# Doctoral Degrees (Applied Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10413/7094

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Item Mathematical modelling of the Ebola virus disease.(2024) Abdalla, Suliman Jamiel Mohamed.; Govinder, Keshlan Sathasiva.; Chirove, Faraimunashe.Despite the numerous modelling efforts to advise public health physicians to understand the dynamics of the Ebola virus disease (EVD) and control its spread, the disease continued to spread in Africa. In the current thesis, we systematically review previous EVD models. Further, we develop novel mathematical models to explore two important problems during the 2018-2020 Kivu outbreak: the impact of geographically targeted vaccinations (GTVs) and the interplay between the attacks on Ebola treatment centres (ETCs) and the spread of EVD. In our systematic review, we identify many limitations in the modelling literature and provide brief suggestions for future work. Our modelling findings underscore the importance of considering GTVs in areas with high infections. In particular, we find that implementing GTVs in regions with high infections so that the total vaccinations are increased by 60% decreases the cumulative cases by 15%. On the other hand, we need to increase the vaccinations to more than 1000% to achieve the 15% decrease in EVD cases if we implement GTVs in areas with low infections. On the impact of the attacks on ETCs, we find that due to the attacks on ETCs, the cumulative cases increased by more than 17% during the 2018-2020 Kivu outbreak. We also find that when 10% of the hospitalised individuals flee the attacks on ETCs after spending only three days under treatment, the cumulative cases increased by more than 30% even if these individuals all returned to the ETCs three days later. On the other hand, if only half of these individuals returned to ETCs for treatment, the cumulative cases increase by approximately 50%. Further, when these patients spend one more day in the community, after which they all return to ETCs, the cumulative cases rise by an additional 10%. Global sensitivity analysis also confirmed these findings. To conclude, our literature systematic review is used to identify many critical factors which were overlooked in previous EVD models. Our modelling findings show that the attacks on ETCs can be destructive to the efforts of EVD response teams. Hence, it is important for decision-makers to tackle the reasons for community distrust and address the roots of the hostility towards ETCs. We also find that GTVs can be used to contain the spread of EVD when ring vaccinations, contact tracing and antiviral treatments cannot successfully control the spread of EVD.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 New families of exact solutions for compact stars.(2018) Kileba Matondo, Didier.; Maharaj, Sunil Dutt.; Ray, Subharthi.In this thesis we present new families of exact solutions to the Einstein and Einstein- Maxwell eld equations which are relevant in the description of highly compact stellar objects. We rst impose a linear equation of state to generate exact solutions in terms of elementary functions which contain earlier quark models, including those of Thirukkanesh and Maharaj (Class. Quantum Grav. 25, 235001 (2008)) and Mafa Takisa and Maharaj (Astrophys. Space Sci. 354, 463 (2013)). Secondly, we nd exact solutions in terms of elementary functions, Bessel and modi ed Bessel functions through the Finch and Skea geometry which satisfy all criteria for physical acceptability. From these models we regain the uncharged model of Finch and Skea (Class. Quantum Grav. 6, 467 (1989)) and the charged model of Hansraj and Maharaj (Int. J. Mod. Phys. D 15, 1311 (2006)) as particular cases. Thirdly, we nd new exact stellar models by imposing a symmetry condition on spacetime, namely a conformal Killing vector. We nd solutions to the eld equations with the help of the gravitational potentials related explicitly by the conformal vector established by Manjonjo et al (Eur. Phys. J. Plus 132, 62 (2017)). For each approach, we select a particular model to study the physical features and then masses and radii with accurate ranges consistent with observed numerical values of compact objects such as SAX J1808.4-3658, LMC X-4, SMC X-1, EXO 1785, Cen X-3, 4U1820-30, PSR J1903+327, Vela X-1 and PSR J1614-2230 are generated. The physical features in all cases are studied comprehensively, and we show that our solutions are stable, well behaved and have realistic physical features.Item Star formation as a function of environment in the MeerKAT Galaxy Clusters Legacy Survey=Ulwazibunkanyezi Njengethuluzi Lokuhlola isiqoqzinkanyezi iMeerKAT Galaxy Clusters Legacy.