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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 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. / Thesis (Ph.D.)-University of Natal, 1995

We investigate the role of Lie symmetries in generating solutions to differential equations that arise in particular physical systems. We first provide an overview of the Lie analysis and review the relevant symmetry analysis of differential equations in general. The Lie symmetries of some simple ordinary differential equations are found t. illustrate the general method. Then we study the properties of particular ordinary differential equations that arise in astrophysics and cosmology using the Lie analysis of differential equations. Firstly, a system of differential equations arising in the model of a relativistic star is generated and a governing nonlinear equation is obtained for a linear equation of state. A comprehensive symmetry analysis is performed on this equation. Secondly, a second order nonlinear ordinary differential equation arising in the model of the early universe is described and a detailed symmetry analysis of this equation is undertaken. Our objective in each case is to find explicit Lie symmetry generators that may help in analysing the model. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2005.

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Mode of access: World Wide Web. / Thesis (Ph. D.)--University of Hawaii at Manoa, 2004. / Includes bibliographical references (leaves 159-164). / Electronic reproduction. / Also available by subscription via World Wide Web / x, 164 leaves, bound ill. 29 cm

146 leaves : tables ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematical Physics, 1979

Thesis (Ph.D.) -- University of Adelaide, Department of Mathematical Physics, 1979.

Diss.--Paris, 2003. / Literaturverz. S. [159] - 162.

Thesis (Honors)--College of William and Mary, 2008. / "This thesis includes work that was published by C.D. Carone and A. Rastogi, Phys. Rev. D." Includes bibliographical references (leaf 85). Also available via the World Wide Web.

Thesis (M. Phil.)--University of Hong Kong, 2008. / Includes bibliographical references (leaf 57-58) Also available in print.