Ce travail est consacré à l'étude du comportement en temps grand d'équations aux dérivées partielles de type parabolique. Plus particulièrement, on s'intéresse à des équations non linéaires de type diffusion, qui interviennent dans de nombreux modèles issus de la physique (par exemple l'équation des milieux poreux) ou de la biologie (par exemple le modèle de Patlak-Keller-Segel pour la chimiotaxie). Dans les chapitres I et II on s'intéresse à une amélioration de l'inégalité de Sobolev à travers son inégalité duale, l'inégalité de Hardy-Littlewood-Sobolev, dans le cadre du laplacien ordinaire et du laplacien fractionnaire, respectivement. Le chapitre III est un passage en revue de l'inégalité d'Onofri, qui joue le rôle de l'inégalité de Sobolev pour la dimension deux. De nouveaux résultats sont apportés, dont certains sont étendus aux variétés riemanniennes au chapitre IV. Enfin, le chapitre V traite des états stationnaires de deux modèles paraboliques, utilisés pour l'étude du déplacement de foules et la modélisation en biologie (chimiotaxie). / This work is dedicated to the study of the large time behaviour of some parabolic type partial differential equations. More specifically, we look into non linear diffusion equations that appear in a number of models arising in physics (e.g. the porous medium equation) or biology (e.g. the Patlak-Keller-Segel model for chemotaxis)Chapters I and II deal with an improved Sobolev inequality by means of its dual, the Hardy-Littlewood-Sobolev inequality, in the framework of the standard and fractional Laplacian, respectively. Chapter III is a review of the Onofri inequality,which acts as the Sobolev inequality for dimension two. New results are provided, and some of them are extended to Riemannian manifolds in Chapter IV. Finally, Chapter V deals with the stationary states of two parabolic models, used for thestudy of crowd motion and modeling in biologie (chemotaxis).

In this thesis we will discuss the well-posedness and long-time dynamics of curved beam and thermoelastic plates. First, we considered the Bresse system with nonlinear damping and forcing terms. For this model we show the Timoshenko system as a singular limit of the Bresse system as the arch curvature l goes to 0 and under suitable assumptions on the nonlinearity we prove the existence of a smooth global attractor with finite fractal dimension and exponential attractors as well. We also compare the Bresse system with the Timoshenko system, in the sense of upper-semicontinuity of their attractors as l → 0. Second, we study a full von Karman system, this model accounts for vertical and in plane displacements. For this system we add a nonlinear thermal coupling and free boundary conditions. It is shown that the system, without any mechanical dissipation imposed on vertical displacements, admits a global attractor which is also smooth and of finite fractal dimension. / Neste trabalho iremos discutir a existência, unicidade, dependência contínua e a dinâmica a longo prazo das soluções de um sistema de equações que modela a vibração de vigas curvas e um modelo de placas termoelásticas. Primeiro consideramos o modelo de Bresse com dissipação não linear e forças externas. Provamos que o sistema de Timoshenko pode ser obtido como limite do sistema de Bresse quando o arco de curvatura l tende para zero e sob algumas hipóteses, mostramos a existência de um atrator global com dimensão fractal finita. Também comparamos o sistema de Bresse com o sistema de Timoshenko no sentido da semicontinuidade de seus atratores quando o parâmetro l → 0. Na segunda parte estudamos o sistema de full Von Karmam. Neste modelo adicionamos efeitos térmicos e condições de fronteira do tipo livre. Mostramos que esse problema, sem dissipação mecânica no deslocamento vertical, também possui um atrator global regular com dimensão infinita.

