Masters Degrees (Pure Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10413/7121

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Analysis of mixed convection in an air filled square cavity.
(2010) Ducasse, Deborah S.; Sibanda, Precious.
A steady state two-dimensional mixed convection problem in an air filled square unit cavity has been numerically investigated. Two different cases of heating are investigated and compared. In the first case, the bottom wall was uniformly heated, the side walls were linearly heated and the top moving wall was heated sinusoidally. The second case differed from the first in that the side walls were instead uniformly cooled. This investigation is an extension of the work by Basak et al. [6, 7] who investigated mixed convection in a square cavity with similar boundary conditions to the cases listed above with the exception of the top wall which was well insulated. In this dissertation, their work is extended to include a sinusoidally heated top wall. The nonlinear coupled equations are solved using the Penalty Galerkin Finite Element Method. Stream function and isotherm results are found for various values of the Reynolds number and the Grashof number. The strength of the circulation is seen to increase with increasing Grashof number and to decrease with increasing Reynolds number for both cases of heating. A comparison is made between the stream function and isotherm results for the two cases. The results for the rate of heat transfer in terms of the Nusselt number are discussed. Both local and average Nusselt number results are presented and discussed. The average Nusselt number is found using Simpson's 1/3rd rule. The rate of heat transfer is found to be higher at all four walls for the case of cooled side walls than that of linearly heated side walls.
Centre manifold theory with an application in population modelling.
(2009) Phongi, Eddy Kimba.; Banasiak, Jacek.
There are basically two types of variables in population modelling, global and local variables. The former describes the behavior of the entire population while the latter describes the behavior of individuals within this population. The description of the population using local variables is more detailed, but it is also computationally costly. In many cases to study the dynamics of this population, it is sufficient to focus only on global variables. In applied sciences, to achieve this, the method of aggregation of variables is used. One of methods used to mathematically justify variables aggregation is the centre manifold theory. In this dissertation we provide detailed proofs of basic results of the centre manifold theory and discuss some examples of applications in population modelling.
Continuous symmetries of difference equations.
(2011) Nteumagne, Bienvenue Feugang.; Govinder, Keshlan Sathasiva.
We consider the study of symmetry analysis of difference equations. The original work done by Lie about a century ago is known to be one of the best methods of solving differential equations. Lie's theory of difference equations on the contrary, was only first explored about twenty years ago. In 1984, Maeda [42] constructed the similarity methods for difference equations. Some work has been done in the field of symmetries of difference equations for the past years. Given an ordinary or partial differential equation (PDE), one can apply Lie algebra techniques to analyze the problem. It is commonly known that the number of independent variables can be reduced after the symmetries of the equation are obtained. One can determine the optimal system of the equation in order to get a reduction of the independent variables. In addition, using the method, one can obtain new solutions from known ones. This feature is interesting because some differential equations have apparently useless trivial solutions, but applying Lie symmetries to them, more interesting solutions are obtained. The question arises when it happens that our equation contains a discrete quantity. In other words, we aim at investigating steps to be performed when we have a difference equation. Doing so, we find symmetries of difference equations and use them to linearize and reduce the order of difference equations. In this work, we analyze the work done by some researchers in the field and apply their results to some examples. This work will focus on the topical review of symmetries of difference equations and going through that will enable us to make some contribution to the field in the near future.
Ermakov systems : a group theoretic approach.
(1993) Govinder, Keshlan Sathasiva.; Leach, Peter Gavin Lawrence.
The physical world is, for the most part, modelled using second order ordinary differential equations. The time-dependent simple harmonic oscillator and the Ermakov-Pinney equation (which together form an Ermakov system) are two examples that jointly and separately describe many physical situations. We study Ermakov systems from the point of view of the algebraic properties of differential equations. The idea of generalised Ermakov systems is introduced and their relationship to the Lie algebra sl(2, R) is explained. We show that the 'compact' form of generalized Ermakov systems has an infinite dimensional Lie algebra. Such algebras are usually associated only with first order equations in the context of ordinary differential equations. Apart from the Ermakov invariant which shares the infinite-dimensional algebra of the 'compact' equation, the other three integrals force the dimension of the algebra to be reduced to the three of sl(2, R). Subsequently we establish a new class of Ermakov systems by considering equations invariant under sl(2, R) (in two dimensions) and sl(2, R) EB so(3) (in three dimensions). The former class contains the generalized Ermakov system as a special case in which the force is velocity-independent. The latter case is a generalization of the classical equation of motion of the magnetic monopole which is well known to possess the conserved Poincare vector. We demonstrate that in fact there are three such vectors for all equations of this type.
