Most real world phenomena is modeled by ordinary and/or partial differential equations. Most of these equations are highly nonlinear and exact solutions are not always possible. Exact solutions always give a good account of the physical nature of the phenomena modeled. However, existing analytical methods can only handle a limited range of these equations. Semi-numerical and numerical methods give approximate solutions where exact solutions are impossible to find. However, some common numerical methods give low accuracy and may lack stability. In general, the character and qualitative behaviour of the solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computational efficient and robust. In this study we introduce innovative techniques for finding solutions of highly nonlinear coupled boundary value problems. These techniques aim to combine the strengths of both analytical and numerical methods to produce efficient hybrid algorithms. In this work, the homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral methods are well known for their high levels of accuracy. The new spectral homotopy analysis method is further improved by using a more accurate initial approximation to accelerate convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral methods are used to solve the linearised equations. The new techniques were used to solve mathematical models in fluid dynamics. The thesis comprises of an introductory Chapter that gives an overview of common numerical methods currently in use. In Chapter 2 we give an overview of the methods used in this work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional squeezing flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter 4 the methods were used to find solutions of the laminar heat transfer problem in a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem due to a shrinking sheet with a chemical reaction, were solved using the new methods. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.

We investigate techniques to generate new classes of exact solutions to the Einstein- Maxwell field equations which represent the gravitational field of charged perfect fluid spherically symmetric distributions of matter. Historically, a large number of solutions have been proposed but only a small number have been demonstrated to satisfy elementary conditions for physical acceptability. Firstly we examine the case of the constant density and constant electric field charged fluid sphere and show empirically that such configurations of matter are unlikely to exist as basic physical requirements are violated. We then make an ansatz relating the fluid's electric field intensity to one of the gravitational potentials thereby simplifying the system of partial differential equations. This prescription yields an algorithmic process to generate new classes of exact solutions. We present a number of new solutions and comment on their viability as stellar models. Graphical plots generated by symbolic software of the main dynamical and geometrical quantities verify that one of our models is suitable to represent a physically relevant distribution of charged matter in the form of a spherical shell. In particular, positive definiteness of energy density and pressure are guaranteed, a pressure free hypersurface denoting the boundary of the star exists, the sound speed is shown to be sub-luminal and the energy conditions are satisfied everywhere in the interior of the star. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2012.

A group G is said to be 2-generated if G = (x, y), for some non-trivial elements x, y E G. In this thesis we investigate three special types of 2-generations of the sporadic simple groups. A group G is a (l, rn, n )-generated group if G is a quotient group of the triangle group T(l, rn, n) = (x, y, zlx1 = ym = zn = xyz = la). Given divisors l, rn, n of the order of a sporadic simple group G, we ask the question: Is G a (l, rn, n)-generated group? Since we are dealing with simple groups, we may assume that III +l/rn + l/n < 1. Until recently interest in this type of generation had been limited to the role it played in genus actions of finite groups. The problem of determining the genus of a finite simple group is tantamount to maximizing the expression III +l/rn +Iln for which the group is (l,rn,n)-generated. Secondly, we investigate the nX-complementary generations of the finite simple groups. A finite group G is said to be nX-complementary generated if, given an arbitrary non-trivial element x E G, there exists an element y E nX such that G = (x, y). Our interest in this type of generation is motivated by a conjecture (Brenner-Guralnick-Wiegold [18]) that every finite simple group can be generated by an arbitrary non-trivial element together with another suitable element. It was recently proved by Woldar [181] that every sporadic simple group G is pAcomplementary generated, where p is the largest prime divisor of IGI. In an attempt to further the theory of X-complementary generations of the finite simple groups, we pose the following problem. Which conjugacy classes nX of the sporadic simple groups are nX-complementary generated conjugacy classes. In this thesis we provide a complete solution to this problem for the sporadic simple groups HS, McL, C03, Co2 , Jt , J2 , J3 , J4 and Fi 22 · We partially answer the question on (l, rn, n)-generation for the said sporadic groups. A finite non-abelian group G is said to have spread r iffor every set {Xl, X2, ' , "xr } of r non-trivial distinct elements, thpre is an element y E G such that G = (Xi, y), for all i. Our interest in this type of 2-generation comes from a problem by BrennerWiegold [19] to find all finite non-abelian groups with spread 1, but not spread 2. Every sporadic simple group has spread 1 (Woldar [181]) and we show that every sporadic simple group has spread 2. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 1997.

