Philosophiae Doctor - PhD / There is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them. / South Africa

Philosophiae Doctor - PhD / In this thesis, we solve some time-dependent partial differential equations, and systems of such equations, that governs reaction-diffusion models in biology. we design and implement some novel exponential time differencing schemes to integrate stiff systems of ordinary differential equations which arise from semi-discretization of the associated partial differential equations. We split the semi-linear PDE(s) into a linear, which contains the highly stiff part of the problem, and a nonlinear part, that is expected to vary more slowly than the linear part. Then we introduce higher-order finite difference approximations for the spatial discretization. Resulting systems of stiff ODEs are then solved by using exponential time differencing methods. We present stability properties of these methods along with extensive numerical simulations for a number of different reaction-diffusion models, including single and multi-species models. When the diffusivity is small many of the models considered in this work are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured by our proposed numerical schemes. Hence, the schemes that we have designed in this thesis are dynamically consistent. Finally, in many cases, we have compared our results with those obtained by other researchers.

[pt] Comumente as análises de estabilidade de taludes são realizadas mediante métodos determinísticos, os quais visam o cálculo de um fator de segurança único assumindo os valores dos parâmetros de resistência como representativos e fixos. Estes métodos não conseguem avaliar as incertezas existentes nas propriedades do solo e tampouco indicam a proporção de influência que tem cada parâmetro de resistência no valor do fator de segurança. Os métodos probabilísticos, com base nas teorias de probabilidade, confiabilidade e estatística, permitem estimar a influência dessas incertezas nos cálculos determinísticos, com a possibilidade de prever mais amplamente o risco de insucesso associado a um projeto geotécnico de estabilidade de taludes. O presente trabalho estuda a aplicação de três métodos probabilísticos (Monte Carlo, Hipercubo Latino e Estimativas Pontuais Alternativas) na avaliação de estabilidade de taludes, com auxílio de métodos de equilíbrio limite no cálculo do fator de segurança. Com objetivo de inferir o impacto das variáveis aleatórias nas estimativas de probabilidade e confiabilidade, bem como da importância de uma quantificação adequada dos valores de desvio padrão, são realizadas comparações dos resultados obtidos com métodos probabilísticos e determinísticos (método das fatias, método dos elementos finitos) discutindo as principais vantagens, dificuldades e limitações nas aplicações dos mesmos em problemas de estabilidade de taludes de solo. / [en] Slope stability analyses are usually carried out using deterministic methods, which aim the calculation of a single safety factor assuming the values of the shear strength parameters as representative and fixed. These methods fail to assess the uncertainties in soil properties and do not indicate the proportion of influence that each resistance parameter has on the final value of the safety factor. The probabilistic methods, based on probability, reliability and statistical theories, allow the estimation of the influence of these uncertainties on the deterministic calculations, with the possibility to broadly predict the risk of failure associated with a geotechnical slope stability project. This dissertation studies the application of three probabilistic methods (Monte Carlo, Latin Hypercube, and Alternative Point Estimates) in the evaluation of slope stability, with aid of limit equilibrium methods for the calculation of safety factors. In order to infer the impact of random variables on the estimates of probability and reliability, as well as the importance of an adequate quantification of the standard deviation values, comparisons are made among the results obtained with probabilistic and deterministic methods (limit equilibrium method, finite element method), discussing the main advantages, difficulties and limitations in their application to soil slope stability problems.

This thesis deals with creation of breakout prediction model implemented via artificial neural network in MATLAB, hence it contains theoretical description of artificial neural networks and also presents how the networks were used for the breakout prediction problem. In addition, models of heat conduction through the mold wall were created and used for generating neccesary data for proper training of artificial neural network. The thesis contains proper description of used numerical methods for solving the heat conduction problems. In the thesis are presented partial results and also results of testing created artificial neural network with usage of real data.

The master's thesis deals with possible ways of accelerating numerical computations, which are present in problems related to heat conduction in solids. The thesis summarizes basic characteristics of heat transfer phenomena with emphasis on heat conduction. Theoretical principles of control volume method are utilized to convert a direct heat conduction problem into a sparse linear system. Relevant fundamentals from the field of inverse heat conduction problems are presented with reference to intensive computations of direct problems of such kind. Numerical methods which are well-suited to find a solution of direct heat conduction problems are described. Remarks on practical implementation of time-efficient computations are made in relation with a two-dimensional heat conduction model. The results are compared and discussed with respect to obtained computational time for several tested methods.

