Masters Degrees (Pure Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10413/7121

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Continuous symmetries of difference equations.
(2011) Nteumagne, Bienvenue Feugang.; Govinder, Keshlan Sathasiva.
We consider the study of symmetry analysis of difference equations. The original work done by Lie about a century ago is known to be one of the best methods of solving differential equations. Lie's theory of difference equations on the contrary, was only first explored about twenty years ago. In 1984, Maeda [42] constructed the similarity methods for difference equations. Some work has been done in the field of symmetries of difference equations for the past years. Given an ordinary or partial differential equation (PDE), one can apply Lie algebra techniques to analyze the problem. It is commonly known that the number of independent variables can be reduced after the symmetries of the equation are obtained. One can determine the optimal system of the equation in order to get a reduction of the independent variables. In addition, using the method, one can obtain new solutions from known ones. This feature is interesting because some differential equations have apparently useless trivial solutions, but applying Lie symmetries to them, more interesting solutions are obtained. The question arises when it happens that our equation contains a discrete quantity. In other words, we aim at investigating steps to be performed when we have a difference equation. Doing so, we find symmetries of difference equations and use them to linearize and reduce the order of difference equations. In this work, we analyze the work done by some researchers in the field and apply their results to some examples. This work will focus on the topical review of symmetries of difference equations and going through that will enable us to make some contribution to the field in the near future.
Ermakov systems : a group theoretic approach.
(1993) Govinder, Keshlan Sathasiva.; Leach, Peter Gavin Lawrence.
The physical world is, for the most part, modelled using second order ordinary differential equations. The time-dependent simple harmonic oscillator and the Ermakov-Pinney equation (which together form an Ermakov system) are two examples that jointly and separately describe many physical situations. We study Ermakov systems from the point of view of the algebraic properties of differential equations. The idea of generalised Ermakov systems is introduced and their relationship to the Lie algebra sl(2, R) is explained. We show that the 'compact' form of generalized Ermakov systems has an infinite dimensional Lie algebra. Such algebras are usually associated only with first order equations in the context of ordinary differential equations. Apart from the Ermakov invariant which shares the infinite-dimensional algebra of the 'compact' equation, the other three integrals force the dimension of the algebra to be reduced to the three of sl(2, R). Subsequently we establish a new class of Ermakov systems by considering equations invariant under sl(2, R) (in two dimensions) and sl(2, R) EB so(3) (in three dimensions). The former class contains the generalized Ermakov system as a special case in which the force is velocity-independent. The latter case is a generalization of the classical equation of motion of the magnetic monopole which is well known to possess the conserved Poincare vector. We demonstrate that in fact there are three such vectors for all equations of this type.
Lie symmetries of junction conditions for radiating stars.
(2011) Abebe, Gezahegn Zewdie.; Govinder, Keshlan Sathasiva.; Maharaj, Sunil Dutt.
We consider shear-free radiating spherical stars in general relativity. In particular we study the junction condition relating the pressure to the heat flux at the boundary of the star. This is a nonlinear equation in the metric functions. We analyse the junction condition when the spacetime is conformally flat, and when the particles are travelling in geodesic motion. We transform the governing equation using the method of Lie analysis. The Lie symmetry generators that leave the equation invariant are identifed and we generate the optimal system in each case. Each element of the optimal system is used to reduce the partial differential equation to an ordinary differential equation which is further analysed. As a result, particular solutions to the junction condition are presented. These exact solutions can be presented in terms of elementary functions. Many of the solutions found are new and could be useful in the modelling process. Our analysis is the first comprehensive treatment of the boundary condition using a symmetry approach. We have shown that this approach is useful in generating new results.

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Browsing Masters Degrees (Pure Mathematics) by Author "Govinder, Keshlan Sathasiva."