# Extensions and generalisations of Lie analysis.

## UKZN ResearchSpace

 dc.contributor.advisor Leach, Peter Gavin Lawrence. dc.creator Govinder, Kesh S. dc.date.accessioned 2012-01-23T12:47:26Z dc.date.available 2012-01-23T12:47:26Z dc.date.created 1995 dc.date.issued 1995 dc.identifier.uri http://hdl.handle.net/10413/4878 dc.description Thesis (Ph.D.)-University of Natal, 1995 en dc.description.abstract The Lie theory of extended groups applied to differential equations is arguably one of the most successful methods in the solution of differential equations. In fact, the theory unifies a number of previously unrelated methods into en a single algorithm. However, as with all theories, there are instances in which it provides no useful information. Thus extensions and generalisations of the method (which classically employs only point and contact transformations) are necessary to broaden the class of equations solvable by this method. The most obvious extension is to generalised (or Lie-Backlund) symmetries. While a subset of these, called contact symmetries, were considered by Lie and Backlund they have been thought to be curiosities. We show that contact transformations have an important role to play in the solution of differential equations. In particular we linearise the Kummer-Schwarz equation (which is not linearisable via a point transformation) via a contact transformation. We also determine the full contact symmetry Lie algebra of the third order equation with maximal symmetry (y'''= 0), viz sp(4). We also undertake an investigation of nonlocal symmetries which have been shown to be the origin of so-called hidden symmetries. A new procedure for the determination of these symmetries is presented and applied to some examples. The impact of nonlocal symmetries is further demonstrated in the solution of equations devoid of point symmetries. As a result we present new classes of second order equations solvable by group theoretic means. A brief foray into Painleve analysis is undertaken and then applied to some physical examples (together with a Lie analysis thereof). The close relationship between these two areas of analysis is investigated. We conclude by noting that our view of the world of symmetry has been clouded. A more broad-minded approach to the concept of symmetry is imperative to successfully realise Sophus Lie's dream of a single unified theory to solve differential equations. dc.language.iso en en dc.subject Lie algebras. en dc.subject Lie groups. en dc.subject Theses--Mathematics. en dc.title Extensions and generalisations of Lie analysis. en dc.type Thesis en