(2023) Kesebonye, Kabelo Calvin.; Hilton, Matthew James.; Knowles, Kenda Leigh.Probing the star formation (SF) activity of cluster galaxies paves an important path towards the understanding of cluster evolution. This thesis presents the study of star formation rates (SFR) in clusters using dustunbiased radio luminosities from the MeerKAT Galaxy Clusters Legacy Survey (MGCLS). Our radio data is complemented by optical data from the Dark Energy Camera Legacy Survey (DECaLS), for photometric redshifts, and also Sunyaev-Zel’dovich (SZ) effect-derived cluster masses from the Atacama Cosmology Telescope (ACT). We present the first statistical study of SFR in clusters using MeerKAT-detected galaxies which takes advantage of MeerKAT’s large field of view to investigate the relation between SF activity and cluster environments out to 2R200. Using radio diffuse emission in the form of haloes and relics as a proxy for cluster merger activity, we divide our cluster sample between disturbed/merger clusters and relaxed clusters. We observe a higher fraction of star-forming galaxies (fSF) in disturbed clusters than in relaxed clusters. Disturbed clusters also have higher masses (M200) and total SFR (ΣSFR) in contrast to relaxed clusters. On analysing the redshift evolution of the massnormalised ΣSFR, we observe a ≈ 4× decline in the SF activity of clusters from the redshift of 0.35 to 0.15, corresponding to ≈ 2 Gyr in look-back time. Our result is roughly consistent with the one from cluster studies that used infrared-derived SFR (≈ 5× decline) at a similar redshift slice as our sample. We use a subsample of double relic-hosting clusters to investigate the relation between cluster SF activity and the time that has passed since the merger started (tmerger) estimated from the relic distances from cluster cores. We observe an anti-correlation between ΣSFR and merger, suggesting that younger mergers have a higher SF activity. However, we see no clear correlation in the mass-normalised ΣSFR with tmerger. We also investigate for differences in the SF activity of galaxies closer to radio relics and those away from the relics and observe no significant differences between the two populations. Iqoqa Ukuhlola kokusebenza kolwazi-bunkanyezi ( star formation -SF) kwesixhobo sezinkanyezi cluster galaxies kucabe indlela emqoka ukuqonda kokukhula kwalamaqoqo (cluster evolution). Lolu cwaningo lwethula ukuhlolwa lwe star (formation rates -SFR) kuma-clusters kusetshenziswa i dust-unbiased radio luminosities kuMeerKAT Galaxy Clusters Legacy Survey (MGCLS). Imininingo ye-radio isebenzisana ne-optical data yeDark Energy Camera Legacy Survey (DECaLS), ukuhlola iphotometric redshifts, ne Sunyaev-Zel'dovich (SZ) nomthelela odalwa yicluster masses esuka kwiAtacama Cosmology Telescope (ACT). Kube sekwethulwa ucwaningo lokuqala lwezibalo lweSFR in clusters kusetshenziswa iMeerKAT-detected galaxies. Kusetshenziswe ithuba lobukhulu bomkhakha lokubonakala kahle kweMeerKAT ukuhlola ubudlelwane phakathi kwe-SFR nemvelo yamaclusters ezinhlobonhlobo zedynamical states. Kusetshenziswe iradio diffuse emission eyizinhlobo zehaloes nerelics njengegunya lokuhlola ukuhlangana kwamacluster , amasampula ama-cluster ahlukaniswe phakathi kwamadisturbed/merger clusters namarelaxed clusters. Okutholakele ukuthi ihigher fraction of star-forming galaxies (fSF) kumaclusters aphazamisekile (disturbed clusters) kunama clusters akhululekile (relaxed clusters). Amaclusters aphazamisekile anama masi aphezulu (M200) kanti isamba sonke siwuSFR (∑SFR) uma eqhathaniswa namaclusters aphazamisekile . Sekucutshungulwa iredshift evolution yemass-normalised ∑SFR, kutholakale ukuthi aphindwe kawu ≅4 times decline ekusebenzeni kwe SF esukela kwiredshift of 0.35 to 0.15, ahlobana ne ≅2 Gyr kwi look-back time. Imiphumela ilinganiselwe ngokuhambisana nezifundo zamacluster kusetshenziswa i-infrared-derived SFR (≅5 times decline) ngokufanayo neredshift slice njengamasampula ocwaningo lolu .Kusetshenziswe elinye isampula esigabeni sesibili ukuze kwenziwe idouble relic-hosting clusters ukuhlola ubudlelwane okusebenza kwama clusters (SF activity) kanye nesikhathi esisuke sidlulile ehlanganiswe (t_merger) kulinganiselwe kusukela ebangeni lerelic distances kucluster ewumsuka (cluster cores). Kutholakale ukungadlelani phakathi kwe ∑SFR ne t_merger, okuchaza ukuthi ukuhlanganiswa kwalawo asemancane kusebenzisa izinga eliphezulu (SF activity). Yize-kunjalo , akubonakali ubudlelwane obucacile kumass-normalised ∑SFR ne t_merger. Kuphinde kwahlolwa ukusebenza okungefani kumagalaxies asondelene neradio relics nalawo aqhelile kumarelics kwase kwangatholakala mehluko otheni kule miphakathi yomibili.Item Study of singularly perturbed models and its applications in ecology and epidemiology.