# Doctoral Degrees (Applied Mathematics)

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Item Partial exchangeability and related topics.(1991) North, Delia Elizabeth.; Dale, Andrew Ian.Show more Partial exchangeability is the fundamental building block in the subjective approach to the probability of multi-type sequences which replaces the independence concept of the objective theory. The aim of this thesis is to present some theory for partially exchangeable sequences of random variables based on well-known results for exchangeable sequences. The reader is introduced to the concepts of partially exchangeable events, partially exchangeable sequences of random variables and partially exchangeable o-fields, followed by some properties of partially exchangeable sequences of random variables. Extending de Finetti's representation theorem for exchangeable random variables to hold for multi-type sequences, we obtain the following result to be used throughout the thesis: There exists a o-field, conditional upon which, an infinite partially exchangeable sequence of random variables behaves like an independent sequence of random variables, identically distributed within types. Posing (i) a stronger requirement (spherical symmetry) and (ii) a weaker requirement (the selection property) than partial exchangeability on the infinite multi-type sequence of random variables, we obtain results related to de Finetti's representation theorem for partially exchangeable sequences of random variables. Regarding partially exchangeable sequences as mixtures of independent and identically distributed (within types) sequences, we (i) give three possible expressions for the directed random measures of the partially exchangeable sequence and (ii) look at three possible expressions for the o-field mentioned in de Finetti's representation theorem. By manipulating random measures and using de Finetti's representation theorem, we point out some concrete ways of constructing partially exchangeable sequences. The main result of this thesis follows by extending de Finetti's represen. tation theorem in conjunction with the Chatterji principle to obtain the following result: Given any a.s. limit theorem for multi-type sequences of independent random variables, identically distributed within types, there exists an analogous theorem satisfied by all partially exchangeable sequences and by all sub-subsequences of some subsequence of an arbitrary dependent infinite multi-type sequence of random variables, tightly distributed within types. We finally give some limit theorems for partially exchangeable sequences of random variables, some of which follow from the above mentioned result.Show more Item Aspects of graph vulnerability.(1994) Day, David Peter.; Swart, Hendrika Cornelia Scott.; Oellermann, Ortrud Ruth.Show more This dissertation details the results of an investigation into, primarily, three aspects of graph vulnerability namely, l-connectivity, Steiner Distance hereditatiness and functional isolation. Following the introduction in Chapter one, Chapter two focusses on the l-connectivity of graphs and introduces the concept of the strong l-connectivity of digraphs. Bounds on this latter parameter are investigated and then the l-connectivity function of particular types of graphs, namely caterpillars and complete multipartite graphs as well as the strong l-connectivity function of digraphs, is explored. The chapter concludes with an examination of extremal graphs with a given l-connectivity. Chapter three investigates Steiner distance hereditary graphs. It is shown that if G is 2-Steiner distance hereditary, then G is k-Steiner distance hereditary for all k≥2. Further, it is shown that if G is k-Steiner distance hereditary (k≥ 3), then G need not be (k - l)-Steiner distance hereditary. An efficient algorithm for determining the Steiner distance of a set of k vertices in a k-Steiner distance hereditary graph is discussed and a characterization of 2-Steiner distance hereditary graphs is given which leads to an efficient algorithm for testing whether a graph is 2-Steiner distance hereditary. Some general properties about the cycle structure of k-Steiner distance hereditary graphs are established and are then used to characterize 3-Steiner distance hereditary graphs. Chapter four contains an investigation of functional isolation sequences of supply graphs. The concept of the Ranked supply graph is introduced and both necessary and sufficient conditions for a sequence of positive nondecreasing integers to be a functional isolation sequence of a ranked supply graph are determined.Show more Item Modelling the physical dynamics of estuaries for management purposes.