The flow of fluids in a network is of practical importance in gas, oil and water transport for industrial and domestic use. When the flow dynamics are understood, one may be interested in the control of the flow formulated as follows: given some fluid properties at a final time, can one determine the initial flow properties that lead to the desired flow properties? In this thesis, we first consider the flow of a multiphase gas, described by the drift flux model, in a network of pipes and that of water, modeled by the shallow water equations, in a network of rivers. These two models are systems of partial differential equations of first order generally referred to as systems of conservation laws. In particular, our contribution in this regard can be summed up as follows: For the drift-flux model, we consider the flow in a network of pipes seen mathematically as an oriented graph. We solve the standard Riemann problem and prove a well posedness result for the Riemann problem at a junction. This result is obtained using coupling conditions that describe the dynamics at the intersection of the pipes. Moreover, we present numerical results for standard pipes junctions. The numerical results and the analytical results are in agreement. This is an extension for multiphase flows of some known results for single phase flows. Thereafter, the shallow water equations are considered as a model for the flow of water in a network of canals. We analyze coupling conditions at the confluence of rivers, precisely the conservation of mass and the equality of water height at the intersection, and implement these results for some classical river confluences. We also consider the case of pooled stepped chutes, a geometry frequently utilized by dams to spill floodwater. Here we consider an approach different from the engineering community in the sense that we resolve the dynamics by solving a Riemann problem at the dam for the shallow water equations with some suitable coupling conditions. Secondly, we consider an optimization problem constrained by the Euler equations with a flow-matching objective function. Differently from the existing approaches to this problem, we consider a linear approximation of the flow equation in the form of the microscopic Lattice Boltzmann Equations (LBE). We derive an adjoint calculus and the optimality conditions from the microscopic LBE. Using multiscale analysis, we obtain an equivalent macroscopic result at the hydrodynamic limit. Our numerical results demonstrate the ability of our method to solve challenging problems in fluid mechanics. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

On-farm trials techniques have become an integral part of research aimed at improving agricultural production especially in subsistence farming. The poor performance of certain technologies on the farmers' fields known to have performed well on stations have been of concern. Traditionally, on-farm trials are meant to address such discrepancies. The main problems associated with on-farm trials in most developing countries are high variability and inappropriate application of statistical knowledge known to work on station to on-farm situation. Characterisation of various on-farm variability and orientation of existing statistical methods may lead to improved agricultural research. Characterization of the various forms of variability in on-farm trials was conducted. Based on these forms of variability, estimation procedures and their strength have been assessed. Special analytical tools for handling non-replicated experiments known to be common to on-farm trials are presented. The above stated procedures have been illustrated through a review of Uganda case. To understand on-farm variability require grouping of sources of variability into agronomic, animal and socioeconomic components. This led to a deeper understanding of levels of variability and appropriate estimation procedures. The mixed model, modified stability analysis and additive main effects and multiplicative interaction methods have been found to play a role in on-farm trials. Proper approach to on-farm trials and application of appropriate statistical tools will lead to efficient results that will subsequently enhance agricultural production especially under subsistence farming. / Thesis (M.Sc.)-University of Natal, Pietermaritzburg, 2002. / Rockefeller Foundation and Makerere University.

Generalized linear mixed models (GLMMs) accommodate the study of overdispersion and correlation inherent in hierarchically structured data. These models are an extension of generalized linear models (GLMs) and linear mixed models (LMMs). The linear predictor of a GLM is extended to include an unobserved, albeit realized, vector of Gaussian distributed random effects. Conditional on these random effects, responses are assumed to be independent. The objective function for parameter estimation is an integrated quasi-likelihood (IQL) function which is often intractable since it may consist of high-dimensional integrals. Therefore, an exact maximum likelihood analysis is not feasible. The penalized quasi-likelihood (PQL) function, derived from a first-order Laplace expansion to the IQL about the optimum value of the random effects and under the assumption of slowly varying weights, is an approximate technique for statistical inference in GLMMs. Replacing the conditional weighted quasi-deviance function in the Laplace-approximated IQL by the generalized chi-squared statistic leads to a corrected profile quasilikelihood function for the restricted maximum likelihood (REML) estimation of dispersion components by Fisher scoring. Evaluation of mean parameters, for fixed dispersion components, by iterative weighted least squares (IWLS) yields joint estimates of fixed effects and random effects. Thus, the PQL criterion involves repeated fitting of a Gaussian LMM with a linked response vector and a conditional iterated weight matrix. In some instances, PQL estimates fail to converge to a neighbourhood of their true values. Bias-corrected PQL estimators (CPQL) have hence been proposed, using asymptotic analysis and simulation. The pseudo-likelihood algorithm is an alternative estimation procedure for GLMMs. Global score statistics for hypothesis testing of overdispersion, correlation and heterogeneity in GLMMs has been developed as well as individual score statistics for testing null dispersion components separately. A conditional mean squared error of prediction (CMSEP) has also been considered as a general measure of predictive uncertainty. Local influence measures for testing the robustness of parameter estimates, by inducing minor perturbations into GLMMs, are recent advances in the study of these models. Commercial statistical software is available for the analysis of GLMMs. / Thesis (M.Sc.)-University of Natal, Durban, 2004.