(2017) Seuneu Tchamga, Milaine Sergine.; Banasiak, Jacek.In recent years the demand for a more accurate description of real life processes and advances in experimental techniques have resulted in construction of very complex mathematical models, consisting of tens, hundreds, if not thousands, of highly coupled di erential equations. The sheer size and complexity of such models often preclude any robust, theoretical or numerical, analysis of them. Fortunately, often such models describe phenomena occurring on vastly di erent time or size scales. We focused on complex processes with two time/size scales described by systems of ordinary di erential equations. In such a case, there is a small parameter that multiplies one or more derivatives. Using the Tikhonov Theorem, we have been able to understand the asymptotic behaviour of the solution to some complex epidemiological models. Furthermore, we present analysis based on the Butuzov theorem, which, for the purpose of the discussed models, was generalized to two dimensional non-autonomous problems. We applied the developed theory on an ecological model with interactions given by the mass action law.Item Investigation of the South African public TVET colleges’ engineering official mathematics curriculum for entry level artisans.(2023) Mazibuko, Godfrey Nkululeko.; Maharaj, Aneshkumar.One of the main objectives of any mathematics curriculum is to equip students with the necessary thinking skills for real-world problems. As the world evolves every day of our lives, so do the people living in it. Hence, the same exceptional functioning curriculum used in previous years is highly possible to be dysfunctional in the current days. Therefore, time and again curriculum evaluation is essential for both Basic and Higher education. However, before the actual curriculum evaluation, one should identify or develop suitable evaluation tool/s. In that regard, this study focused on the evaluation of the public Technical and Vocational Education and Training (TVET) colleges’ mathematics curriculum from N1 to N2. Initially, the intention of the current study was to collect data across all KwaZulu-Natal TVET Colleges, which was unsuccessful due to a lack of cooperation from some of the TVET colleges’ gatekeepers. The study was only able to access the eMnambithi TVET College data set, where 47 students participated. Two aspects were evaluated, namely, the participating students’ attainment of the curriculum objectives and the ability of the curriculum to equip students with high order thinking skills (HOTS). The Tyler’s objective model was adopted to evaluate the effectiveness of the curriculum to train students for the attainment of the curriculum’s objectives. That was done using the pre- and post-assessments method as stated by the pioneer of that model. The results indicated that the curriculum was most likely to be incapable of equipping the students for the attainment of its own objectives. Further on, this study used the Susceptible-Infected-Recovered (SIR) model to develop a new model called the Susceptible Vaccinated-Healthy-Infected-Recovered (SVHIR) model. The SVHIR model was used to evaluate the effectiveness of the curriculum to equip students with HOTS. Also, the results obtained from the SVHIR model indicated that the curriculum was most likely to be incapable of equipping the students with HOTS. It was also found that the students’ ability to attain the curriculum objectives and their HOTS have a strong linear relationship. The latter implied that fully equipping students with HOTS should enable them to better attain the curriculum objectives. The convenience sampling supports the need to conduct a future study that covers all the TVET colleges that did not respond to the researcher’s request for access on time. Further pursuance will give more clarity and findings that may or may not differ that much with the ones of this reported study.Item New exact solutions for neutral and charged shear-free relativistic fluids.(2022) Gumede, Sfundo Cebolenkosi.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.We study shear-free gravitating fluids in general relativity. We first analyse the integrability of the Emden-Fowler equation that governs the behaviour of shear-free neutral perfect fluid distributions. We find a new exact solution and generate a new first integral. The first integral is subject to an integrability condition which can be expressed as a third order differential equation whose solution can be expressed in terms of elementary functions and elliptic integrals. We extend this approach to include the effect of the electromagnetic charge. The Einstein-Maxwell system for a charged shear-free matter can be reduced to a generalized Emden-Fowler equation. We integrate this equation and find a new first integral. For this solution to exist two integral equations arise as integrability conditions. The integrability conditions can be solved to find new solutions. In both cases the first integrals are given parametrically. Our investigations suggest that complexity of a self-gravitating fluid is related to the existence of a first integral. For both neutral and charged fluids the general form of the parametric solution depends