Thesis (MSc (Applied Mathematics))--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: In recent years the Laplace inversion method has emerged as a viable alternative method for the numerical solution of PDEs. Effective methods for the numerical inversion are based on the approximation of the Bromwich integral. In this thesis, a numerical study is undertaken to compare the efficiency of the Laplace inversion method with more conventional time integrator methods. Particularly, we consider the method-of-lines based on MATLAB’s ODE15s and the Crank-Nicolson method. Our studies include an introductory chapter on the Laplace inversion method. Then we proceed with spectral methods for the space discretization where we introduce the interpolation polynomial and the concept of a differentiation matrix to approximate derivatives of a function. Next, formulas of the numerical differentiation formulas (NDFs) implemented in ODE15s, as well as the well-known second order Crank-Nicolson method, are derived. In the Laplace method, to compute the Bromwich integral, we use the trapezoidal rule over a hyperbolic contour. Enhancement to the computational efficiency of these methods include the LU as well as the Hessenberg decompositions. In order to compare the three methods, we consider two criteria: The number of linear system solves per unit of accuracy and the CPU time per unit of accuracy. The numerical results demonstrate that the new method, i.e., the Laplace inversion method, is accurate to an exponential order of convergence compared to the linear convergence rate of the ODE15s and the Crank-Nicolson methods. This exponential convergence leads to high accuracy with only a few linear system solves. Similarly, in terms of computational cost, the Laplace inversion method is more efficient than ODE15s and the Crank-Nicolson method as the results show. Finally, we apply with satisfactory results the inversion method to the axial dispersion model and the heat equation in two dimensions. / AFRIKAANSE OPSOMMING: In die afgelope paar jaar het die Laplace omkeringsmetode na vore getree as ’n lewensvatbare alternatiewe metode vir die numeriese oplossing van PDVs. Effektiewe metodes vir die numeriese omkering word gebasseer op die benadering van die Bromwich integraal. In hierdie tesis word ’n numeriese studie onderneem om die effektiwiteit van die Laplace omkeringsmetode te vergelyk met meer konvensionele tydintegrasie metodes. Ons ondersoek spesifiek die metode-van-lyne, gebasseer op MATLAB se ODE15s en die Crank-Nicolson metode. Ons studies sluit in ’n inleidende hoofstuk oor die Laplace omkeringsmetode. Dan gaan ons voort met spektraalmetodes vir die ruimtelike diskretisasie, waar ons die interpolasie polinoom invoer sowel as die konsep van ’n differensiasie-matriks waarmee afgeleides van ’n funksie benader kan word. Daarna word formules vir die numeriese differensiasie formules (NDFs) ingebou in ODE15s herlei, sowel as die welbekende tweede orde Crank-Nicolson metode. Om die Bromwich integraal te benader in die Laplace metode, gebruik ons die trapesiumreël oor ’n hiperboliese kontoer. Die berekeningskoste van al hierdie metodes word verbeter met die LU sowel as die Hessenberg ontbindings. Ten einde die drie metodes te vergelyk beskou ons twee kriteria: Die aantal lineêre stelsels wat moet opgelos word per eenheid van akkuraatheid, en die sentrale prosesseringstyd per eenheid van akkuraatheid. Die numeriese resultate demonstreer dat die nuwe metode, d.i. die Laplace omkeringsmetode, akkuraat is tot ’n eksponensiële orde van konvergensie in vergelyking tot die lineêre konvergensie van ODE15s en die Crank-Nicolson metodes. Die eksponensiële konvergensie lei na hoë akkuraatheid met slegs ’n klein aantal oplossings van die lineêre stelsel. Netso, in terme van berekeningskoste is die Laplace omkeringsmetode meer effektief as ODE15s en die Crank-Nicolson metode. Laastens pas ons die omkeringsmetode toe op die aksiale dispersiemodel sowel as die hittevergelyking in twee dimensies, met bevredigende resultate.

In the thesis we study particular sequences of invariant differ- ential operators of first and second order which live on homogeneous spaces of a particular type of parabolic geometries. We show that they form a reso- lution of the kernel of the first operator and that they descend to resolutions of overdetermined, constant coefficient, first order systems of PDE's called the k-Dirac operators. This gives uniform description of resolutions of the k-Dirac operator studied in Clifford analysis. We give formula for second order operators which appear in the resolutions. 1

Let g be a complex semisimple Lie algebra, Vλ an irreducible g-module with high weight λ, pI a standard parabolic subalgebra of g with Levi factor £I and nil radical nI, and H*(nI, Vλ) the cohomology group of Λn'I ⊗Vλ. We describe the decomposition of H*(nI, Vλ) into irreducible £1-modules.

