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

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.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-10-24T13:02:25Z No. of bitstreams: 1 williammassayukisakaguchiyamashita.pdf: 6136709 bytes, checksum: 40c2acb18069362d16c30a65e17521d1 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-11-23T12:20:00Z (GMT) No. of bitstreams: 1 williammassayukisakaguchiyamashita.pdf: 6136709 bytes, checksum: 40c2acb18069362d16c30a65e17521d1 (MD5) / Made available in DSpace on 2018-11-23T12:20:00Z (GMT). No. of bitstreams: 1 williammassayukisakaguchiyamashita.pdf: 6136709 bytes, checksum: 40c2acb18069362d16c30a65e17521d1 (MD5) Previous issue date: 2018-08-17 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / A incidência global dos vírus da Dengue e, mais recentemente, do Zika, Chikungunya e Febre Amarela, tem aumentado o interesse em estudar e compreender a dinâmica populacional do mosquito. Essas doenças são predominantemente disseminadas pelo Aedes aegypti nos países tropicais e subtropicais do mundo. Compreender essa dinâmica é importante para a saúde pública nos países, onde as condições climáticas e ambientais são favoráveis para a propagação destas doenças. Por essa razão, modelos que estudam a dinâmica populacional em uma cidade são de suma importância. Este trabalho discute a modelagem numérica da dinâmica populacional do mosquito Aedes aegypti em uma vizinhança urbana de uma cidade. Em um primeiro momento, apresentamos os resultados teóricos preliminares de modelos unidimensionais. Em seguida, propomos um modelo bidimensional utilizando equações diferenciais parciais. Este modelo permite incorporar fatores externos (vento e inseticidas químicos) e dados topográficos (ruas, blocos de construção, parques, florestas e praias). O modelo proposto foi testado em exemplos envolvendo duas cidades brasileiras (o centro da cidade de Juiz de Fora e a Praia de Copacabana no Rio de Janeiro). / The global incidence of the Dengue virus and, more recently, the Zika, Chikungunya and Yellow Fever, has increased interest in studying and understanding the population dynamics of the mosquito. These diseases are predominantly disseminated by Aedes aegypti in the tropical and subtropical countries of the world. Understanding this dynamics is important for public health in countries, where climatic and environmental conditions are favorable for the spread of these diseases. For this reason, models that study the population dynamics in a city are of short importance. This work discusses the numerical modeling of the population dynamics of the mosquito Aedes aegypti in an urban neighborhood of a city. First, we present the preliminary theoretical results of one-dimensional models. Next, we propose a two-dimensional model using partial differential equations. This model allows incorporating external factors (wind and chemical insecticides) and topographic data (streets, building blocks, parks, forests and beaches). The proposed model was tested in examples involving two Brazilian cities (the city center of Juiz de Fora and Copacabana Beach in Rio de Janeiro).

Orientador: Ademir Pastor Ferreira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-24T23:55:39Z (GMT). No. of bitstreams: 1 Cunha_AlyssonTobiasRibeiro_D.pdf: 2613588 bytes, checksum: a1484c40a841c1479e707e39620338b7 (MD5) Previous issue date: 2014 / Resumo: O resumo poderá ser visualizado no texto completo da tese digital / Abstract: The abstract is available with the full electronic digital document / Doutorado / Matematica / Doutor em Matemática

