Doctoral Degrees (Applied Mathematics)

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2-generations of the sporadic simple groups.
(1997) Ganief, Moegamad Shahiem.; Moori, Jamshid.
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.
Age structured models of mathematical epidemiology.
(2013) Massoukou, Rodrigue Yves M'pika.; Banasiak, Jacek.
We consider a mathematical model which describes the dynamics for the spread of a directly transmitted disease in an isolated population with age structure, in an invariant habitat, where all individuals have a finite life-span, that is, the maximum age is finite, hence the mortality is unbounded. We assume that infected individuals do not recover permanently, meaning that these diseases do not convey immunity (these could be: common cold, influenza, gonorrhoea) and the infection can be transmitted horizontally as well as vertically from adult individuals to their newborns. The model consists of a nonlinear and nonlocal system of equations of hyperbolic type. Note that the above-mentioned model has been already analysed by many authors who assumed a constant total population. With this assumption they considered the ratios of the density and the stable age profile of the population, see [16, 31]. In this way they were able to eliminate the unbounded death rate from the model, making it easier to analyse by means of the semigroup techniques. In this work we do not make such an assumption except for the error estimates in the asymptotic analysis of a singularly perturbed problem where we assume that the net reproduction rate R ≤ 1. For certain particular age-dependent constitutive forms of the force of infection term, solvability of the above-mentioned age-structured epidemic model is proven. In the intercohort case, we use the semigroup theory to prove that the problem is well-posed in a suitable population state space of Lebesgue integrable vector valued functions and has a unique classical solution which is positive, global in time and has the property of continuous dependence on the initial data. Further, we prove, under additional regularity conditions (composed of specific assumptions and compatibility conditions at the origin), that the solution is smooth. In the intracohort case, we have to consider a suitable population state space of bounded vector valued functions on which the (unbounded) population operator cannot generate a strongly continuous semigroup which, therefore, is not suitable for semigroup techniques–any strongly continuous semigroup on the space of bounded vector valued functions is uniformly continuous, see [6, Theorem 3.6]. Since, for a finite life-span of the population, the space of bounded vector valued functions is a subspace densely and continuously embedded in the state space of Lebesgue integrable vector valued functions, thus we can restrict the analysis of the intercohort case to the above-mentioned space of bounded vector valued functions. We prove that this state space is invariant under the action of the strongly continuous semigroup generated by the (unbounded) population operator on the state space of Lebesgue integrable vector valued functions. Further, we prove the existence and uniqueness of a mild solution to the problem. In general, different time scales can be identified in age-structured epidemiological models. In fact, if the disease is not terminal, the process of getting sick and recovering is much faster than a typical demographical process. In this work, we consider the case where recovering is much faster than getting sick, giving birth and death. We consider a convenient approach that carries out a preliminary theoretical analysis of the model and, in particular, identifies time scales of it. Typically this allows separation of scales and aggregation of variables through asymptotic analysis based on the Chapman-Enskog procedure, to arrive at reduced models which preserve essential features of the original dynamics being at the same time easier to analyse.
Analysis of multiple control strategies for pre-exposure prophylaxis and post-infection interventions on HIV infection.
(2016) Afassinou, Komi.; Chirove, Faraimunashe.; Govinder, Keshlan Sathasiva.
Abstract available in PDF file.
Analysis of nonlinear Benjamin equation posed on the real line.
(2022) Aluko, Olabisi Babatope.; Paramasur, Nabendra.; Shindin, Sergey Konstantinovich.
The thesis contains a comprehensive theoretical and numerical study of the nonlinear Benjamin equation posed in the real line. We explore wellposedness of the problem in weighted settings and provide a detailed study of existence, regularity and orbital stability of traveling wave solutions. Further, we present a comprehensive study of the Malmquist-Takenaka-Christov (MTC) computational basis and employ it for the numerical treatment of the nonstaionary and the stationary Benjamin equations.
Analysis of shear-free spherically symmetric charged relativistic fluids.
(2011) Kweyama, Mandlenkosi Christopher.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.
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.
An analysis of symmetries and conservation laws of some classes of PDEs that arise in mathematical physics and biology.