on a cubic and quartic polynomial respectively. The special case of repeated roots leads to simplification and this regains earlier results. We also study relativistic charged shear-free gravitating fluids in higher dimensions. Two classes of exact solutions to the Einstein-Maxwell equations are found. We obtain these solutions by reducing the Einstein-Maxwell equations to a single second order nonlinear partial differential equation containing two arbitrary functions. This generalizes the condition of pressure isotropy to higher dimensions; the new condition is functionally different from four dimensions. The new exact solutions obtained in higher dimensions reduce to known results in four dimensions. The presence of higher dimensions affects the dynamics of relativistic fluids in general relativity.Item Aspects of trapped surfaces and spacetime singularities.(2019) Sherif, Abbas Mohamed.; Maharaj, Sunil Dutt.; Goswami, Rituparno.Abstract available in PDF.Item Generalized radiating stellar models with cosmological constant and electric charge.(2019) Mahomed, Abu Bakr.; Maharaj, Sunil Dutt.; Narain, Rivendra Basanth.A general matter distribution, with the addition of the cosmological constant and electric charge, for the interior spacetime of a spherically symmetric radiating star undergoing gravitational collapse is considered in this investigation. The matching of the metric potentials and extrinsic curvature for the interior spacetime to the Vaidya exterior spacetime leads to the junction condition that relates the radial pressure to the heat flux. The presence of the cosmological constant and electric charge changes the nature of the problem significantly. Using Einstein-Maxwell field equations we express the junction condition as a Riccati equation in one of the metric potentials. In general this Riccati equation is not integrable. Special cases for particular matter distributions result in new classes of exact solutions to the Riccati equation. Previous results are also regained in this process. A transformation, called the horizon function, is then introduced to transform the Riccati equation into a simpler form. Several new classes of exact solutions are also found for the transformed Riccati equation. A new transformation called the generalized horizon function is introduced. This transformation preserves the form of the Riccati equation. The generalized horizon function leads to a transformed generalized Riccati equation. It is also possible to obtain earlier models by making assumptions on certain parameters. New models arise by restricting the values of parameters. The classes of solutions found can be given both implicitly and explicitly. The horizon function, and its generalization, can be obtained explicitly for all models.Item Fixed point theory in various generalized metric-type spaces.(2022) Jele, Thokozani Cyprian Martin.; Singh, Virath Sewnath.; Singh, Pravin.In the theory of fixed points, there are numerous articles dealing with generalization of the basic Banach contraction mapping principle. There has been two lines of approach. The first one is concerned with generalizations of the contractive conditions on the mapping space. The other line of investigation deals with various generalizations of the metric spaces and the results that can be obtained in these new frameworks, referred to as metric-type spaces. In this thesis, we elected for the latter approach by providing a more general framework for a b-metric space , G-metric space and S-metric space. In this thesis, we proved that these new metric-type spaces equipped with various contractions type mappings have unique fixed points and provide numerous examples of each metric-type spaced mentioned.Item Probing the nature of dark energy with 21-cm intensity mapping.(2020) Yohana, Elimboto Mwiki.; Ma, Yin-Zhe.Two approaches to measure the BAOs (baryon acoustic oscillations) with optical and radio telescopes, namely; galaxy redshift and intensity mapping (IM) surveys have been introduced and discussed in the literature. Among the two methods, the galaxy redshift survey has been used to great effect and is based on the detection and survey of millions of individual galaxies and measuring their redshifts by comparing templates of the spectral energy distributions of the light emitted from the galaxies with optical lines. IM is novel but a robust approach that focuses on surveys of extremely large volumes of galaxies without resolving each individual galaxy and can efficiently probe scales over redshift ranges inaccessible to the current galaxy redshift surveys. However, the IM survey has promisingly shown to have better overall sensitivity to the BAOs than the galaxy redshift survey but has a number of serious issues to be quantified. The most obvious of these issues is the presence of foreground contaminants from the Milky Way galaxy and extragalactic point sources