(1996) Slinger, Jill Hillary.; Hearne, John W.Show more South African estuaries are characterised by highly variable inflows owing to the semi-arid nature of the land mass which they drain. The interaction of this variability with that of the marine environment (seasonality, high wave events, synoptic effects) gives rise to the distinctive character of South African estuaries. In general, they are small, micro-tidal, bar-built systems with strong flood tidal dominance. Approximately half of the 273 systems along the coast exhibit intermittent closure of the mouth, while a number can become hypersaline during dry periods. In view of the increasing development pressures on the rivers and estuaries of South Africa and their strong dependence on freshwater flow for the maintenance of their character and functioning, and the need for justifiable, scientifically-based decision making regarding the freshwater requirements of estuaries is evident. This study was initiated to address this issue by first developing a model to simulate the physical dynamics of South African estuaries over time scales from months to years, so enabling prediction of the medium to long term consequences of alterations in the freshwater inflow on the abiotic components of an estuary. Thereafter, the efficacy of management policies involving water releases and mouth breachings could be evaluated in terms of their success in maintaining the character and functioning of an estuary. A semi-empirical estuarine systems model incorporating seven state variables, namely water volume, salt content, stratification, circulation, tidal flushing, freshwater flushing and the height of the sill at the mouth, was formulated and implemented on two case studies. Estuarine physics concepts were incorporated dynamically in the model in a novel manner. For instance, the bulk densimetric Froude number and the Estuarine Richardson number are used in the simulation of the stratification-circulation states, while the Ackers and White sediment transport formula was modified to yield results which agreed with field observations of the closure and breaching of the mouth of the Great Brak Estuary. Additionally, tidal exchange through the mouth was modelled phenomenologically and successfully calibrated against observations for both case studies. Model results were found to be fairly robust to uncertainties in parameter values. However, most encouraging of all is that behaviour known to occur in shallow estuaries, such as modulation of the n11.:.m water level by low frequency forcing and the generation of overtides, was reproduced by the estuarine systems model although it was not specifically included in the model formulation. The model is thus considered to reliably predict the physical dynamics of South African estuaries over time scales of months to years. A number of management policies involving freshwater allocations, water releases and breachings of the mouth (where appropriate) were tested on the two case studies, namely the Great Brak Estuary, a small, temporarily open system, and the permanently open Kromme Estuary. The results indicate an increase in marine dominance as freshwater flow to the estuaries decreases. The variability in the estuarine environment declines and the systems become more inert to freshwater flooding and more sensitive to marine forcing. By applying the estuarine systems modelling approach, the performance of different management policies could be evaluated in comparison with reference policies. Accordingly, for both case studies, preferred management policies which utilize the present total annual allocations to the estuaries more beneficially could be indicated. Further management applications included the use of the estuarine systems model in a linked system of abiotic and biotic models to facilitate more comprehensive prediction of the consequences of freshwater abstraction and so more informed assessment of estuarine freshwater requirements. The estuarine systems model results were critical in enabling the prediction of the faunal and floral responses in the intermittently closed Great Brak Estuary as it is presently the only abiotic model capable of simulating the closure and breaching of the estuary mouth over a number of years. It is anticipated that further developments will occur in biological prediction in the near future and that this could require developments or adaptations to the estuarine systems model, particularly when details of the type of information required for biological prediction becomes known. Additionally, the use of the estuarine systems model in a strategic management sense is suggested. It could play a role as a screening tool for regional water resource planning, while the preliminary quantification of the extent of anthropogenic influence in expediting the movement of estuaries towards the later successionary stage of a coastal lagoon is a powerful indication of the level of prediction which could become possible in the future. Thus enhanced management decision making is now possible on a site specific basis and at a more strategic water resources planning level.Show more Item 2-generations of the sporadic simple groups.(1997) Ganief, Moegamad Shahiem.; Moori, Jamshid.Show more 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.Show more Item Developing an integrated decision support system for an oil refinery.