A Técnica do Super-Passo pode melhorar significantemente a performance do Método de Euler Explícito, reduzindo a restrição existente ao passo no tempo. A técnica é descrita para a equação do calor linear. É mostrada a simplicidade de sua implementação para o caso do Método de Euler Explícito. A perfomance da Técnica do Super-Passo é comparada aos Métodos de Euler Explícito e Implícito, e Crank-Nicolson. Os resultados obtidos sugerem que o Super-Passo pode melhorar a eficiência do Método de Euler Explícito em aproximadamente uma ordem de grandeza reduzindo o tempo de processamento, enquanto que o erro produzido pela Técnica do Super-Passo é comparável ao produzido pelo Método de Euler Implícito. / The Super-Time-Stepping Technique can significantly increase the performance of the Explícit Euler Method, reducing the existing tie step restriction. The Techinique is described for a linear heat equation. The simplicity of this implementation for the case of the Explícit Euler Method is shown. The perfomance of the Technique of the Super-Time-Stepping is compared to Explicit and Implicit Euler, and Crank-Nicolson Methods. The obtained results suggest that the Technique of the Super-Time-Stepping potentially increases the efficiency of the Explicit Euler Method by 2 factor of 3 regarding the processing time, while the error produced is comparable to that produced by the Implicit Euler Method.

Neste trabalho estudamos questões relativas a existência, unicidade, dependência contínua, continuidade, taxas de decaimento e comportamento assintótico de soluções para uma classe de equações de placas lineares e não lineares. No primeiro capítulo revisamos alguns conteúdos e colecionamos uma série de resultados provenientes da teoria geral de análise funcional, semigrupos lineares e atratores, os quais serão aplicados ao longo desta tese. Nos dois próximos capítulos abordamos uma equação da placa de quarta ordem dissipativa com perturbações não lineares do tipo p- Laplaciano e localmente Lipschitz e com memória. No segundo capítulo provamos a estabilidade exponencial de energia correspondente ao problema homogêneo com memória de segunda ordem. Em seguida, no terceiro capítulo estabelecemos resultados que comprovam a existência de um atrator global com dimensão fractal finita para o sistema dinâmico associado ao problema com história de deslocamento relativo que equivale ao problema original. Finalmente, no quarto capítulo tratamos um modelo viscoelástico de placas de Mindlin-Timoshenko de segunda ordem. Nesta ocasião, consideramos essecialmente dois casos, o primeiro quando o sistema é totalmente dissipativo e, em seguida, quando o sistema é parcialmente dissipativo. No primeiro caso, determinamos que o semigrupo linear associado ao problema é analítico e, como consequência, é exponencialmente estável. No segundo caso, mostramos que o semigrupo perde decaimento exponencial e analiticidade, no entanto, provamos que as soluções possuem decaimento do tipo polinomial / In this work we study some questions concerning with existence, uniqueness, continuous dependence, continuity, rates of decay and asymptotic behavior of solutions for a class of linear and nonlinear plate equations. In the first chapter we review some concepts and collect a series of results provided from general theory of functional analysis, linear semigroups and attractors which will be applied throughout this thesis. In the next two chapters we discuss a damped plate equation of fourth order with nonlinear perturbations of the lower order of p-Laplacian type and locally Lipschitz, and a memory term. In the second chapter we prove the exponential stability of energy corresponding to the homogeneous problem with memory of second order. Then in the third chapter we establish some results that allow us to prove the existence of a global attractor with finite fractal dimension for dynamical system associated to the problem with relative displacement history which is equivalent to the original problem. Finally, in the fourth chapter we deal with a viscoelastic Mindlin-Timoshenko plate model of second order. At this moment we consider essentially two cases. The first one when the system is fully damped, then when the system is partially damped. In the first case we show that the semigroup associated to the Mindlin-Timoskenko system is analytic, which in particular implies exponential decay. In the second case we show that such semigroup loses exponential decay, also loses analyticity. However, we prove in this last case that the solutions have decay of polynomial type