Evolutionary dynamics of coexisting species.
(2000) Muir, Peter William.; Apaloo, Joseph.; Hearne, John W.
Ever since Maynard-Smith and Price first introduced the concept of an evolutionary stable strategy (ESS) in 1973, there has been a growing amount of work in and around this field. Many new concepts have been introduced, quite often several times over, with different acronyms by different authors. This led to other authors trying to collect and collate the various terms (for example Lessard, 1990 & Eshel, 1996) in order to promote better understanding ofthe topic. It has been noticed that dynamic selection did not always lead to the establishment of an ESS. This led to the development ofthe concept ofa continuously stable strategy (CSS), and the claim that dynamic selection leads to the establishment of an ESSif it is a CSS. It has since been proved that this is not always the case, as a CSS may not be able to displace its near neighbours in pairwise ecological competitions. The concept of a neighbourhood invader strategy (NIS) was introduced, and when used in conjunction with the concept of an ESS, produced the evolutionary stable neighbourhood invader strategy (ESNIS) which is an unbeatable strategy. This work has tried to extend what has already been done in this field by investigating the dynamics of coexisting species, concentrating on systems whose dynamics are governed by Lotka-Volterra competition models. It is proved that an ESNIS coalition is an optimal strategy which will displace any size and composition of incumbent populations, and which will be immune to invasions by any other mutant populations, because the ESNIS coalition, when it exists, is unique. It has also been shown that an ESNIS coalition cannot exist in an ecologically stable state with any finite number of strategies in its neighbourhood. The equilibrium population when the ESNIS coalition is the only population present is globally stable in a n-dimensional system (for finite n), where the ESNIS coalition interacts with n - 2 other strategies in its neighbourhood. The dynamical behaviour of coexisting species was examined when the incumbent species interacted with various invading species. The different behaviour ofthe incumbent population when invaded by a coalition using either an ESNIS or an NIS phenotype underlines the difference in the various strategies. Similar simulations were intended for invaders who were using an ESS phenotype, but unfortunately the ESS coalition could not be found. If the invading coalition use NIS phenotypes then the outcome is not certain. Some, but not all of the incumbents might become extinct, and the degree to which the invaders flourish is very dependent on the nature ofthe incumbents. However, if the invading species form an ESNIS coalition, one is certain of the outcome. The invaders will eliminate the incumbents, and stabilise at their equilibrium populations. This will occur regardless of the composition and number of incumbent species, as the ESNIS coalition forms a globally stable equilibrium point when it is at its equilibrium populations, with no other species present. The only unknown fact about the outcome in this case is the number ofgenerations that will pass before the system reaches the globally stable equilibrium consisting ofjust the ESNIS. For systems whose dynamics are not given by Lotka-Volterra equations, the existence ofa unique, globally stable ESNIS coalition has not been proved. Moreover, simulations of a non Lotka-Volterra system designed to determine the applicability ofthe proof were inconclusive, due to the ESS coalition not having unique population sizes. Whether or not the proof presented in this work can be extended to non Lotka-Volterra systems remains to be determined.
Filter characterisations of the extendibility of continuous functions.
(1991) Maltby, Gavin Richard.; Swart, Johan.
Abstract available in PDF.
First integrals for the Bianchi universes : supplementation of the Noetherian integrals with first integrals obtained by using Lie symmetries.
(1997) Pantazi, Hara.; Leach, Peter Gavin Lawrence.
No abstract available.
Iterative algorithms for approximating solutions of some optimization problems in Hadamard spaces.
(2019) Ogwo, Grace Nnennaya.; Mewomo, Oluwatosin Temitope.
Abstract available in PDF.
Iterative algorithms for approximating solutions of variational inequality problems and monotone inclusion problems.
(2017) Chinedu, Izuchukwu.; Mewomo, Oluwatosin Temitope.