(1998) Azizi, Abbas.; Hearne, John W.Show more This thesis considers the problem of residue upgrading operations in an oil refinery. Visbreaking is a residue-upgrading process that improves profitability of a refinery. The economics of converting the heavy residue into the lighter and more valuable streams, coupled with the installation of a modem visbreaker unit at the Engen Refinery in Durban, provides sufficient motives to develop a mathematical model to simulate the unit's capability and estimate the economics of the visbreaking process and fuel oil operations. Furthermore, the proposed model should provide a crude-dependent visbreaking yield that can be used in the refinery's global linear programme (LP), employed to evaluate and select the crude and to optimise refinery's operations. Traditionally, kinetically based models have been used to simulate and study the refining reaction processes. In this case, due to the complexity of the process and some unknown reactions, the performances of existing visbreaking simulators are not fully satisfactory. Consequently, a neural network model of the visbreaking process and fuel oil blending operation is developed. The proposed model is called the adaptive visbreaker paradigm, since it is formed using neuroengineering, a technique that fabricates empirically-based neural network models. The network operates in supervised mode to predict the visbreaking yields and the residue quality. It was observed that due to the fluctuation in the quality of feedstock, and plant operating conditions, the prediction accuracy of the model needs to be improved. To improve the system's predictability, a network reciprocation procedure has been devised. Network reciprocation is a mechanism that controls and selects the input data used in the training of a neural network system. Implementation of the proposed procedure results in a considerable improvement in the performance ofthe network. 3 To facilitate the interaction between the simulation and optimisation routines, an integrated system to incorporate the fuel oil blending with the neurally-based module is constructed. Under an integrated system, the economics of altering the models' decision variables can be monitored. To account for the visbreakability of the various petroleum crudes, the yield predicted by the adaptive visbreaker paradigm should enter into the visbreaker,s sub-model of the global refinery LP. To achieve this, a mechanism to calculate and update the visbreaking yields of various crude oils is also developed. The computational results produced by the adaptive visbreaker paradigm prove that the economics of the visbreaking process is a multi-dimensional variable, greatly influenced by the feed quality and the unit's operating condition. The results presented show the feasibility of applying the proposed model to predict the cracking reaction yields. Furthermore, the model allows a dynamic monitoring of the residue properties as applicable to fuel oil blending optimisation. In summary, the combination of the proposed models forms an integrated decision support system suitable for studying the visbreaking and associated operations, and to provide a visbreaking yield pattern that can be incorporated into the global refinery LP model. Using an integrated decision support system, refinery planners are able to see through the complex interactions between business and the manufacturing process by performing predictive studies using these models.Show more Item Interative approaches to convex feasibility problems.(2001) Pillay, Paranjothi.; Xu, Hongjun.; O'Hara, John Gerard.Show more Solutions to convex feasibility problems are generally found by iteratively constructing sequences that converge strongly or weakly to it. In this study, four types of iteration schemes are considered in an attempt to find a point in the intersection of some closed and convex sets. The iteration scheme Xn+l = (1 - λn+1)y + λn+1Tn+lxn is first considered for infinitely many nonexpansive maps Tl , T2 , T3 , ... in a Hilbert space. A result of Shimizu and Takahashi [33] is generalized, and it is shown that the sequence of iterates converge to Py, where P is some projection. This is further generalized to a uniformly smooth Banach space having a weakly continuous duality map. Here the iterates converge to Qy, where Q is a sunny nonexpansive retraction. For this same iteration scheme, with finitely many maps Tl , T2, ... , TN , a complementary result to a result of Bauschke [2] is proved by introducing a new condition on the sequence of parameters (λn). The iterates converge to Py, where P is the projection onto the intersection of the fixed point sets of the Tis. Both this result and Bauschke's result [2] are then generalized to a uniformly smooth Banach space, and to a reflexive Banach space having a weakly continuous duality map and having Reich's property. Now the iterates converge to Qy, where Q is the unique sunny nonexpansive retraction onto the intersection of the fixed point sets of the Tis. For a random map r : N {I, 2, ... ,N}, the iteration scheme xn+l = Tr(n+l)xn is considered. In a finite dimensional Hilbert space with Tr(n) = Pr(n) , the iterates converge to a point in the intersection of the fixed point sets of the PiS. In an arbitrary Banach space, under certain conditions on the mappings, the iterates converge to a point in the intersection of the fixed point sets of the Tis. For the scheme xn+l = (1- λn+l)xn+λn+lTr(n+l)xn, in a finite dimensional Hilbert space the iterates converge to a point in the intersection of the fixed point sets of the Tis, and in an infinite dimensional Hilbert space with the added assumption that the random map r is quasi-cyclic, then the iterates converge weakly to a point in the intersection of the fixed point sets of the Tis. Lastly, the minimization of a convex function θ is considered over some closed and convex subset of a Hilbert space. For both the case where θ is a quadratic function and for the general case, first the unique fixed points of some maps Tλ are shown to converge to the unique minimizer of θ and then an algorithm is proposed that converges to this unique minimizer.Show more Item Wadley's problem with overdispersion.