Interval analysis is an essential tool in the construction of validated numerical solutions of Initial Value Problems (IVP) for Ordinary (ODE) and Partial (PDE) Differential Equations. A validated solution typically consists of guaranteed lower and upper bounds for the exact solution or set of exact solutions in the case of uncertain data, i.e. it is an interval function (enclosure) containing all solutions of the problem. IVP for ODE: The central point of discussion is the wrapping effect. A new concept of wrapping function is introduced and applied in studying this effect. It is proved that the wrapping function is the limit of the enclosures produced by any method of certain type (propagate and wrap type). Then, the wrapping effect can be quantified as the difference between the wrapping function and the optimal interval enclosure of the solution set (or some norm of it). The problems with no wrapping effect are characterized as problems for which the wrapping function equals the optimal interval enclosure. A sufficient condition for no wrapping effect is that there exist a linear transformation, preserving the intervals, which reduces the right-hand side of the system of ODE to a quasi-isotone function. This condition is also necessary for linear problems and "near" necessary in the general case. Hyperbolic PDE: The Initial Value Problem with periodic boundary conditions for the wave equation is considered. It is proved that under certain conditions the problem is an operator equation with an operator of monotone type. Using the established monotone properties, an interval (validated) method for numerical solution of the problem is proposed. The solution is obtained step by step in the time dimension as a Fourier series of the space variable and a polynomial of the time variable. The numerical implementation involves computations in Fourier and Taylor functoids. Propagation of discontinuo~swaves is a serious problem when a Fourier series is used (Gibbs phenomenon, etc.). We propose the combined use of periodic splines and Fourier series for representing discontinuous functions and a method for propagating discontinuous waves. The numerical implementation involves computations in a Fourier hyper functoid. / Mathematical Sciences / D. Phil. (Mathematics)

This paper develops a numerical algorithm that produces finite element solutions for a broad class of partial differential equations. The method is based on steepest descent methods in the Sobolev space H¹(Ω). Although the method may be applied in more general settings, we consider only differential equations that may be written as a first order quasi-linear system. The method is developed in a Hilbert space setting where strong convergence is established for part of the iteration. We also prove convergence for an inner iteration in the finite element setting. The method is demonstrated on Burger's equation and the Navier-Stokes equations as applied to the square cavity flow problem. Numerical evidence suggests that the accuracy of the method is second order,. A documented listing of the FORTRAN code for the Navier-Stokes equations is included.

Neste trabalho estudam-se modelos matemáticos e computacionais do transporte de múltiplas espécies em cenários de contaminação subsuperficial, que constituem meios porosos saturados. Deste modo, combinam-se os efeitos de sorção em condições de equilíbrio e não-equilíbrio, representadas pelas isotermas do tipo de Freundlich, com os processos de biodegradação dados pelas cinéticas de Monod. Matematicamente, o problema é descrito por um sistema de equações diferenciais parciais não lineares de convecção-difusão-reação acopladas pelos termos de reação. Do ponto de vista numérico esta é a primeira vez na literatura que são tratadas sorções não-lineares em condições de equilíbrio e não-equilíbrio acopladas com biodegradação não linear no transporte fortemente convectivo de múltiplas espécies. Primeiramente, apresenta-se um algoritmo completamente acoplado, no sentido que em cada equação do sistema os operadores diferenciais são discretizados e resolvidos simultaneamente como um sistema completo. Em seguida, desenvolve-se um método de decomposição de operadores,o qual trata de uma maneira seqüencial os termos de elementos finitos estabilizado combinado com um esquema de diferenças finitas na aproximação das equações de transporte. Para tratar-se as não linearidades, em geral, foram usados algoritmos do tipo Newton-Raphson e um método de Runge-Kutta na segunda etapa de decomposição de operadores. Os diversos resultados obtidos das simulações computacionais demonstraram a eficiência das metodologias propostas, apresentando de forma clara a influência dos efeitos não lineares de sorção sobre a biodegradação na localização espacial e na forma da pluma. O conhecimento destas interações é bastante importante para ajudar nas escolhas de biorremediação in situ em regiões subsuperficiais contaminadas.