In this work, we introduce and study an iterative algorithm independent of the operator norm for approximating a common solution of split equality variational inequality prob- lem and split equality xed point problem. Using our algorithm, we state and prove a strong convergence theorem for approximating an element in the intersection of the set of solutions of a split equality variational inequality problem and the set of solutions of a split equality xed point problem for demicontractive mappings in real Hilbert spaces. We then considered nite families of split equality variational inequality problems and proposed an iterative algorithm for approximating a common solution of this problem and the multiple-sets split equality xed point problem for countable families of multivalued type-one demicontractive-type mappings in real Hilbert spaces. A strong convergence re- sult of the sequence generated by our proposed algorithm to a solution of this problem was also established. We further extend our study from the frame work of real Hilbert spaces to more general p-uniformly convex Banach spaces which are also uniformly smooth. In this space, we introduce an iterative algorithm and prove a strong convergence theorem for approximating a common solution of split equality monotone inclusion problem and split equality xed point problem for right Bregman strongly nonexpansive mappings. Finally, we presented numerical examples of our theorems and applied our results to study the convex minimization problems and equilibrium problems.
Iterative approximation of solutions of some optimization problems in Banach spaces.
(2018) Oyewole, Olawale Kazeem.; Mewomo, Oluwatosin Temitope.
Let C be a nonempty closed convex subset of a q-uniformly smooth Banach space X which admits a weakly sequentially continuous generalized duality mapping. In this dissertation, we study the approximation of the zero of a strongly accretive operator A : X ! X which is also a xed point of a k-strictly pseudo-contractive self mapping T of C: Also, we introduce a U-mapping for nite family of mixed equilibrium problems involving 􀀀 relaxed monotone operators. We prove a strong convergence theorem for nding a common solution of nite family of these equilibrium problems in a uniformly smooth and strictly convex Banach space. We present some applications of this theorem and a numerical example. Furthermore, due to the faster rate of convergence of inertial type algorithm, we propose an inertial type iterative algorithm and prove a weak convergence theorem of the scheme to a solution of split variational inclusion problems involving accretive operators in Banach spaces. We give some applications and a numerical example to show the relevance of our result. Our results in this dissertation extend and improve some recent results in the literature.
Iterative schemes for approximating common solutions of certain optimization and fixed point problems in Hilbert spaces.
(2021) Olona, Musa Adewale.; Mewomo, Oluwatosin Temitope.
In this dissertation, we introduce a shrinking projection method of an inertial type with self-adaptive step size for finding a common element of the set of solutions of Split Gen- eralized Equilibrium Problem (SGEP) and the set of common fixed points of a countable family of nonexpansive multivalued mappings in real Hilbert spaces. The self-adaptive step size incorporated helps to overcome the difficulty of having to compute the operator norm while the inertial term accelerates the rate of convergence of the propose algorithm. Under standard and mild conditions, we prove a strong convergence theorem for the sequence generated by the proposed algorithm and obtain some consequent results. We apply our result to solve Split Mixed Variational Inequality Problem (SMVIP) and Split Minimiza- tion Problem (SMP), and present numerical examples to illustrate the performance of our algorithm in comparison with other existing algorithms. Moreover, we investigate the problem of finding common solutions of Equilibrium Problem (EP), Variational Inclusion Problem (VIP)and Fixed Point Problem (FPP) for an infinite family of strict pseudo- contractive mappings. We propose an iterative scheme which combines inertial technique with viscosity method for approximating common solutions of these problems in Hilbert spaces. Under mild conditions, we prove a strong theorem for the proposed algorithm and apply our results to approximate the solutions of other optimization problems. Finally, we present a numerical example to demonstrate the efficiency of our algorithm in comparison with other existing methods in the literature. Our results improve and complement contemporary results in the literature in this direction.
Lie symmetries of junction conditions for radiating stars.
(2011) Abebe, Gezahegn Zewdie.; Govinder, Keshlan Sathasiva.; Maharaj, Sunil Dutt.
We consider shear-free radiating spherical stars in general relativity. In particular we study the junction condition relating the pressure to the heat flux at the boundary of the star. This is a nonlinear equation in the metric functions. We analyse the junction condition when the spacetime is conformally flat, and when the particles are travelling in geodesic motion. We transform the governing equation using the method of Lie analysis. The Lie symmetry generators that leave the equation invariant are identifed and we generate the optimal system in each case. Each element of the optimal system is used to reduce the partial differential equation to an ordinary differential equation which is further analysed. As a result, particular solutions to the junction condition are presented. These exact solutions can be presented in terms of elementary functions. Many of the solutions found are new and could be useful in the modelling process. Our analysis is the first comprehensive treatment of the boundary condition using a symmetry approach. We have shown that this approach is useful in generating new results.
Locally conformal almost kenmotsu manifolds.
(2019) Maduna, Snethemba Hlobisile.; Massamba, Fortuné.