which strongly dominate the neutral hydrogen (Hi) signal of our interest. Under this study, we are interested to realize the IM approach, pave the pathway, and optimize the scientific outputs of future radio experiments. We, therefore, carry out simulations and present forecasts of the cosmological constraints by employing Hi IM technique with three near-term radio telescopes by assuming 1 year of observational time. The telescopes considered here are Five-hundred-meter Aperture Spherical radio Telescope (FAST), BAOs In Neutral Gas Observations (BINGO), and Square Kilometre Array Phase I (SKA-I) single-dish experiments. We further forecast the combined constraints of the three radio telescopes with Planck measurements. In order to tackle the foreground challenge, we develop strategies to model various sky components and employ an approach to clean them from our Milky Way galaxy and extragalactic point sources by considering a typical single-dish radio telescope. Particularly, the Principal Component Analysis foreground separation approach considered can indeed recover the cosmological Hi signal to high precision. We show that, although the approach may face some challenges, it can be fully realized on the selected range of angular scales.Item Role of Weyl tensor and spacetime shear in relativistic fluids.(2021) Mayala, Roger Mbonga.; Maharaj, Sunil Dutt.; Goswami, Rituparno.The main gravitational theory in which we develop this work is general relativity. We study the role of the Weyl tensor in general relativistic fluid motion including the e↵ects of spacetime shear. Firstly we consider conformally flat perturbations on the Friedmann Lemaitre RobertsonWalker (FLRW) spacetime containing a general matter field. Working with the linearised field equations, we find some important geometrical properties of matter shear and vorticity, and show how they interact with the thermodynamic quantities in the absence of any free gravity powered by the Weyl curvature. We demonstrate that the matter shear obeys a transverse traceless tensor wave equation and the vorticity obeys a vector wave equation in this linearised regime. These shear and vorticity waves replace the gravitational waves in the sense that they causally carry information about local change in the curvature of these spacetimes. We also study the heat transport equation in this case, and show how this varies from the Newtonian case. Secondly we show that a general but shear-free perturbation of homogeneous and isotropic universes are necessarily silent, without any gravitational waves. We prove this in two steps. First, we establish that a shear-free perturbation of these universes are acceleration-free and the fluid flow geodesics of the background universe map onto themselves in the perturbed universe. This e↵ect then decouples the evolution equations of the electric and magnetic part of the Weyl tensor in the perturbed spacetimes and the magnetic part no longer contains any tensor modes. Although the electric part, that drives the tidal forces, does have tensor modes sourced by the anisotropic stress, these modes have homogeneous oscillations at every point on a time slice without any wave propagation. This analysis shows the critical role of the shear tensor in generating cosmological gravitational waves.Item Epidemiological modelling of foot and mouth disease control in cattle: incorporating time and spatial spread of disease dynamics.(2019) Tessema, Kassahun Mengist.; Chirove, Faraimunashe.; Sibanda, Precious.Foot and mouth disease (FMD) is a contagious animal viral infection that can spread rapidly if the disease is not monitored and controlled. Therefore, protecting livestock and controlling foot and mouth disease is important for preventing economic losses. Much of the global burden of economic losses due to foot and mouth disease falls on the worlds poorest countries that mostly depend upon the health of their livestock. In these countries, the availability of FMD also has an impact on the overall herd fertility, modifying the herd structure and affecting the selection of breeds. Modelling the dynamics of FMD using mathematical analysis and simulations can assist to monitor and control the spread of the disease. In this thesis, we develop, study, and analyse models of foot and mouth disease in cattle by incorporate vaccination that does not induce rapid protection, time delays, both time and spatial spread with different control strategies. The results show that even though vaccines may not induce rapid protection the combining of a high rate of vaccination and low loss of vaccine protection rate may be successful in reducing the foot and mouth burden provided critical vaccination thresholds are taken into consideration. The results also show that control strategies play a significant role in moving the animals into protected routes of infection than leaving more animals