(2009) Leask, Kerry Leigh.; Haines, Linda Margaret.; Matthews, Glenda Beverley.Show more Wadley’s problem frequently emerges in dosage-mortality data and is one in which the number of surviving organisms is observed but the number initially treated is unknown. Data in this setting are also often overdispersed, that is the variability within the data exceeds that described by the distribution modelling it. The aim of this thesis is to explore distributions that can accommodate overdispersion in a Wadley’s problem setting. Two methods are essentially considered. The first considers adapting the beta-binomial and multiplicative binomial models that are frequently used for overdispersed binomial-type data to a Wadley’s problem setting. The second strategy entails modelling Wadley’s problem with a distribution that is suitable for modelling overdispersed count data. Some of the distributions introduced can be used for modelling overdispersed count data as well as overdispersed doseresponse data from a Wadley context. These models are compared using goodness of fit tests, deviance and Akaike’s Information Criterion and their properties are explored.Show more Item Relativistic radiating stars with generalised atmospheres.(2010) Govender, Gabriel.; Maharaj, Sunil Dutt.Show more In this dissertation we construct radiating models for dense compact stars in relativistic astrophysics. We first utilise the standard Santos (1985) junction condition to model Euclidean stars. By making use of the heuristic Euclidean condition and a linear transformation in the gravitational potentials, we generate a particular exact solution in closed form to the nonlinear stellar boundary condition. Earlier models of spherical nonadiabatic gravitational collapse are then extended by considering the effect of radial perturbations in the matter and metric variables, on the evolution of the stellar fluid and the dynamics of the collapse process. The governing equation describing the temporal behaviour of the model is solved on the stellar surface. The model becomes static in the later stages of collapse. The Santos junction condition is then generalised to describe a radiating star which has a two-fluid atmosphere, consisting of a radiation field and a string fluid. We show that in the appropriate limit when the string energy density goes to zero, the standard result is regained. An exact solution to the generalised boundary condition is found. The generalised boundary condition is extended to hold in the case when the shear is nonvanishing. We demonstrate that our results can be used to model the flow of a string fluid in terms of a diffusion transport process.Show more Item Applications of Lie symmetries to gravitating fluids.(2011) Msomi, Alfred Mvunyelwa.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.Show more This thesis is concerned with the application of Lie's group theoretic method to the Einstein field equations in order to find new exact solutions. We analyse the nonlinear partial differential equation which arises in the study of non- static, non-conformally flat fluid plates of embedding class one. In order to find the group invariant solutions to the partial differential equation in a systematic and comprehensive manner we apply the method of optimal subgroups. We demonstrate that the model admits linear barotropic equations of state in several special cases. Secondly, we study a shear-free spherically symmetric cosmological model with heat flow. We review and extend a method of generating solutions developed by Deng. We use the method of Lie analysis as a systematic approach to generate new solutions to the master equation. Also, general classes of solution are found in which there is an explicit relationship between the gravitational potentials which is not present in earlier models. Using our systematic approach, we can recover known solutions. Thirdly, we study generalised shear-free spherically symmetric models with heat flow in higher dimensions. The method of Lie generates new solutions to the master equation. We obtain an implicit solution or we can reduce the governing equation to a Riccati equation.Show more Item Fischer Clifford matrices and character tables of certain groups associated with simple groups O+10(2) [the simple orthogonal group of dimension 10 over GF (2)], HS and Ly.