Mathematical modeling of R5 and X4 HIV : from within host dynamics to the epidemiology of HIV infection.
Manda, Edna C.; Chirove, Faraimunashe.
Most existing models have considered the immunological processes occurring within the host and the epidemiological processes occurring at population level as decoupled systems. We present a new model using continuous systems of non linear ordinary differential equations by directly linking the within host dynamics capturing the interactions between Langerhans cells, CD4+ T-Cells, R5 HIV and X4 HIV and the without host dynamics of a basic compartmental HIV/AIDS, susceptible, infected, AIDS model. The model captures the biological theories of the cells that take part in HIV transmission. The study incorporates in its analysis the differences in time scales of the fast within host dynamics and the slow without host dynamics. In the mathematical analysis, important thresholds, the reproduction numbers, were computed which are useful in predicting the progression of the infection both within the host and without the host. The study results showed that the model exhibits four within host equilibrium points inclusive of three endemic equilibria whose effects translate into different scenarios at the population level. All the endemic equilibria were shown to be globally stable using Lyapunov functions and this is an important result in linking the within host dynamics to the population dynamics, because the disease free equilibrium point ceases to exist. The linked models had no effect on the basic reproduction numbers of the within host dynamics but on the basic reproduction number of the population dynamics. The effects of linking were observed on the endemic equilibrium points of both the within host and population dynamics. Therefore, linking the two dynamics leads to the increase in the viral load within the host and increase in the epidemic levels in the population dynamics.
Noether's theorem and first integrals of ordinary differential equations.
(1997) Moyo, Sibusiso.; Leach, Peter Gavin Lawrence.
The Lie theory of extended groups is a practical tool in the analysis of differential equations, particularly in the construction of solutions. A formalism of the Lie theory is given and contrasted with Noether's theorem which plays a prominent role in the analysis of differential equations derivable from a Lagrangian. The relationship between the Lie and Noether approach to differential equations is investigated. The standard separation of Lie point symmetries into Noetherian and nonNoetherian symmetries is shown to be irrelevant within the context of nonlocality. This also emphasises the role played by nonlocal symmetries in such an approach.
On amenability properties of some closed ideals of B(X)
(2018) Buthelezi, Thabo Njabulo.; Mewomo, Oluwatosin Temitope.
Abstract available in PDF file.
On free convection and heat transfer in a micropolar fluid flow past a moving semi-infinite plate.
(2012) Tessema, Kassahun Mengist.; Sibanda, Precious.
In this dissertation we investigate free convective heat and mass transfer in micropolar fluid flow past a moving semi-infinite vertical porous plate in the presence of a magnetic field. The aim of this study was to use recent semi-numerical methods such as the successive linearisation method and the spectral-homotopy analysis method to study the effects of viscous heating and the effects of different fluid parameters. The governing boundary layer equations for linear momentum, angular momentum (microrotation), temperature and concentration profiles are transformed to a system of ordinary differential equations and solved using the successive linearisation method and the spectral-homotopy analysis method. The accuracy of the solutions was determined by comparison with numerical approximations obtained using the Matlab bvp4c solver. The influences of the micropolar parameter, Darcy number, Prandtl number, Schmidt number, magnetic parameter, heat absorption parameter, Soret and Dufour numbers, local Reynolds number and Grashof number on velocity, microrotation, temperature and concentration profiles were determined. The results obtained are presented graphically and in tabular form.
On Huppert Conjecture for some quasi-simple groups.
(2014) Majozi, Philani Rodney.; TongViet, Hung P.; Massamba, Fortuné.
Abstract available in PDF file.
On pseudo-amenability of C(X;A) for norm irregular Banach algebra A.
(2017) Adiele, Ugochukwu.; Mewomo, Oluwatosin Temitope.
Abstract available in PDF file.
On the geometry of CR-manifolds.
(2015) Mazibuko, Langelihle.; Massamba, Fortuné.
We study two classes of CR-submanifolds in Kählerian and cosymplectic manifolds. More precisely, we compare the geometry of CR-submanifolds of the above two underlying smooth manifolds. We derive expressions relat- ing the sectional curvatures, the necessary and sufficient conditions for the integrability of distributions. Further, we study totally umbilical, totally geodesic and foliation geometry of the CR-submanifolds of both spaces and found many interesting results. We prove that, under some condition, there are classes CR submanifold in cosymplectic space forms which are in the classes extrinsic spheres. Examples are given throughout the thesis.

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