Implementation in PROGRAM LIE

Application to an Extended Emden-Fowler-Type Equation

Discussion