(2016) Okeke, Justina Ebele.; Narain, Rivendra Basanth.; Govinder, Keshlan Sathasiva.
In this thesis, the symmetry properties and the conservation laws for a number of well-known PDEs which occur in certain areas of mathematical physics are studied. We focus on wave equations that arise in plasma physics, solid physics and fluid mechanics. Firstly, we carry out analyses for a class of non-linear partial differential equations, which describes the longitudinal motion of an elasto-plastic bar and anti-plane shearing deformation. In order to systematically explore the mathematical structure and underlying physics of the elasto-plastic flow in a medium, we generate all the geometric vector fields of the model equations. Using the classical Lie group method, it is shown that this equation does not admit space dilation type symmetries for a speci fic parameter value. On the basis of the optimal system, the symmetry reductions and exact solutions to this equation are derived. The conservation laws of the equation are constructed with the help of Noether's theorem We also consider a generalized Boussinesq (GB) equation with damping term which occurs in the study of shallow water waves and a system of variant Boussinesq equations. The conservation laws of these systems are derived via the partial Noether method and thus demonstrate that these conservation laws satisfy the divergence property. We illustrate the use of these conservation laws by obtaining several solutions for the equations through the application of the double reduction method, which encompasses the association of symmetries and conservation laws. A similar analysis is performed for the generalised Gardner equation with dual power law nonlinearities of any order. In this case, we derive the conservation laws of the system via the Noether approach after increasing the order and by the use of the multiplier method. It is observed that only the Noether's approach gives a uni ed treatment to the derivation of conserved vectors for the Gardner equation and can lead to local or an in finite number of nonlocal conservation laws. By investigating the solutions using symmetry analysis and double reduction methods, we show that the double reduction method yields more exact solutions; some of these solutions cannot be recovered by symmetry analysis alone. We also illustrate the importance of group theory in the analysis of equations which arise during investigations of reaction-diffusion prey-predator mechanisms. We show that the Lie analysis can help obtain different types of invariant solutions. We show that the solutions generate an interesting illustration of the possible behavioural patterns.
Applications of embedding theory in higher dimensional general relativity.
(2012) Moodley, Jothi.; Amery, Gareth.
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.
Applications of Lie symmetries to gravitating fluids.
(2011) Msomi, Alfred Mvunyelwa.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.
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.
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.
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.
Aspects of connectedness in metric frames.
(2019) Rathilal, Cerene.; Pillay, Paranjothi.; Baboolal, Dharmanand.
Abstract available in PDF file.
Aspects of graph vulnerability.
(1994) Day, David Peter.; Swart, Hendrika Cornelia Scott.; Oellermann, Ortrud Ruth.
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.
Aspects of trapped surfaces and spacetime singularities.
(2019) Sherif, Abbas Mohamed.; Maharaj, Sunil Dutt.; Goswami, Rituparno.
Abstract available in PDF.
Asymptotic analysis of singularly perturbed dynamical systems.
(2011) Goswami, Amartya.; Banasiak, Jacek.
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.
Bivariate pseudospectral collocation algorithms for nonlinear partial differential equations.
(2016) Magagula, Vusi Mpendulo.; Motsa, Sandile Sydney.; Sibanda, Precious.
Abstract available in PDF file.
Bounds on distance-based topological indices in graphs.
(2012) Morgan, Megan Jane.; Mukwembi, Simon.; Swart, Hendrika Cornelia Scott.
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.
The C-Band All Sky Survey commissioning and data analysis.
(2017) Heilgendorff, Heiko Martin.; Chiang, Hsin Cynthia.; Sievers, Jonathan Leroy.
