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1 
On new and improved seminumerical techniques for solving nonlinear fluid flow problems.Makukula, Zodwa Gcinaphi. January 2012 (has links)
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.
Seminumerical 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 seminumerical
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 quasilinearisation 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 twodimensional
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 ReinerRivlin 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 twodimensional 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 KwaZuluNatal, Pietermaritzburg, 2012.

2 
New classes of exact solutions for charged perfect fluids.Mthethwa, Thulani Richard. 28 March 2014 (has links)
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 subluminal and the energy conditions are satisfied everywhere in the
interior of the star. / Thesis (M.Sc.)University of KwaZuluNatal, Durban, 2012.

3 
2generations pf the sporadic simple groups.Ganief, Moegamad Shahiem. January 1997 (has links)
A group G is said to be 2generated if G = (x, y), for some nontrivial elements x, y E G. In this thesis we investigate three special types of 2generations 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 nXcomplementary generations of the finite simple groups. A finite group G is said to be nXcomplementary generated if, given an arbitrary nontrivial 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 (BrennerGuralnickWiegold [18]) that every finite simple group can be generated by an arbitrary nontrivial 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 Xcomplementary generations of the finite simple groups, we pose the following problem. Which conjugacy classes nX of the sporadic simple groups are nXcomplementary 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 nonabelian group G is said to have spread r iffor every set {Xl, X2, ' , "xr } of r nontrivial distinct elements, thpre is an element y E G such that G = (Xi, y), for all i. Our interest in this type of 2generation comes from a problem by BrennerWiegold [19] to find all finite nonabelian 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.

4 
Conservation laws models in networks and multiscale flow optimization.Ngnotchouye, Jean Medard Techoukouegno. January 2011 (has links)
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
driftflux 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 flowmatching 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 KwaZuluNatal, Pietermaritzburg, 2011.

5 
Assessment of variability in onfarm trials : a Uganda case.January 2002 (has links)
Onfarm 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, onfarm trials are meant to address such discrepancies. The main problems associated with onfarm trials in most developing countries are high
variability and inappropriate application of statistical knowledge known to work on station to onfarm situation. Characterisation of various onfarm variability and orientation of existing statistical methods may lead to improved agricultural research. Characterization of the various forms of variability in onfarm trials was conducted. Based on these forms of variability, estimation procedures and their strength have been assessed. Special analytical tools for handling nonreplicated experiments known to be
common to onfarm trials are presented. The above stated procedures have been illustrated through a review of Uganda case. To understand onfarm 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 onfarm
trials. Proper approach to onfarm 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.

6 
A classical approach for the analysis of generalized linear mixed models.January 2004 (has links)
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 quasilikelihood (IQL) function which is often intractable since it may consist of highdimensional integrals. Therefore, an exact maximum likelihood analysis is not feasible. The penalized quasilikelihood (PQL) function, derived from a firstorder 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 quasideviance function in the Laplaceapproximated IQL by the generalized chisquared 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. Biascorrected PQL estimators (CPQL) have hence been proposed, using asymptotic analysis and simulation. The pseudolikelihood 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.

7 
Initial conditions of the universe : signatures in the cosmic microwave background and baryon acoustic oscillations.Kasanda, Simon Muya. January 2012 (has links)
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
largescale 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 photonbaryon 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 semianalytic 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 KwaZuluNatal, Westville, 2012.

8 
Matrix models of population theory.Abdalla, Suliman Jamiel Mohamed. 12 May 2014 (has links)
Nonnegative matrices arise naturally in population models. In this thesis, we first study Perron
Frobenius theory of nonnegative 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 nonlinear 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 nonincreasing 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 KwaZuluNatal, Durban, 2013.

9 
Vertexcriticality of the domination parameters of graphsRoux, Adriana 03 1900 (has links)
Thesis (MSc (Mathematical Sciences))University of Stellenbosch, 2011. / Includes bibliography. / Please refer to full text to view abstract.

10 
Bounds on distancebased topological indices in graphs.Morgan, Megan Jane. January 2012 (has links)
This thesis details the results of investigations into bounds on some distancebased
topological indices.
The thesis consists of six chapters. In the first chapter we define the standard
graph theory concepts, and introduce the distancebased 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 wellstudied 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 distancebased 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 KwaZuluNatal, Westville, 2012.

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