# Applied Mathematics

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Item 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.Item 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.Item Algebraizing deductive systems.(1995) Van Alten, Clint Johann.; Raftery, James Gordon.; Sturm, Teo.Abstract available in PDF.Item Analysis and numerical solutions of fragmentation equation with transport.(2012) Wetsi, Poka David.; Banasiak, Jacek.; Shindin, Sergey Konstantinovich.Fragmentation equations occur naturally in many real world problems, see [ZM85, ZM86, HEL91, CEH91, HGEL96, SLLM00, Ban02, BL03, Ban04, BA06] and references therein. Mathematical study of these equations is mostly concentrated on building existence and uniqueness theories and on qualitative analysis of solutions (shattering), some effort has be done in finding solutions analytically. In this project, we deal with numerical analysis of fragmentation equation with transport. First, we provide some existence results in Banach and Hilbert settings, then we turn to numerical analysis. For this approximation and interpolation theory for generalized Laguerre functions is derived. Using these results we formulate Laguerre pseudospectral method and provide its stability and convergence analysis. The project is concluded with several numerical experiments.Item Analysis of models arising from heat conduction through fins using Lie symmetries and Tanh method.(2021) Bulunga, Vusi Andile.; Mhlongo, Mfanafikile Don.Abstract available in PDF.Item 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.Item 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.Item 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.Item 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.Item Application of the wavelet transform for sparse matrix systems and PDEs.(2009) Karambal, Issa.; Paramasur, Nabendra.; Singh, Pravin.We consider the application of the wavelet transform for solving sparse matrix systems and partial differential equations. The first part is devoted to the theory and algorithms of wavelets. The second part is concerned with the sparse representation of matrices and well-known operators. The third part is directed to the application of wavelets to partial differential equations, and to sparse linear systems resulting from differential equations. We present several numerical examples and simulations for the above cases.Item 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.Item 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.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.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.Item Aspects of connectedness in metric frames.(2019) Rathilal, Cerene.; Pillay, Paranjothi.; Baboolal, Dharmanand.Abstract available in PDF file.Item 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.Item Aspects of trapped surfaces and spacetime singularities.(2019) Sherif, Abbas Mohamed.; Maharaj, Sunil Dutt.; Goswami, Rituparno.Abstract available in PDF.Item Assessment of variability in on-farm trials : a Uganda case.(2002) Lapaka, Odong Thomas; Njuho, Peter Mungai.On-farm trials techniques have become an integral part of research aimed at improving agricultural production especially in subsistence farming. The poor performance of certain technologies on the farmers' fields known to have performed well on stations have been of concern. Traditionally, on-farm trials are meant to address such discrepancies. The main problems associated with on-farm trials in most developing countries are high variability and inappropriate application of statistical knowledge known to work on station to on-farm situation. Characterisation of various on-farm variability and orientation of existing statistical methods may lead to improved agricultural research. Characterization of the various forms of variability in on-farm trials was conducted. Based on these forms of variability, estimation procedures and their strength have been assessed. Special analytical tools for handling non-replicated experiments known to be common to on-farm trials are presented. The above stated procedures have been illustrated through a review of Uganda case. To understand on-farm variability require grouping of sources of variability into agronomic, animal and socioeconomic components. This led to a deeper understanding of levels of variability and appropriate estimation procedures. The mixed model, modified stability analysis and additive main effects and multiplicative interaction methods have been found to play a role in on-farm trials. Proper approach to on-farm trials and application of appropriate statistical tools will lead to efficient results that will subsequently enhance agricultural production especially under subsistence farming.Item 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.Item Bayesian analysis of cosmological models.(2010) Moodley, Darell.; Moodley, Kavilan.; Sealfon, C.In this thesis, we utilise the framework of Bayesian statistics to discriminate between models of the cosmological mass function. We first review the cosmological model and the formation and distribution of galaxy clusters before formulating a statistic within the Bayesian framework, namely the Bayesian razor, that allows model testing of probability distributions. The Bayesian razor is used to discriminate between three popular mass functions, namely the Press-Schechter, Sheth-Tormen and normalisable Tinker models. With a small number of particles in the simulation, we find that the simpler model is preferred due to the Occam’s razor effect, but as the size of the simulation increases the more complex model, if taken to be the true model, is preferred. We establish criteria on the size of the simulation that is required to decisively favour a given model and investigate the dependence of the simulation size on the threshold mass for clusters, and prior probability distributions. Finally we outline how our method can be extended to consider more realistic N-body simulations or be applied to observational data.Item Bivariate pseudospectral collocation algorithms for nonlinear partial differential equations.(2016) Magagula, Vusi Mpendulo.; Motsa, Sandile Sydney.; Sibanda, Precious.Abstract available in PDF file.