into the unprotected route of infection. We also capture the effects of prophylactic vaccination, reactive vaccination, prophylactic treatment, reactive culling and the effects of time delay. The results of foot and mouth disease with two-time delays show that the burden of infection decreases significantly when unprotected animals delay maximally their time to show clinical symptoms, and at the same time by increasing the effectiveness of the control strategies. The study also explores the effects of spatial diffusion, quarantine of clinically infected animals and shedding of foot and mouth disease virus into the environment. Analysis of foot and mouth disease control models suggests that implementing of an effective combination of control strategies, limiting the movement of susceptible animals and the shedding of FMDV protects animals from foot and mouth disease burden.Item A numerical study of heat and mass transfer in non-Newtonian nanofluid models.(2019) Mthethwa, Hloniphile Mildred Sithole.; Sibanda, Precious.; Motsa, Sandile Sydney.A theoretical study of boundary layer flow, heat and mass transport in non-Newtonian nanofluids is presented. Because of the diversity in the physical structure and properties of non-Newtonian fluids, it is not possible to describe their behaviour using a single constitutive model. In the literature, several constitutive models have been proposed to predict the behaviour and rheological properties of non-Newtonian fluids. The question of interest is how the fluid physical parameters affect the boundary layer flow, and heat and mass transfer in various nanofluids. In this thesis, nanofluid models in various geometries and subject to different boundary conditions are constructed and analyzed. A range of fluid models from simple to complex are studied, leading to highly nonlinear and coupled differential equations, which require advanced numerical methods for their solution. This thesis is a conjoin between mathematical modeling of non-Newtonian nanofluid flows and numerical methods for solving differential equations. Some recent spectral techniques for finding numerical solutions of nonlinear systems of differential equations that model fluid flow problems are used. The numerical methods of primary interest are spectral quasilinearization, local linearization and bivariate local linearization methods. Consequently, one of the objectives of this thesis is to test the accuracy, robustness and general validity of these methods. The dependency of heat and mass transfer, and skin friction coefficients on the physical parameters is quantified and discussed. Results show that nanofluids and physical parameters have an important and significant impact on boundary layer flows, and on heat and mass transfer processes.Item Modelling the physical dynamics of estuaries for management purposes.(1996) Slinger, Jill Hillary.; Hearne, John W.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.Item On thermal convective instability in rotating fluids.(2018) Noreldin, Osman A. I.; Sibanda, Precious.Abstract available on the PDF.Item On paired decoupled quasi-linearization methods for solving nonlinear systems of differential equations that model boundary layer fluid flow problems.(2018) Otegbeye, Olumuyiwa.; Motsa, Sandile Sydney.Two numerical methods, namely the spectral quasilinearization method (SQLM) and the spectral local linearization method (SLLM), have been found to be highly efficient methods for solving boundary layer flow problems that are modeled using systems of differential equations. Conclusions have been drawn that the SLLM gives highly accurate results but requires more iterations than the SQLM to converge to a consistent solution. This leads to the problem of figuring out how to improve on the rate of convergence of the SLLM while maintaining its high accuracy. The objective of this thesis is to introduce a method that makes use of quasilinearization in pairs of equations to decouple large systems of differential equations. This numerical method, hereinafter called the paired quasilinearization method (PQLM) seeks to break down a large coupled nonlinear system of differential equations into smaller linearized pairs of equations. We describe the numerical algorithm for general systems of both ordinary and partial differential equations. We also describe the implementation of spectral methods to our respective numerical algorithms. We use MATHEMATICA to carry out the numerical analysis of the PQLM throughout the thesis and MATLAB for investigating the influence of various parameters on the flow profiles in Chapters 4, 5 and 6. We begin the thesis by defining the various terminologies, processes and methods that are applied throughout the course of the study. We apply the proposed paired methods to systems of ordinary and partial differential equations that model boundary layer flow problems. A comparative