(2011) Seretlo, Thekiso Trevor.; Moori, Jamshid.Show more The character table of any finite group provides a considerable amount of information about a group and the use of character tables is of great importance in Mathematics and Physical Sciences. Most of the maximal subgroups of finite simple groups and their automorphisms are extensions of elementary abelian groups. Various techniques have been used to compute character tables, however Bernd Fischer came up with the most powerful and informative technique of calculating character tables of group extensions. This method is known as the Fischer-Clifford Theory and uses Fischer-Clifford matrices, as one of the tools, to compute character tables. This is derived from the Clifford theory. Here G is an extension of a group N by a finite group G, that is G = N.G. We then construct a non-singular matrix for each conjugacy class of G/N =G. These matrices, together with partial character tables of certain subgroups of G, known as the inertia groups, are used to compute the full character table of G. In this dissertation, we discuss Fischer-Clifford theory and apply it to both split and non-split extensions. We first, under the guidance of Dr Mpono, studied the group 27:S8 as a maximal subgroup of 27:SP(6,2), to familiarize ourselves to Fischer-Clifford theory. We then looked at 26:A8 and 28:O+8 (2) as maximal subgroups of 28:O+8 (2) and O+10(2) respectively and these were both split extensions. Split extensions have also been discussed quite extensively, for various groups, by different researchers in the past. We then turned our attention to non-split extensions. We started with 24.S6 and 25.S6 which were maximal subgroups of HS and HS:2 respectively. Except for some negative signs in the first column of the Fischer-Clifford matrices we used the Fisher-Clifford theory as it is. The Fischer-Clifford theory, is also applied to 53.L(3, 5), which is a maximal subgroup of the Lyon's group Ly. To be able to use the Fisher-Clifford theory we had to consider projective representations and characters of inertia factor groups. This is not a simple method and quite some smart computations were needed but we were able to determine the character table of 53.L(3,5). All character tables computed in this dissertation will be sent to GAP for incorporation.Show more Item Analysis of shear-free spherically symmetric charged relativistic fluids.(2011) Kweyama, Mandlenkosi Christopher.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.Show more We study the evolution of shear-free spherically symmetric charged fluids in general relativity. This requires the analysis of the coupled Einstein-Maxwell system of equations. Within this framework, the master field equation to be integrated is yxx = f(x)y2 + g(x)y3 We undertake a comprehensive study of this equation using a variety of ap- proaches. Initially, we find a first integral using elementary techniques (subject to integrability conditions on the arbitrary functions f(x) and g(x)). As a re- sult, we are able to generate a class of new solutions containing, as special cases, the models of Maharaj et al (1996), Stephani (1983) and Srivastava (1987). The integrability conditions on f(x) and g(x) are investigated in detail for the purposes of reduction to quadratures in terms of elliptic integrals. We also obtain a Noether first integral by performing a Noether symmetry analy- sis of the master field equation. This provides a partial group theoretic basis for the first integral found earlier. In addition, a comprehensive Lie symmetry analysis is performed on the field equation. Here we show that the first integral approach (and hence the Noether approach) is limited { more general results are possible when the full Lie theory is used. We transform the field equation to an autonomous equation and investigate the conditions for it to be reduced to quadrature. For each case we recover particular results that were found pre- viously for neutral fluids. Finally we show (for the first time) that the pivotal equation, governing the existence of a Lie symmetry, is actually a fifth order purely differential equation, the solution of which generates solutions to the master field equation.Show more Item Applications of symmetry analysis of partial differential and stochastic differential equations arising from mathematics of finance.(2011) Nwobi, Felix Noyanim.; O'Hara, John Gerard.; Leach, Peter Gavin Lawrence.Show more In the standard modeling of the pricing of options and derivatives as generally understood these days the underlying process is taken to be a Wiener Process or a Levy Process. The stochastic process is modeled as a stochastic differential equation. From this equation a partial differential equation is obtained by application of the Feynman-Kac Theorem. The resulting partial differential equation is of Hamilton-Jacobi-Bellman type. Analysis of the partial differential equations arising from Mathematics of Finance using the methods of the Lie Theory of Continuous Groups has been performed over the last twenty years, but it is only in recent years that there has been a concerted effort to make full use of the Lie theory. We propose an extension of Mahomed and Leach's (1990) formula for the nth-prolongation of an nth-order ordinary differential equation to the nth-prolongation of the generator of an hyperbolic partial differential equation with p dependent and k independent variables. The symmetry analysis of this partial differential equation shows that the associated Lie algebra is {sl(2,R)⊕W₃}⊕s ∞A₁ with 12 optimal systems. A modeling approach based upon stochastic volatility for modeling prices in the deregulated Pennsylvania State Electricity market is adopted for application. We propose a dynamic linear model (DLM) in which switching structure for the measurement matrix is incorporated into a two-state Gaussian mixture/first-order autoregressive (AR (1)) configuration in a nonstationary independent process defined by time-varying probabilities. The estimates of maximum likelihood of the parameters from the "modified" Kalman filter showed a significant mean-reversion rate of 0.9363 which translates to a half-life price of electricity of nine months. Associated with this mean-reversion is the high measure of price volatility at 35%. Within the last decade there has been some work done upon the symmetries of stochastic differential equations. Here empirical results contradict earliest normality hypotheses on log-return series in favour of asymmetry of the probability distribution describing the process. Using the Akaike Information Criterion (AIC) and the Log-likelihood estimation (LLH) methods as selection criteria, the normal inverse Gaussian (NIG) outperformed four other candidate probability distributions among the class of Generalized Hyperbolic (GH) distributions in describing the heavy tails present in the process. Similarly, the Skewed Student's t (SSt) is the best fit for Bonny Crude Oil and Natural Gas log-returns. The observed volatility measures of these three commodity prices were examined. The Weibull distribution gives the best fit both electricity and crude oil data while the Gamma distribution is selected for natural gas data in the volatility profiles among the five candidate probability density functions (Normal, Lognormal, Gamma, Inverse Gamma and the Inverse Gaussian) considered.Show more Item Double-diffusive convection flow in porous media with cross-diffusion.(2011) Awad, Faiz G.; Sibanda, Precious.Show more In this thesis we study double-diffusive convection and cross-diffusion effects in flow through porous media. Fluid flows in various flow geometries are investigated and the governing equations are solved analytically and numerically using established and recent techniques such as the Keller-box method, the spectral-homotopy analysis method and the successive linearisation method. The effects of the governing parameters such as the Soret, Dufour, Lewis, Rayleigh and the Peclet numbers and the buoyancy ratio on the fluid properties, and heat and mass transfer at the surface are determined. The accuracy, computational efficiency and validity of the new methods is established. This study consists of five published and one submitted paper whose central theme is the study of double-diffusive convection in porous media. A secondary theme is the application of recent numerical semi-numerical methods in the solution of nonlinear boundary value problems, particularly those that arise in the study of fluid flow problems. Paper 1. An investigation of the quiescent state in a Maxwell fluid with double-diffusive convection in porous media using linear stability analysis is presented. The fluid motion is modeled using the modified Darcy-Brinkman law. The critical Darcy- Rayleigh numbers for the onset of convection are obtained and numerical simulations carried out to show the effects of the Soret and Dufour parameters on the critical Darcy-Rayleigh numbers. For some limiting cases, known results in the literature are recovered. Paper 2. We present an investigation of heat and mass transfer in a micropolar fluid with cross-diffusion effects. Approximate series solutions of the governing non-linear differential equations are obtained using the homotopy analysis method (HAM). A comparison is made between the results obtained using the HAM and the numerical results obtained using the Matlab bvp4c numerical routine. Paper 3. The spectral homotopy analysis method (SHAM) as a new improved version of the homotopy analysis method is introduced. The new technique is used to solve the MHD Jeffery-Hamel problem for a convergent or divergent channel. We show that the SHAM improves the applicability of the HAM by removing the restrictions associated with the HAM as well as accelerating the convergence rate. Paper 4. We present a study of free and forced convection from an inverted cone in porous media with diffusion-thermo and thermo-diffusion effects. The highly nonlinear governing equations are solved using a novel successive linearisation method (SLM). This method combines a non-perturbation technique with the Chebyshev spectral collection method to produce an algorithm with accelerated and assured convergence. Comparison of the results obtained using the SLM, the Runge-Kutta together with a shooting method and the Matlab bvp4c numerical routine show the accuracy and computational efficiency of the SLM. Paper 5. Here we study cross-diffusion effects and convection from inverted smooth and wavy cones. In the case of a smooth cone, the highly non-linear governing equations are solved using the successive linearisation method (SLM), a shooting method together with a Runge-Kutta of order four and the Matlab bvp4c numerical routine. In the case of the wavy cone the governing equations are solved using the Keller-box method. Paper 6. We examine the problem of mixed convection, heat and mass transfer along a semi-infinite plate in a fluid saturated porous medium subject to cross-diffusion and radiative heat transfer. The governing equations for the conservation of momentum, heat and solute concentration transfer are solved using the successive linearisation method, the Keller-box technique and the Matlab bvp4c numerical routine.Show more Item Asymptotic analysis of singularly perturbed dynamical systems.