Orientador: Marcelo da Silva Montenegro / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-16T14:07:09Z (GMT). No. of bitstreams: 1 Melo_AlisonMarceloVanDerLaan_M.pdf: 595460 bytes, checksum: f3496cc25c882ea841e02b15bffe5256 (MD5) Previous issue date: 2010 / Resumo: Observação: O resumo, na íntegra poderá ser visualizado no texto completo da tese digital / Abstract: Note: The complete abstract is available with the full electronic digital thesis or dissertations / Mestrado / Matematica / Mestre em Matemática

Orientador: Marcelo da Silva Montenegro / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T06:27:08Z (GMT). No. of bitstreams: 1 Ottoboni_RafaelRodrigo_M.pdf: 900733 bytes, checksum: ddb6b509b4ec4392f5b2145085b216b5 (MD5) Previous issue date: 2007 / Resumo: Por apresentar basicamente fórmulas, o resumo na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: The complete abstract is available with the full electronic digital thesis or dissertations / Mestrado / Mestre em Matemática

FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Um concentrador solar parabÃlico à um tipo de refletor usado para aumentar a intensidade da radiaÃÃo em uma superfÃcie. Para que ocorra a reflexÃo adequada da radiaÃÃo solar, o refletor parabÃlico deve constantemente rastrear a posiÃÃo do sol de tal forma que os raios solares sejam refletidos em um tubo que se encontra na posiÃÃo de foco da parÃbola. A proposta deste trabalho à desenvolver um sistema eletrÃnico de controle de baixo custo para concentradores parabÃlicos aplicados à refrigeraÃÃo e proporcionar autonomia ao equipamento. TambÃm à apresentado um sistema mecÃnico para acionamento do rastreador. Por ser parte de um sistema de refrigeraÃÃo, o tubo colocado no foco da parÃbola deve sofrer processos de aquecimento e resfriamento. O sistema à constituÃdo de uma placa de controle com um microcontrolador em seu nÃcleo, sensores de luminosidade e sensores limitadores de movimento e um motor de passo responsÃvel pelo acionamento da estrutura mÃvel do concentrador solar. AtravÃs do sistema desenvolvido, pode-se fazer com que sejam atingidas temperaturas em torno de 180ÂC na superfÃcie externa do tubo coletor colocado na posiÃÃo de foco da parÃbola / A parabolic solar concentrator is a type of reflector utilized to increase the radiation intensity on a surface. In order to have a proper reflection of solar radiation, the parabolic reflector should constantly track the sunâs position so that solar rays are converged to a tube that is at the focus position of the parabola. The purpose of this study is to develop an electronic control system for low-cost parabolic trough concentrators applied to refrigeration process, and to provide autonomy to the equipment. It also is presented a mechanical system to guide the tracker. As part of a refrigeration device, the tube placed at the parabolaâs focus should experiment heating and cooling processes. The electronic system consists of a control plate with a microprocessor at its core, motion limiting and light sensors, and a step motor responsible for move the solar concentrator structure. Through this developed system, temperatures close to 180oC can be reached on external surface of the heat adsorption tube, which is placed on the position of parabolaâs focus

As the communication deficit in the Arctic region is enormous especially above 75 [Deg] N latitude, the concern and opportunity of providing reliable & efficient connectivity in the Arctic region has beenduly noted & understood by N66 Connect AB (N66). This case study documents a comprehensive research which implements system engineering approach for establishment of a Ground Station (GS) at Svalbard, Norway with sole focus of connecting the inaccessible geographical region lying in the Arctic with rest of the world. Several GS system & subsystem are studied and comparative analysis is made on how the communication can be established with the N66 Connect AB (N66)’s potential clients and its satellites that are to be deployed in September, 2018.The case study resulted in analysis of several risks involved during development & operation of the GS,the hardware, software & operational architecture, the features of GS’s system capable of meeting N66’s objectives and the market potential of the service after GS operations.

Dans la première partie, on démontre des inégalités de Carleman pour des systèmes paraboliques. Au chapitre 1, on démontre des inégalités de stabilité pour un système parabolique 2 x 2 en utilisant des inégalités de Carleman avec une seule observation. Il s'agit d'un problème inverse pour l'identification des coefficients et les conditions initiales du système. Le chapitre2 est consacré aux inégalités de Carleman pour des systèmes paraboliques dont les coefficients de diffusion sont de classe C1 par morceaux ou à variations bornées. A la fin, on donne quelques applications à la contrôlabilité à zéro. La seconde partie est consacrée à l'étude d'un problème de diffraction d'ondes acoustiques dans un demi-plan homogène. Il s'agit d'un problème aux limites associé à l'équation de Helmholtz dans le demi-plan supérieur avec une donnée de Neumann non homogène au bord. On apporte des éléments de réponse sur la question d'unicité et d'existence des solutions pour certaines classes de la donnée au bord. / In the first part, we prove Carleman estimates for parabolic systems. In chapter1, we prove stability inequalities for 2 x 2 parabolic system using Carleman estimates with one observation. It is concerns to the identification of the coefficients and initial conditions of the system. The chapter2 is devoted to th Carleman estimates of parabolic systems for which the diffusion coefficients are assumed to be ofclass piecewise C1 or with bounded variations. In the end, we give some applications to the null controllability. The second part is devoted to the study of the scattering problem of acoustics waves in a homogeneous half-plane. It is about a boundary value problem associated to the Helmholtz equation in theupper half-plane with a nonhomogeneous Neumann boundary data. We provide some answers to the question of uniqueness and existence of solutions for some classes of the boundary data.