study is carried out on the different possible combinations made for each example in order to determine the most suitable pairing needed to generate the most accurate solutions. We test convergence speed using the infinity norm of solution error. We also test their accuracies by using the infinity norm of the residual errors. We also compare our method to the SLLM to investigate if we have successfully improved the convergence of the SLLM while maintaining its accuracy level. Influence of various parameters on fluid flow is also investigated and the results obtained show that the paired quasilinearization method (PQLM) is an efficient and accurate method for solving boundary layer flow problems. It is also observed that a small number of grid-points are needed to produce convergent numerical solutions using the PQLM when compared to methods like the finite difference method, finite element method and finite volume method, among others. The key finding is that the PQLM improves on the rate of convergence of the SLLM in general. It is also discovered that the pairings with the most nonlinearities give the best rate of convergence and accuracy.Item New solutions for a radiating star.(2018) Zitha, Vusi Monde.; Maharaj, Sunil Dutt.; Govender, Megandren.Item Mathematical models for heat and mass transfer in nanofluid flows.(2018) Ahamed, Sami Musa Sulieman.; Sibanda, Precious.The behaviour and evolution of most physical phenomena is often best described using mathematical models in the form of systems of ordinary and partial differential equations. A typical example of such phenomena is the flow of a viscous impressible fluid which is described by the Navier-Stokes equations, first derived in the nineteenth century using physical approximations and the principles of mass and momentum conservation. The flow of fluids, and the growth of flow instabilities has been the subject of many investigations because fluids have wide uses in engineering and science, including as carriers of heat, solutes and aggregates. Conventional heat transfer fluids used in engineering applications include air, water and oil. However, each of these fluids has an inherently low thermal conductivity that severely limit heat exchange efficiency. Suspension of nanosized solid particles in traditional heat transfer fluids significantly increases the thermophysical properties of such fluids leading to better heat transfer performance. In this study we present theoretical models to investigate the flow of unsteady nanofluids, heat and mass transport in porous media. Different flow configurations are assumed including an inclined cylinder, a moving surface, a stretching cone and the flow of a polymer nanocomposite modeled as an Oldroyd-B fluid. The nanoparticles assumed include copper, silver and titanium dioxide with water as the base fluid. Most recent boundary-layer nanofluid flow studies assume that the nanoparticle volume fraction can be actively controlled at a bounding solid surface, similar to temperature controls. However, in practice, such controls present significant challenges, and may, in practice, not be possible. In this study the nanoparticle flux at the boundary surface is assumed to be zero. Unsteadiness in fluid flows leads to complex system of partial differential equations. These transport equations are often highly nonlinear and cannot be solved to find exact solutions that describe the evolution of the physical phenomena modeled. A large number of numerical or semi-numerical techniques exist in the literature for finding solutions of nonlinear systems of equations. Some of these methods may, however be subject to certain limitations including slow convergence rates and a small radius of convergence. In recent years, innovative linearization techniques used together with spectral methods have been suggested as suitable tools for solving systems of ordinary and partial differential equations. The techniques which include the spectral local linearization method, spectral relaxation method and the spectral quasiliearization method are used in this study to solve the transport equations, and to determine how the flow characteristics are impacted by changes in certain important physical and fluid parameters. The findings show that these methods give accurate solutions and that the speed of convergence of solutions is comparable with methods such as the Keller-box, Galerkin, and other finite difference or finite element methods. The study gives new insights, and result on the influence of certain events, such as internal heat generation, velocity slip, nanoparticle thermophoresis and random motion on the flow structure, heat and mass transfer rates and the fluid properties in the case of a nanofluid.Item On the linearization of systems of differential equations.(2019) Mkhize, Thembisile Gloria.; Govinder, Keshlan Sathasiva.; Moyo, Sibusiso.; Meleshko, Sergey V.