In vitro tissue engineering is a method for developing living and functional tissues external to the body, often within a device called a bioreactor to control the chemical and mechanical environment. However, the quality of bone tissue engineered products is currently inadequate for clinical use as the implant cannot bear weight. In an effort to improve the quality of the construct, hydrostatic pressure, the pressure in a fluid at equilibrium that is required to balance the force exerted by the weight of the fluid above, has been investigated as a mechanical stimulus for promoting extracellular matrix deposition and mineralisation within bone tissue. Thus far, little research has been performed into understanding the response of bone tissue cells to mechanical stimulation. In this thesis we investigate an in vitro bone tissue engineering experimental setup, whereby human mesenchymal stem cells are seeded within a collagen gel and cultured in a hydrostatic pressure bioreactor. In collaboration with experimentalists a suite of mathematical models of increasing complexity is developed and appropriate numerical methods are used to simulate these models. Each of the models investigates different aspects of the experimental setup, from focusing on global quantities of interest through to investigating their detailed local spatial distribution. The aim of this work is to increase understanding of the underlying physical processes which drive the growth and development of the construct, and identify which factors contribute to the highly heterogeneous spatial distribution of the mineralised extracellular matrix seen experimentally. The first model considered is a purely temporal model, where the evolution of cells, solid substrate, which accounts for the initial collagen scaffold and deposited extracellular matrix along with attendant mineralisation, and fluid in response to the applied pressure are examined. We demonstrate that including the history of the mechanical loading of cells is important in determining the quantity of deposited substrate. The second and third models extend this non-spatial model, and examine biochemically and biomechanically-induced spatial patterning separately. The first of these spatial models demonstrates that nutrient diffusion along with nutrient-dependent mass transfer terms qualitatively reproduces the heterogeneous spatial effects seen experimentally. The second multiphase model is used to investigate whether the magnitude of the shear stresses generated by fluid flow, can qualitatively explain the heterogeneous mineralisation seen in the experiments. Numerical simulations reveal that the spatial distribution of the fluid shear stress magnitude is highly heterogeneous, which could be related to the spatial heterogeneity in the mineralisation seen experimentally.

The retina is the ocular tissue responsible for the detection of light. Its extensive demand for oxygen, coupled with a concomitant elevated supply, renders this tissue prone to both hypoxia and hyperoxia. In this thesis, we construct mathematical models of the retina, formulated as systems of reaction-diffusion equations, investigating its oxygen-related dynamics in healthy and diseased states. In the healthy state, we model the oxygen distribution across the human retina, examining the efficacy of the protein neuroglobin in the prevention of hypoxia. It has been suggested that neuroglobin could prevent hypoxia, either by transporting oxygen from regions where it is rich to those where it is poor, or by storing oxygen during periods of diminished supply or increased uptake. Numerical solutions demonstrate that neuroglobin may be effective in preventing or alleviating hypoxia via oxygen transport, but that its capacity for oxygen storage is essentially negligible, whilst asymptotic analysis reveals that, contrary to the prevailing assumption, neuroglobin's oxygen affinity is near optimal for oxygen transport. A further asymptotic analysis justifies the common approximation of a piecewise constant oxygen uptake across the retina, placing existing models upon a stronger theoretical foundation. In the diseased state, we explore the effect of hyperoxia upon the progression of the inherited retinal diseases, known collectively as retinitis pigmentosa. Both numerical solutions and asymptotic analyses show that this mechanism may replicate many of the patterns of retinal degeneration seen in vivo, but that others are inaccessible to it, demonstrating both the strengths and weaknesses of the oxygen toxicity hypothesis. It is shown that the wave speed of hyperoxic degeneration is negatively correlated with the local photoreceptor density, high density regions acting as a barrier to the spread of photoreceptor loss. The effects of capillary degeneration and treatment with antioxidants or trophic factors are also investigated, demonstrating that each has the potential to delay, halt or partially reverse photoreceptor loss. In addition to answering questions that are not accessible to experimental investigation, these models generate a number of experimentally testable predictions, forming the first loop in what has the potential to be a fruitful experimental/modelling cycle.