In this thesis, we investigate the signatures of isocurvature initial conditions in the cosmic microwave background (CMB) through the temperature and polarization anisotropies, and in the large-scale structure distribution through the baryon acoustic oscillations (BAO). The first part of this thesis is a brief review of the standard cosmological model with its underlying linear cosmological perturbation theory. We supplement it with a general discussion on the initial conditions of the primordial fluctuations. In the third chapter, we review the evolution of the perturbations in the adiabatic model. We focus on the evolution of adiabatic perturbations in the photons and baryons from the epoch of initial conditions to the photon-baryon decoupling, as these determine the main features of the primary CMB anisotropies and of the baryon acoustic oscillations. The fourth chapter recalls the theory of the CMB anisotropies in the adiabatic model. We consider the perturbations from the last scattering surface and evolve them through the line of sight integral to get the adiabatic CMB power spectrum. We review the effect of different cosmological parameters on the adiabatic CMB temperature spectrum. In the fifth chapter, we investigate the observational signatures of the isocurvature perturbations in the CMB anisotropies. We first derive simple semi-analytic expressions for the evolution of the photon and baryon perturbations prior to decoupling for the four isocurvature regular modes and show that these modes excite different harmonics which couple differently to Silk damping and alter the form and evolution of acoustic waves. We study the impact of different cosmological parameters on the CMB angular power spectrum through the line of sight integral and find that the impact of the physical baryon and matter densities in isocurvature models differ the most from their effect in adiabatic models. In the last two chapters, we explore in detail the effect of allowing for small amplitude admixtures of general isocurvature perturbations in addition to the dominant adiabatic mode, and their effect on the baryon acoustic oscillations. The sixth chapter focuses on the distortion of the standard ruler distance and the degradation of dark energy constants due to the inclusion of isocurvature perturbations, while the seventh chapter discusses in more detail the sensitivity of BAO dark energy constraints to general isocurvature perturbations. We stress the role played by Silk damping on the BAO peak features in breaking the degeneracy in the peak location for the different isocurvature modes and show how more general initial conditions impact our interpretation of cosmological data in dark energy studies. We find that the inclusion of these additional isocurvature modes leads to a significant increase in the Dark Energy Task Force figure of merit when considered in conjunction with CMB data. We also show that the incorrect assumption of adiabaticity has the potential to substantially bias our estimates of the dark energy parameters. We find that the use of the large scale structure data in conjunction with CMB data significantly improves our ability to measure the contributions of different modes to the initial conditions. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.

Non-negative matrices arise naturally in population models. In this thesis, we first study Perron- Frobenius theory of non-negative irreducible matrices. We use this theory to investigate the asymptotic behaviour of discrete time linear autonomous models. Then we discuss an application for this in age structured population. Furthermore, we study Liapunov stability of a general non-linear autonomous model. We consider a general nonlinear autonomous model that arises in structured population. We assume that the associated nonlinear matrix of this model is non-increasing at all density levels. Then, we show the existence of global extinction. In addition, we show the stability condition of the extinction equilibrium of the this model in the Liapunov sense. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2013.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2011. / Includes bibliography. / Please refer to full text to view abstract.

This thesis details the results of investigations into bounds on some distance-based topological indices. The thesis consists of six chapters. In the first chapter we define the standard graph theory concepts, and introduce the distance-based graph invariants called topological indices. We give some background to these mathematical models, and show their applications, which are largely in chemistry and pharmacology. To complete the chapter we present some known results which will be relevant to the work. Chapter 2 focuses on the topological index called the eccentric connectivity index. We obtain an exact lower bound on this index, in terms of order, and show that this bound is sharp. An asymptotically sharp upper bound is also derived. In addition, for trees of given order, when the diameter is also prescribed, tight upper and lower bounds are provided. Our investigation into the eccentric connectivity index continues in Chapter 3. We generalize a result on trees from the previous chapter, proving that the known tight lower bound on the index for a tree in terms of order and diameter, is also valid for a graph of given order and diameter. In Chapter 4, we turn to bounds on the eccentric connectivity index in terms of order and minimum degree. We first consider graphs with constant degree (regular graphs). Došlić, Saheli & Vukičević, and Ilić posed the problem of determining extremal graphs with respect to our index, for regular (and more specifically, cubic) graphs. In addressing this open problem, we find upper and lower bounds for the index. We also provide an extremal graph for the upper bound. Thereafter, the chapter continues with a consideration of minimum degree. For given order and minimum degree, an asymptotically sharp upper bound on the index is derived. In Chapter 5, we turn our focus to the well-studied Wiener index. For trees of given order, we determine a sharp upper bound on this index, in terms of the eccentric connectivity index. With the use of spanning trees, this bound is then generalized to graphs. Yet another distance-based topological index, the degree distance, is considered in Chapter 6. We find an asymptotically sharp upper bound on this index, for a graph of given order. This proof definitively settles a conjecture posed by Tomescu in 1999. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.