The C-Band All Sky Survey (C-BASS) is a ground based radio survey that scans the entire sky in Stokes I, Q and U at a central frequency of 5 GHz with a 1 GHz bandwidth and an angular resolution of 0.73 . The experiment consists of two telescopes, one at Owens Valley Radio Observatory in California and the other at the SKA support base in Klerefontein, South Africa. The primary aim of this experiment is to produce high fidelity maps of the entire sky in Stokes I, Q and U. These maps will be used by CMB experiments for the removal of Galactic foreground radiation, via component separation, and will provide vital aid in the search for the primordial CMB B-mode polarized signal. C-BASS also aims to probe the Galactic magnetic field using synchrotron radiation and will search for new areas of anomalous microwave emission. In this thesis, I present the contribution that I have made to the C-BASS experiment. I contributed to C-BASS instrumentation development by working extensively on the commissioning of the southern telescope; in particular, I developed an optical pointing system and refined the automated analysis process. I contributed to the development of the C-BASS data analysis pipeline for both the northern and southern telescopes, with the development of a new RFI flagging method, work on map making techniques and convergence, and self-consistency tests. The northern survey is complete and data analysis is at an advanced stage. The southern instrument is undergoing commissioning on site and will soon begin survey operations. My contributions to the project have improved the processed data quality in both surveys and will aid in the successful completion of the southern survey.
Computational study of high order numerical schemes for fluid-structure interaction in gas dynamics.
(2013) Pedro, Jose Caluyna.; Sibanda, Precious.
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.
Conformal symmetry and applications to spherically symmetric spacetimes.
(2018) Manjonjo, Addial Mackingtosh.; Maharaj, Sunil Dutt.; Moopanar, Selvandren.
In this thesis we study static spherically symmetric spacetimes with a spherical conformal symmetry and a nonstatic conformal factor. We analyse the general solution of the conformal Killing vector equation subject to integrability conditions which impose restrictions on the metric functions. The Weyl tensor is used to characterise the conformal geometry. An explicit relationship between the gravitational potentials for both conformally and nonconformally at cases is obtained. The Einstein equations can then be written in terms of a single gravitational potential. Previous results of conformally invariant static spheres are special cases of our solutions. For isotropic pressure we can find all metrics explicitly and show that the models always admit a barotropic equation of state. We show that this treatment contains well known metrics such Schwarzschild (interior), Tolman, Kuchowicz, Korkina and Orlyanskii, Patwardhan and Vaidya, and Buchdahl and Land. For anisotropic pressures the solution of the fluid equations is found in general. We then consider an astrophysical application of conformal symmetries. We investigate spherical exact models for compact stars with anisotropic pressures and a conformal symmetry. We generate a new anisotropic solution to the Einstein field equations. We demonstrate that this exact solution produces a relativistic model of a compact star. The model generates stellar radii and masses consistent with PSR J1614-2230, Vela X1, PSR J1903+327 and Cen X-3. A detailed physical examination shows that the model is regular, well behaved and stable. The mass-radius limit and the surface red shift are consistent with observational constraints.
Developing an integrated decision support system for an oil refinery.
(1998) Azizi, Abbas.; Hearne, John W.
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.
Differential equations for relativistic radiating stars.
(2013) Abebe, Gezahegn Zewdie.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.
We consider radiating spherical stars in general relativity when they are conformally flat, geodesic with shear, and accelerating, expanding and shearing. We study the junction conditions relating the pressure to the heat flux at the boundary of the star in each case. The boundary conditions are nonlinear partial differential equations in the metric functions. We transform the governing equations to ordinary differential equations using the geometric method of Lie. The Lie symmetry generators that leave the equations invariant are identified, and we generate the optimal system in each case. Each element of the optimal system is used to reduce the partial differential equations to ordinary differential equations which are further analyzed. As a result, particular solutions to the junction conditions are presented for all types of radiating stars. New exact solutions, which are group invariant under the action of Lie point infinitesimal symmetries, are found. Our solutions contain families of traveling wave solutions, self-similar variables, and other forms with different combinations of the spacetime variables. The gravitational potentials are given in terms of elementary functions, and the line elements can be given explicitly in all cases. We show that the Friedmann dust model is regained as a special case in particular solutions. We can connect our results to earlier investigations and we show explicitly that our models are generalizations. Some of our solutions satisfy a linear equation of state. We also regain previously obtained solutions for the Euclidean star as a special case in our accelerating model. Our results highlight the importance of Lie symmetries of differential equations for problems arising in relativistic astrophysics.

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