Les systèmes non linéaires sont décrits pour la plupart avec des équations aux dérivées partiellesqui les caractérisent, comme la chaine de pendules couplés, la chaine de protéines comportant des molécules avec liaisons hydrogène, les réseaux atomiques ...etc. Ces modèles comportent le plus souvent des interactions inter particulaires anharmoniques et des potentiels de substrat déformables. En effet, aux conséquences importantes dues à la non linéarité et à la dispersion, ces autres phénomènes comme l’anharmonicité et la déformabilité conduisent à d’autres propriétés de propagation des ondes solitaires telles que les compactons, les kinks et les antikinks , les peakons , … ainsi qu’à la capacité du système à transmettre un signal. Nous utilisons ici la méthode de bifurcation pour tracer les différents portraits de phases obtenus par variation des paramètres du système. Nous mettons en évidence l’influence du facteur d’anharmonicité sur la transmissivité et la bistabilité du système: Il en ressort que l’amplitude du signal d’entrée qui produit la bistabilité augmente avec la valeur absolue du coefficient d’anharmonicité et la bistabilité est retardée. En tenant compte des propriétés importantes générées par de tels systèmes, il nous a paru intéressant de construire une ligne électrique caractérisée par les mêmes équations, mais en doublant sur un tronçon de 10 cellules la valeur de la capacité par rapport à celles des 10 condensateurs suivants, et en reproduisant ce motif avec une périodicité de 20 cellules. Nous réalisons ainsi un super réseau qui simule un neurone myélinisé. Les types de solitons obtenus semblent mieux adaptés pour décrire le signal électrique qui caractérise l’influx neuronal localisé dans l’espace avec un support compact. / Non-linear systems are almostly described by partial differential equations that characterize them. We have some systems such as the chain of coupled pebdelums, the protein chain comprising molecules with hydrogen bonds, atomic lattice, and so on .These systems are most often characterized by anharmonic inter particulate interactions and and then immersed in deformable potential substrates. In addition to nonlinearity and dispersion, these other phenomena namely anharmonicity and deformability are responsible for certain properties of propagation of solitary waves such as (compactons, kinks and anti-kinks, peackons, ...etc) and also the ability of the systems to transmit a signal . We used the bifurcation method to plot the different phase portraits obtained . For various parameters of such systems , we have highlighted the influence of anharmonicity on transmissivity and bistability of the system: It appears that the amplitude of the input signal which produces bistability increases with anharmonicity and the bistability is delayed.To considering these important properties generated by such systems, it seemed interesting to buildin an electrical line characterized by the same equations of the system. By alternately doubling the capacitance of the capacitors of a section of this line, we have realised a super-lattice that simulates a myelinised neuron. The types of solitons we get from this line are better adapted to describe the electrical signal which characterizes the neuron impulse located in space with a compact support.

Submitted by Maria Cristina (library@lncc.br) on 2017-05-04T15:50:15Z No. of bitstreams: 1 Andrey thesis.pdf: 2906483 bytes, checksum: ac671d92ac073e5725aa0f7a45b24dcf (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2017-05-04T15:50:26Z (GMT) No. of bitstreams: 1 Andrey thesis.pdf: 2906483 bytes, checksum: ac671d92ac073e5725aa0f7a45b24dcf (MD5) / Made available in DSpace on 2017-05-04T15:50:36Z (GMT). No. of bitstreams: 1 Andrey thesis.pdf: 2906483 bytes, checksum: ac671d92ac073e5725aa0f7a45b24dcf (MD5) Previous issue date: 2016-11-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) / The electrical impedance tomography (EIT) problem consists in determining the distribution of the electrical conductivity of a medium subject to a set of current fluxes, from measurements of the corresponding electrical potentials on its boundary. EIT is probably the most studied inverse problem since the fundamental works by Calderón from the eighties. It has many relevant applications in medicine (detection of tumors), geophysics (localization of mineral deposits) and engineering (detection of corrosion in structures). In this work, we are interested in reconstructing a number of anomalies with different electrical conductivity from the background from total as well as partial boundary measurements. Since the EIT problem is written as an over-determined boundary value problem, the idea is to rewrite it as an optimization problem. In particular, a functional measuring the misfit between the boundary measurements and the electrical potentials obtained from the model is minimized with respect to a set of ball-shaped anomalies by using the concept of topological derivatives. The resulting topology optimization algorithm is non-iterative and therefore very robust with respect to noisy data. Finally, in order to show the effectiveness of the devised reconstruction algorithm, some numerical experiments into two spatial dimensions are presented, taking into account total and partial noisy boundary measurements. / O problema de tomografia por impedância elétrica (EIT) consiste em determinar a distribuição da condutividade elétrica de um meio sujeito à um conjunto de fluxos, a partir de medidas dos correspondentes potenciais elétricos sobre sua fronteira. EIT é provavelmente o problema inverso mais estudado desde o trabalho fundamental de Calderón dos anos oitenta. EIT possui muitas aplicações relevantes em medicina (detecção de tumores), geofísica (localização de depósitos minerais) e engenharia (detecção de corrosões em estruturas). Neste trabalho, estamos interessados na reconstrução de um número de anomalias com condutividades elétricas diferentes do meio a partir de medidas totais ou parciais feitas sobre a fronteira do corpo. Uma vez que o problema de EIT é escrito como um problema de valor de contorno sobre-determinado, a ideia é reescrevê-lo como um problema de otimização. Em particular, um funcional de forma que mede a diferença entre as medidas na fronteira e potenciais elétricos obtidos a partir de um modelo é minimizado com respeito a um conjunto de anomalias circulares usando o conceito de derivada topológica. O algoritmo de otimização resultante é não iterativo e muito robusto com respeito à ruído. Finalmente, a fim de mostrar a eficácia do algoritmo de reconstrução proposto, alguns experimentos numéricos em duas dimensões espaciais são apresentados, levando em conta medidas de fronteira totais e parciais corrompidas com ruído.