(2011) Goswami, Amartya.; Banasiak, Jacek.Show more According to the needs, real systems can be modeled at various level of resolution. It can be detailed interactions at the individual level (or at microscopic level) or a sample of the system (or at mesoscopic level) and also by averaging over mesoscopic (structural) states; that is, at the level of interactions between subsystems of the original system (or at macroscopic level). With the microscopic study one can get a detailed information of the interaction but at a cost of heavy computational work. Also sometimes such a detailed information is redundant. On the other hand, macroscopic analysis, computationally less involved and easy to verify by experiments. But the results obtained may be too crude for some applications. Thus, the mesoscopic level of analysis has been quite popular in recent years for studying real systems. Here we will focus on structured population models where we can observe various level of organization such as individual, a group of population, or a community. Due to fast movement of the individual compare of the other demographic processes (like death and birth), the problem is multiple-scale. There are various methods to handle multiple-scale problem. In this work we will follow asymptotic analysis ( or more precisely compressed Chapman–Enskog method) to approximate the microscopic model by the averaged one at a given level of accuracy. We also generalize our model by introducing reducible migration structure. Along with this, considering age dependency of the migration rates and the mortality rates, the thesis o ers improvement of the existing literature.Show more Item On new and improved semi-numerical techniques for solving nonlinear fluid flow problems.(2012) Makukula, Zodwa Gcinaphi.; Sibanda, Precious.; Motsa, Sandile Sydney.Show more 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.Show more Item Bounds on distance-based topological indices in graphs.(2012) Morgan, Megan Jane.; Mukwembi, Simon.; Swart, Hendrika Cornelia Scott.Show more 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.Show more Item Applications of embedding theory in higher dimensional general relativity.(2012) Moodley, Jothi.; Amery, Gareth.Show more The study of embeddings is applicable and signicant to higher dimensional theories of our universe, high-energy physics and classical general relativity. In this thesis we investigate local and global isometric embeddings of four-dimensional spherically symmetric spacetimes into five-dimensional Einstein manifolds. Theorems have been established that guarantee the existence of such embeddings. However, most known explicit results concern embedded spaces with relatively simple Ricci curvature. We consider the four-dimensional gravitational field of a global monopole, a simple non-vacuum space with a more complicated Ricci tensor, which is of theoretical interest in its own right, and occurs as a limit in Einstein-Gauss-Bonnet Kaluza-Klein black holes, and we obtain an exact solution for its embedding into Minkowski space. Our local embedding space can be used to construct global embedding spaces, including a globally at space and several types of cosmic strings. We present an analysis of the result and comment on its signicance in the context of induced matter theory and the Einstein-Gauss-Bonnet gravity scenario where it can be viewed as a local embedding into a Kaluza-Klein black hole. Difficulties in solving the five-dimensional equations for given four-dimensional spaces motivate us to investigate which embedded spaces admit bulks of a specific type. We show that the general Schwarzschild-de Sitter spacetime and the Einstein Universe are the only spherically symmetric spacetimes that can be embedded into an Einstein space with a particular metric form, and we discuss their five-dimensional solutions. Furthermore, we determine that the only spherically symmetric spacetime in retarded time coordinates that can be embedded into a particular Einstein bulk is the general Vaidya-de Sitter solution with constant mass. These analyses help to provide insight to the general embedding problem. We also consider the conformal Killing geometry of a five-dimensional Einstein space that embeds a static spherically symmetric spacetime, and we show how the Killing geometry of the embedded space is inherited by its bulk. The study of embedding properties such as these enables a deeper mathematical understanding of higher dimensional cosmological models and is also of physical interest as conformal symmetries encode conservation laws.Show more Item Initial conditions of the universe : signatures in the cosmic microwave background and baryon acoustic oscillations.(2012) Kasanda, Simon Muya.; Moodley, Kavilan.Show more 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.Show more Item Computational study of high order numerical schemes for fluid-structure interaction in gas dynamics.