<p>Multiple industry applications, including combustion, flow control, and medicine, have leveraged nanosecond pulsed plasma (NPP) discharges to create plasma generated reactive species (PGRS). The PGRS are essential to induce plasma-assisted mechanisms, but the rate of generation and permanence of these species remains complex. Many of the mechanisms surrounding plasma discharge have been discovered through experiments, but a consistent challenge of time scales limits the plasma measurements. Thus, a well-constructed model with experimental research will help elucidate complex plasma physics. The motivation of this work is to construct a feasible physical model within the additional numerical times scale limitations and computational resources. This thesis summarizes the development of a one-moment fluid model for NPP discharges, which are applied due to their efficacy in generating ionized and excited species from vacuum to atmospheric pressure. </p><p>From a pulsed power perspective, the influence of pulse parameters, such as electric field intensity, pulse shape and repetition rate, are critical; however, the effects of these parameters on PGRS remain incompletely characterized. Here, we assess the influence of pulse conditions on the electric field and PGRS computationally by coupling a quasi-one-dimensional model for a parallel plate geometry, with a Boltzmann solver (BOLSIG+) used to improve plasma species characterization. We first consider a low-pressure gas discharge (3 Torr) using a five-species model for argon. <a>We then extend to a 23 species model with a reduced set of reactions for air chemistry remaining at low pressure.</a> The foundations of a single NPP is first discussed to build upon the analysis of repeating pulses. Because many applications use multiple electric pulses (EPs) the need to examine EP parameters is necessary to optimize ionization and PGRS formation. </p><p>The major goal of this study is to understand how the delivered EP parameters scale with the generated species in the plasma. Beginning with a similar scaling study done by Paschen we examine the effects of scaling pressure and gap length when the product remains constant for the two models. This then leads to our study on the relationship of pulsed power for different voltages and pulse widths of EPs. By fixing the energy delivered to the gap for a single pulse we determine that the electron and ion number densities both increased with decreasing pulse duration; however, the rate of this increase of number densities appeared to reach a limit for 3 ns. These results suggest the feasibly of achieving comparable outputs using less expensive pulse generators with higher pulse duration and lower peak voltage. Lastly, we study these outcomes when increasing the number of pulses and discuss the effects of pulse repetition and the electron temperature.</p><p>Future work will extend this parametric study to different geometries (i.e. pin-to-plate, and pin-to pin) and ultimately incorporate this model into a high-fidelity computational fluid dynamics (CFD) model that may be compared to spectroscopic results under quiescent and flowing conditions will be discussed.<br></p>

Philosophiae Doctor - PhD / Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.

Au cours de l’utilisation de certains systèmes propulsifs, tels que les moteurs fusées cryotechniques ou les moteurs Diesel, le point de fonctionnement peut varier sur une large plage de pressions. Ces variations de pression peuvent conduire à un changement de régime thermodynamique si la pression critique du fluide est franchie, l’injection initialement diphasique devenant alors transcritique. Ce changement modifie la topologie de l’écoulement, ainsi que la dynamique du mélange, ce qui impacte le comportement de la flamme. L’objectif de cette thèse est de développer une méthodologie originale capable de traiter au sein du même solveur des écoulements sous-critiques ainsi que supercritiques. Pour cela, une extension du solveur AVBP-RG aux écoulements diphasiques sous-critiques est proposée, basée sur des modèles d’interface diffuse. Les développements nécessaires à l’intégration de ces modèles dans le cadre du solveur aux éléments finis sont effectués. Des simulations numériques multidimensionnelles sont ensuite proposées de manière à confronter le modèle à des données exprérimentales, vis-à-vis desquelles un bon accord est observé. Cesrésultats offrent des perspectives encourageantes vers de futures améliorations du modèle et des applications à des configurations industrielles complexes. / In various industrial combustion devices, such as liquid rocket engines at ignition or Diesel engines during the compression stage, the operating point varies over a wide range of pressures. These pressure variations can lead to a change of thermodynamic regime when the critical pressure is exceeded, switching from two-phase injection to transcritical injection. This change modifies the topology of the flow and the mixing, thereby impacting the flame dynamics. The objective of the present Ph.D thesis is to develop an original methodology able to address both subcritical and supercritical flows within the same solver. To achieve this, an extension of the real gas solver AVBP-RG to subcritical two-phase flows is provided, based on diffuse interface models. The required developments for the integration of such models into the finite-element framework of the solver are provided. Multidimensional numerical simulations are led in order to confront the model with experimental data, with which good agreement is observed. These results offer encouraging perspectives regarding further enhancements of the model and applications to complex industrial cases.

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