Submitted by Maria Cristina (library@lncc.br) on 2017-08-14T19:30:04Z No. of bitstreams: 1 MTS-thesis.pdf: 2646966 bytes, checksum: fc278af06348a899491121677d2bb5b5 (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2017-08-14T19:30:15Z (GMT) No. of bitstreams: 1 MTS-thesis.pdf: 2646966 bytes, checksum: fc278af06348a899491121677d2bb5b5 (MD5) / Made available in DSpace on 2017-08-14T19:30:24Z (GMT). No. of bitstreams: 1 MTS-thesis.pdf: 2646966 bytes, checksum: fc278af06348a899491121677d2bb5b5 (MD5) Previous issue date: 2017-04-11 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) / Neuroscience aims to understand the mechanisms that regulate the nervous system, to fight existing maladier associated with brain functions, to extend the knowledge in human cognitive development, among others. In the present work we study the communication between neurons of a region of the brain with the purpose to construct a mathematical and computationally feasible model that accurately describes how the information is transmitted between neuronal cells. We approached the behavior of neurons through the FiztHugh-Nagumo equations, constructing a discrete model consistent with the continuous model through the strategy of increasing the number of neurons within the considered neural network. Consequently we obtain numerical results characterized by models of differential equations that describe a distribution of an action potential through non-linear equations of the reaction-diffusion-convection type and a convergence study of the discrete model. / A neurociência tem como objetivo entender os mecanismos que regulam o sistema nervoso, para combater os males existentes associados a funções cerebrais, ampliar o conhecimento no desenvolvimento cognitivo humano, etc. No presente trabalho estudamos a comunicação entre neurônios de uma mesma região do cérebro com o propósito na construção de um modelo matemático que descreva de forma acurada e exequível computacionalmente como as informações são transmitidas entre as células neuronais. Abordamos o comportamento dos neurônios através das equações de FiztHugh-Nagumo, construindo um modelo discreto consistente com o modelo contínuo através da estratégia de aumentar cada vez mais a quantidade de neurônios dentro da rede neural considerada. Consequentemente obtemos resultados numéricos caracterizados por modelos de equações diferenciais parciais que descrevem a distribuição de um potencial de ação através de equações não lineares do tipo reação-difusão-convecção e um estudo de convergência do modelo discreto.