(2013) Pedro, Jose Caluyna.; Sibanda, Precious.Show more Solving the fluid-structure interaction (FSI) problems is particularly challenging. This is because the coupling of the fluid and structure may require different solvers in different points of the solution domain, and with different mesh requirements. In this thesis, a partitioned approach is considered. Two solvers are employed to deal with each part of the problem (fluid and structure), where the interaction process is realized via exchanging information from the fluid-structure interface in a staggered fashion. One of the advantages of this approach is that we can take advantage of the existing algorithms that have been used for solving fluid or structural problems, which leads a reduction in the code development time, Hou (2012). However, it requires careful implementation so that spurious results in terms of stability and accuracy can be avoided. We found that most fluid-structure interaction computations through a staggered approach are based on at most second order time integration methods. In this thesis we studied the performance of some high order fluid and structure dynamic methods, when applied in a staggered approach to an FSI problem in a structure prediction way by combining predictors with time integration schemes to obtain stable schemes. Nonlinear Euler equations for gas dynamics were investigated and the analysis was realized through the piston problem. An adapted one-dimensional high order finite volume WENO₃ scheme for nonlinear hyperbolic conservation laws-Dumbser (2007a), Dumbser et al. 2007b)-was considered and a numerical flux was proposed. The numerical results of the proposed method show the non-oscillatory property when compared with traditional numerical methods such as the Local Lax-Friedrichs. So far to our knowledge, the WENO₃₋ as proposed in this work- has not been applied to FSI problems. Thus, it was proposed to discretize the fluid domain in space, and in order to adapt it to a moving mesh was reformulated to couple with an Arbitrary Lagrangian Eulerian (ALE) approach. To integrate in time the structure we started by using Newmark schemes as well as the trapezoidal-rule backward differentiation formulae of order 2 (TR − BDF2). Two study cases were carried out by taking into account the transient effects on the fluid behaviour. In the first case, we only consider the structural mass in the dynamic coupled system and in the second case, a quasi-steady fluid was considered. In order to test the performance of the structural solvers, simulations were carried out, firstly, without the contribution of fluid mass, and then a comparative study of the performance of various structure solvers in a staggered approach framework were realized in order to study the temporal accuracy for the partitioned fluid-structure interaction coupling. For a quasi-steady fluid case, the oscillation frequency of the coupled system was successfully estimated using the TR-BDF2 scheme, and the coupled system was solved for various Courant numbers in a structural predictor fashion. The results showed better performance of the TR-BDF2 scheme. Newmark’s schemes as well as the TR-BDF2 are only second order accuracy. However, the Newmark (average acceleration) is traditionally preferred by researchers as a structure solver in a staggered approach for FSI problems, although higher order schemes do exist. van Zuijlen (2004), in his partitioned algorithm proposed the explicit singly diagonal implicit Runge-Kutta (ESDIRK) family of schemes of order 3 to 5 to integrate both fluid and structure. Therefore in this work, these schemes were considered and applied as structural solvers. Their performance was studied through numerical experiments, and comparisons were realized with the performance of the traditional Newmark’s schemes. The results show that although their computational cost is high, they present a high order of accuracy.Show more Item Modelling waterborne diseases.(2013) Collins, Obiora Cornelius.; Govinder, Keshlan Sathasiva.Show more Waterborne diseases are among the major health problems threatening the life of individuals globally. This thesis investigates the dynamics of waterborne disease under different conditions and consequently determines possible intervention strategies to minimize the spread of the disease. The following problems are addressed: The effects of seasonal variations on the dynamics of waterborne disease together with the possible benefits of control intervention strategies such as vaccination, treatment and provision of clean water under the assumption of a homogeneous population are investigated. Specifically, we determine the optimal use of the intervention strategies to mitigate the spread of the disease. The dynamics of waterborne disease in a multiple socioeconomic class community is explored. Particularly, we investigate the effects of migration of individuals due to socioeconomic reasons on the dynamics of waterborne disease under the assumption of heterogeneous mixing population. We examine the effects of multiple contaminated water sources on the dynamics of waterborne disease under the assumption of homogeneous population. We also consider the problem of minimizing cost and determine the optimal use of vaccination to reduce the spread of infections. The effects of heterogeneity on the transmission dynamics of waterborne disease is explored. Furthermore, we scrutinize use of the control intervention strategies to mitigate the spread of the infections under a heterogeneous population setting.Show more