Conditional symmetry properties for ordinary differential equations
Date
2015-05-07
Authors
Fatima, Aeeman
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
This work deals with conditional symmetries of ordinary di erential equations
(ODEs). We re ne the de nition of conditional symmetries of systems of ODEs
in general and provide an algorithmic viewpoint to compute such symmetries
subject to root di erential equations. We prove a proposition which gives important
and precise criteria as to when the derived higher-order system inherits
the symmetries of the root system of ODEs. We rstly study the conditional
symmetry properties of linear nth-order (n 3) equations subject to root linear
second-order equations. We consider these symmetries for simple scalar higherorder
linear equations and then for arbitrary linear systems. We prove criteria
when the derived scalar linear ODEs and even order linear system of ODEs inherit
the symmetries of the root linear ODEs. There are special symmetries such
as the homogeneity and solution symmetries which are inherited symmetries. We
mention here the constant coe cient case as well which has translations of the
independent variable symmetry inherited. Further we show that if a system of
ODEs has exact solutions, then it admits a conditional symmetry subject to the
rst-order ODEs related to the invariant curve conditions which arises from the
known solution curves. This is even true if the system has no Lie point sym
Description
A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2015.