Orientador: Erivaldo Antonio da Silva / Resumo: Este trabalho apresenta um modelo matemático alternativo aos filtros passa-baixas convencionais no Processamento Digital de Imagens. O modelo de Equação Diferencial Parcial (EDP) foi aplicado em imagens orbitais para extração das feições de interesse e os resultados obtidos foram comparados com os resultados do operador de Sobel e o Gradiente Morfológico. O modelo matemático utilizado no trabalho foi baseado na teoria de EDPs e surge como uma proposta metodológica alternativa para a área de Cartografia. O modelo de EDP consiste em aplicar seletivamente a equação, suavizando adequadamente uma imagem sem perder as bordas e outros detalhes contidos na imagem, principalmente pistas de aeroportos e estradas pavimentadas. / Abstract: This work presents an alternative mathematical model for conventional low-pass filters in Digital Image Processing. The model of Partial Differential Equation (PDE) was applied to orbital image to extract features of interest and the obtained results were compared to over obtained for Sobel operator and Morphological Gradient. The mathematical model used in this work was based on PDE theory and was intented to be on alternative methodology for Cartography area. This model consists in selectivels applying the model of PDE, in order adequatels smooth an image without losing edges and other details on the image, mainls airports tracks and paved roads. / Mestre

The subject of this thesis is a class of fast inpainting methods (image or video) based on the idea of filling the inpainting domain in successive shells from its boundary inwards. Image pixels (or video voxels) are filled by assigning them a color equal to a weighted average of either their already filled neighbors (the ``direct'' form of the method) or those neighbors plus additional neighbors within the current shell (the ``semi-implicit'' form). In the direct form, pixels (voxels) in the current shell may be filled independently, but in the semi-implicit form they are filled simultaneously by solving a linear system. We focus in this thesis mainly on the image inpainting case, where the literature contains several methods corresponding to the {\em direct} form of the method - the semi-implicit form is introduced for the first time here. These methods effectively differ only in the order in which pixels (voxels) are filled, the weights used for averaging, and the neighborhood that is averaged over. All of them are very fast, but at the same time all of them leave undesirable artifacts such as ``kinking'' (bending) or blurring of extrapolated isophotes. This thesis has two main goals. First, we introduce new algorithms within this class, which are aimed at reducing or eliminating these artifacts, and also target a specific application - the 3D conversion of images and film. The first part of this thesis will be concerned with introducing 3D conversion as well as Guidefill, a method in the above class adapted to the inpainting problems arising in 3D conversion. However, the second and more significant goal of this thesis is to study these algorithms as a class. In particular, we develop a mathematical theory aimed at understanding the origins of artifacts mentioned. Through this, we seek is to understand which artifacts can be eliminated (and how), and which artifacts are inevitable (and why). Most of the thesis is occupied with this second goal. Our theory is based on two separate limits - the first is a {\em continuum} limit, in which the pixel width →0, and in which the algorithm converges to a partial differential equation. The second is an asymptotic limit in which h is very small but non-zero. This latter limit, which is based on a connection to random walks, relates the inpainted solution to a type of discrete convolution. The former is useful for studying kinking artifacts, while the latter is useful for studying blur. Although all the theoretical work has been done in the context of image inpainting, experimental evidence is presented suggesting a simple generalization to video. Finally, in the last part of the thesis we explore shell-based video inpainting. In particular, we introduce spacetime transport, which is a natural generalization of the ideas of Guidefill and its predecessor, coherence transport, to three dimensions (two spatial dimensions plus one time dimension). Spacetime transport is shown to have much in common with shell-based image inpainting methods. In particular, kinking and blur artifacts persist, and the former of these may be alleviated in exactly the same way as in two dimensions. At the same time, spacetime transport is shown to be related to optical flow based video inpainting. In particular, a connection is derived between spacetime transport and a generalized Lucas-Kanade optical flow that does not distinguish between time and space.

We investigate the energy levels of a particle confined in the isotropic oscillator potential with a magnetic and spatially varying electric field. Here we are able to exactly solve the Schrodinger equation, using matrix methods, for the first excited states. To this end we find that the spatial gradient of the electric field acts as a magnetic field in certain circumstances. Here we present the changes in the energy levels as functions of the electric field, and other parameters.