Correct modelling of embankment breach formation is essential for an accurate assessment of the associated flood risk. Modelling breach formation due to overflow requires a thorough understanding of the geotechnical processes in unsaturated soils as well as erosion processes under supercritical flow conditions. This thesis describes 1D slope stability analysis performed for unsaturated soils whose moisture content changes with time. The analysis performed shows that sediment-laden gravity flows play an important role in the erosion behaviour of embankments. The thesis also describes a practical, fast breach model based on a simplified description of the physical processes that can be used in modelling and decision support frameworks for flooding. To predict the breach hydrograph, the rapid model distinguishes between breach formation due to headcut erosion and surface erosion in the case of failure due to overflow. The model also predicts the breach hydrograph in the case of failure due to piping. The assumptions with respect to breach flow modelling are reviewed, and result in a new set of breadth-integrated Navier-Stokes equations, that account for wall shear stresses and a variable breadth geometry. The vertical 2D flow field described by the equations can be used to calculate accurately the stresses on the embankment during the early stages of breach formation. Pressure-correction methods are given for solving the 2D Navier-Stokes equations for a variable breadth, and good agreement is found when validating the flow model against analytical solutions.

Made available in DSpace on 2015-03-04T18:51:03Z (GMT). No. of bitstreams: 1 Apresentacao e resumos.pdf: 92299 bytes, checksum: 957bb50a66505cf631cffee02a010cfc (MD5) Previous issue date: 2006-07-14 / Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior / Neste trabalho estudam-se modelos matemáticos e computacionais do transporte de múltiplas espécies em cenários de contaminação subsuperficial, que constituem meios porosos saturados. Deste modo, combinam-se os efeitos de sorção em condições de equilíbrio e não-equilíbrio, representadas pelas isotermas do tipo de Freundlich, com os processos de biodegradação dados pelas cinéticas de Monod. Matematicamente, o problema é descrito por um sistema de equações diferenciais parciais não lineares de convecção-difusão-reação acopladas pelos termos de reação. Do ponto de vista numérico esta é a primeira vez na literatura que são tratadas sorções não-lineares em condições de equilíbrio e não-equilíbrio acopladas com biodegradação não linear no transporte fortemente convectivo de múltiplas espécies. Primeiramente, apresenta-se um algoritmo completamente acoplado, no sentido que em cada equação do sistema os operadores diferenciais são discretizados e resolvidos simultaneamente como um sistema completo. Em seguida, desenvolve-se um método de decomposição de operadores,o qual trata de uma maneira seqüencial os termos de elementos finitos estabilizado combinado com um esquema de diferenças finitas na aproximação das equações de transporte. Para tratar-se as não linearidades, em geral, foram usados algoritmos do tipo Newton-Raphson e um método de Runge-Kutta na segunda etapa de decomposição de operadores. Os diversos resultados obtidos das simulações computacionais demonstraram a eficiência das metodologias propostas, apresentando de forma clara a influência dos efeitos não lineares de sorção sobre a biodegradação na localização espacial e na forma da pluma. O conhecimento destas interações é bastante importante para ajudar nas escolhas de biorremediação in situ em regiões subsuperficiais contaminadas.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / O objetivo desta tese é o desenvolvimento de algoritmos para determinar as soluções, e para determinação de fontes, das equações de Poisson e da condução de calor definidas em uma esfera. Determinamos as formas das equações de Poisson e de calor sobre a esfera, e desenvolvemos métodos iterativos, baseados em uma malha icosaedral e sua respectiva malha dual, para obter as soluções das mesmas. Mostramos que os métodos iterativos convergem para as soluções das equações discretizadas. Empregamos o método de regularização iterada de Alifanov para resolver o problema inverso, de determinação de fonte, definido na esfera. / The objective of this thesis is the development of algorithms to determine the solutions, and for determination of sources of, the equations of Poisson and heat conduction for a sphere. We establish the form of equations of Poisson and heat on the sphere, and developed iterative methods, based on a icosaedral mesh and its dual mesh, to obtain the solutions for them. It is shown that the iterative methods converge to the solutions of the equations discretizadas. It employed the method of settlement of Alifanov iterated to solve the inverse problem, determination of source, set in the sphere.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / O objetivo desta tese é o desenvolvimento de algoritmos para determinar as soluções, e para determinação de fontes, das equações de Poisson e da condução de calor definidas em uma esfera. Determinamos as formas das equações de Poisson e de calor sobre a esfera, e desenvolvemos métodos iterativos, baseados em uma malha icosaedral e sua respectiva malha dual, para obter as soluções das mesmas. Mostramos que os métodos iterativos convergem para as soluções das equações discretizadas. Empregamos o método de regularização iterada de Alifanov para resolver o problema inverso, de determinação de fonte, definido na esfera. / The objective of this thesis is the development of algorithms to determine the solutions, and for determination of sources of, the equations of Poisson and heat conduction for a sphere. We establish the form of equations of Poisson and heat on the sphere, and developed iterative methods, based on a icosaedral mesh and its dual mesh, to obtain the solutions for them. It is shown that the iterative methods converge to the solutions of the equations discretizadas. It employed the method of settlement of Alifanov iterated to solve the inverse problem, determination of source, set in the sphere.