Orientador: Olivâine Santana de Queiroz / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T03:06:43Z (GMT). No. of bitstreams: 1 Hitomi_EduardoEizoAramaki_M.pdf: 5800906 bytes, checksum: 04b93921a20d8ab0f71d4977b9e93e73 (MD5) Previous issue date: 2015 / Resumo: Nesta dissertação realizamos um estudo sobre o fluxo de curvatura média em espaços Euclidianos sob as perspectivas analítica e geométrica. Tratamos inicialmente da existência e regularidade de soluções em tempos pequenos de equações parabólicas quase lineares de segunda ordem em variedades Riemannianas, o que é essencial para garantirmos a existência de uma solução suave em tempo pequeno do fluxo de curvatura média. Em uma segunda parte, passamos a alguns resultados sobre o comportamento no intervalo maximal de existência de uma solução suave da hipersuperfície em evolução, por meio de equações das componentes geométricas associadas e de Princípios de Máximo. Próximo desse tempo maximal, analisamos a formação de singularidades do Tipo I por meio da Fórmula de Monotonicidade de Huisken e de rescalings, e do Tipo II por meio de uma técnica de blow-up devida a Hamilton. Em especial, reservamos o caso de curvas a um capítulo a parte e apresentamos resultados clássicos da teoria de curve-shortening flows / Abstract: In this dissertation we study the mean curvature flow in Euclidean spaces from the analytic and geometric point of view. We deal initially with short-time existence and regularity of a solution for second order quasilinear parabolic equations on Riemannian manifolds, which is essential to guarantee the short-time existence of a smooth solution to the mean curvature flow. In a second part, we present some results concerning the behavior of the evolving hypersurface close to the maximal time of existence of a smooth solution, by means of Maximum Principles and evolution equations of the associated geometric components. Close to this maximal time, we analyse the formation of singularities of Type I by means of rescalings and Huisken's Monotonicity Formula, and of Type II by means of a blow-up technique due to Hamilton. In particular, we reserve the case of curves to a separate chapter, where we present some classical results in curve-shortening flow theory / Mestrado / Matematica / Mestre em Matemática

In recent years power generation from renewable energy has grown substantially both in South Africa and around the world. This growth is set to continue as there is more pressure to reduce the burning of fossil fuels. However, renewable energy power generation suffers from unpredictability, which causes problems when it comes to managing power grids. Concentrated Solar Power (CSP) plants offer a practical solution to store power in the form of thermal energy storage (TES). Thus, the plant can run when there is no solar energy available, leading to a more stable power supply. Unfortunately, CSP plants cost more than other renewables such as photovoltaic and wind power. Thus, there is a need for research into how to bring down the cost of CSP plants. One of the most proven types of CSP is the parabolic trough plant. The most recent innovation is to try and use molten salt as the heat transfer fluid which would reduce the cost of the plant. However, this new technology has not been implemented on a full scale CSP plant and little testing has been done to prove the technology. The HPS2 is a test facility aimed at testing the use of molten salt as a heat transfer fluid (HTF). This test facility, located in Evora Portugal, is being developed by an international consortium led by the German DLR institute of Solar Research. It is one of the first test facilities of its kind where experiments will be conducted to demonstrate the validity of using molten salt as a HTF and a storage medium in a parabolic trough CSP plant. The HPS2 test facility is not yet operational and there is a need for a dynamic thermofluid process model to better understand and predict both its steady state and transient operational behaviour. This dissertation reports on the development of such a dynamic thermofluid process model and the results obtained from it. The process model developed primarily focuses on the steam cycle with the TES incorporated into the model. The physical geometry of each of the components are employed to construct discretized elements for which the conservation of mass, energy, and momentum are applied in a one-dimensional network approach. The economizer and evaporator combined has a helical coil geometry and uses molten salt as a heat transfer fluid, which is unique. Thus, correlations had to be adjusted for the flow characteristics found in the economizer/evaporator. Results from the steady state simulations of the steam cycle show that the molten salt mass flowrate through the steam generation system will have to be reduced from the initially expected value to meet operational requirements. Results of the dynamic simulations show that the test facility will be able to produce a constant power supply despite transient solar conditions and highlights key dynamic responses for operators to be aware of.