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Masters Degrees (Pure Mathematics)

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

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Browsing Masters Degrees (Pure Mathematics) by Subject "Differential equations--Numerical solutions."

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    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.
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    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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    Noether's theorem and first integrals of ordinary differential equations.
    (1997) Moyo, Sibusiso.; Leach, Peter Gavin Lawrence.
    The Lie theory of extended groups is a practical tool in the analysis of differential equations, particularly in the construction of solutions. A formalism of the Lie theory is given and contrasted with Noether's theorem which plays a prominent role in the analysis of differential equations derivable from a Lagrangian. The relationship between the Lie and Noether approach to differential equations is investigated. The standard separation of Lie point symmetries into Noetherian and nonNoetherian symmetries is shown to be irrelevant within the context of nonlocality. This also emphasises the role played by nonlocal symmetries in such an approach.