The nonlinear Schrödinger equation appears in different fields of physics, for example in the theory of Bose-Einstein condensates or in wave propagation models. From a mathematical point of view, the study of this equation is interesting and delicate, notably because it can have a very rich set of solutions with various behaviours.<p><p>In this thesis, we have been interested in standing waves, which satisfy an elliptic partial differential equation. When this equation is seen as a singularly perturbed problem, its solutions concentrate, in the sense that they converge uniformly to zero outside some concentration set, while they remain positive on this set.<p><p>We have obtained three kind of new results. Firstly, under symmetry assumptions, we have found solutions concentrating on a sphere. Secondly, we have obtained the same type of solutions for the Schrödinger-Poisson system. The method consists in applying the mountain pass theorem to a penalized problem. Thirdly, we have proved the existence of solutions of the nonlinear Schrödinger equation concentrating at a local maximum of the potential. These solutions are found by a more general minimax principle. Our results are characterized by very weak assumptions on the potential./<p><p>L'équation de Schrödinger non-linéaire apparaît dans différents domaines de la physique, par exemple dans la théorie des condensats de Bose-Einstein ou dans des modèles de propagation d'ondes. D'un point de vue mathématique, l'étude de cette équation est intéressante et délicate, notamment parce qu'elle peut posséder un ensemble très riche de solutions avec des comportements variés. <p><p>Dans cette thèse ,nous nous sommes intéressés aux ondes stationnaires, qui satisfont une équation aux dérivées partielles elliptique. Lorsque cette équation est vue comme un problème de perturbations singulières, ses solutions se concentrent, dans le sens où elles tendent uniformément vers zéro en dehors d'un certain ensemble de concentration, tout en restant positives sur cet ensemble. <p><p>Nous avons obtenu trois types de résultats nouveaux. Premièrement, sous des hypothèses de symétrie, nous avons trouvé des solutions qui se concentrent sur une sphère. Deuxièmement, nous avons obtenu le même type de solutions pour le système de Schrödinger-Poisson. La méthode consiste à appliquer le théorème du col à un problème pénalisé. Troisièmement, nous avons démontré l'existence de solutions de l'équation de Schrödinger non-linéaire qui se concentrent en un maximum local du potentiel. Ces solutions sont obtenues par un principe de minimax plus général. Nos résultats se caractérisent par des hypothèses très faibles sur le potentiel. / Doctorat en sciences, Spécialisation mathématiques / info:eu-repo/semantics/nonPublished