Orientador: João Frederico da Costa Azevedo Meyer / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T09:35:38Z (GMT). No. of bitstreams: 1 Prestes_ManoelFernandoBiagioni_M.pdf: 6775504 bytes, checksum: 58bb4307baf12ee09b0deeae72c3c8c9 (MD5) Previous issue date: 2011 / Resumo Este estudo visa descrever a evolução de material impactante na Lagoa do Taquaral, tendo sido inclusive apresentada inicialmente, uma descrição desse meio aquático, enfatizando-se os aspectos histórico, climático e geomorfológico nesta contextualização. Para a modelagem do fenômeno evolutivo utilizou-se a equação diferencial parcial clássica de Difusão-Advecção, tradicionalmente empregada na modelagem de fenômenos deste gênero. A discretização espacial do modelo caracteriza-se pelo uso do Método das Diferenças Finitas, sendo que a discretização temporal foi obtida através do Método de Crank-Nicolson. Quanto aos resultados numérico-computacionais obtidos, podemos destacar as três situações-cenário consideradas, conforme a direção predominante dos ventos adotada, com vistas a estabelecer adequados mecanismos de monitoramento, da dispersão de material impactante no meio aquático. Outrossim buscamos, neste trabalho, ferramentas capazes de propiciar estratégias a serem adotadas em políticas de prevenção e contingência, para os problemas gerados pela intervenção antrópica na micro-região em estudo. Ensejamos, ainda, estimular o poder público quanto instituição, a promover um planejamento e manuseio mais adequado do acervo ambiental / Abstract: This work has the purpose of describing the evolutionary behavior of a pollutant in a certain domain, and we have adopted the Taquaral lake as the objective example, which we initially describe in its historic, climatic and geomorphological aspects. In order to mathematically model this situation, we used a classical diffusive-advective partial differential equation. The spatial discretization is undertaken with the use of Second order central Finite Differences, while the discretization in time is done with the Crank-Nicolson Method. Three scenarios were considered, according to predominant wind directions, adopted for the numerical essays. The purpose of this was to create effective computational tools for monitoring pollutant spills and discharges in the aquatic medium. In other words, this work also intends to make available a numerical (and mathematical, as well as computational) tool for evaluating preventive and contingency policies for those polluting problems created by anthropic urban activities, besides stimulating a more precise environmental planning in this kind of situation / Mestrado / Matemática Universitária / Mestre em Matemática Universitária

Orientadores: Marko Antonio Rojas Medar, Maria Angeles Rodriguez Bellido / Tese (doutorado) - Universidade Estadual de Campinas. Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T14:33:11Z (GMT). No. of bitstreams: 1 VillamizarRoa_ElderJesus_D.pdf: 1860214 bytes, checksum: 00b73d5c74c2d58828f10d232dbe32fb (MD5) Previous issue date: 2005 / Resumo: Abordamos vários problemas relativos à existência, unicidade, regularidade e estabilidade de alguns sistemas de equações do tipo Navier-Stokes; Estudamos a existência de soluções ultra fracas para as equações de Boussinesq estacionárias, com condições de fronteira ouco regulares, do tipo Dirichlet para a velocidade e condições mistas do tipo Dirichlet e Neumann para a temperatura. Seguidamente provamos a existência de soluções tempo-periódicas brandas e fortes em domínios não limitados para as equações de Boussinesq de evolução em espaços de Lorentz. Via a teoria de semigrupos provamos resultados de existência global, comportamento assintótico e estabilidade das soluções para as equações de fluidos micropolares. Posteriormente consideramos uma classe de equações não lineares estacionárias abstratas em um espaço de Hilbert separável e mostramos algumas propriedades qualitativas relativas a esse modelo, entre elas, uma propriedade sobre a cardinalidade do conjunto das soluções do modelo abstrato em questão e uma propriedade de dependência contínua das soluções com respeito aos dados do problema. Finalmente provamos a existência de soluções fortes para as equações de Navier-Stokes com densidade variável em domínios espaciais não limitados / Abstract: The main objective of this work is to study the number of solutions of polynomial equations over finite fields. For that we used basic results on Character sums and on the number of solutions of a Quadratic Form. This approach uses elementary techniques even considering the increasing on computations. Therefore this method allowed us to study and determine formulae for the number of solutions of certain polynomial equations well known, without the need of more sophisticated tools. Among the applications of the obtained formulae, we have some examples of plane algebraic curves which number of rational points achieve the Weil bound, that is, maximal curves which are of great interest in code theory. In addition, other examples were obtained of projective manifolds over finite fields which number of points achieve the Weil-Deligne bound / Doutorado / Equações Diferenciais Parciais / Doutor em Matemática

Cette thèse aborde les principaux aspects applicatifs en matière de restauration et amélioration d'images. A travers une approche progressive, deux nouveaux paradigmes sont introduits : la mise en place d'une déconvolution et d'un débruitage simultanés avec une amélioration de cohérence, et la fusion avec débruitage. Ces paradigmes sont définis dans un cadre théorique d'approches EDP - variationnelles. Le premier paradigme représente une étape intermédiaire dans la validation et l'analyse du concept de restauration et d'amélioration combinées, tandis que la deuxième proposition traitant du modèle conjoint fusion-débruitage illustre les avantages de l'utilisation d'une approche parallèle en traitement d'images, par opposition aux approches séquentielles. Ces deux propositions sont théoriquement et expérimentalement formalisées, analysées et comparées avec les approches les plus classiques, démontrant ainsi leur validité et soulignant leurs caractéristiques et avantages. / This thesis addresses key issues of current image restoration and enhancement methodology, and through a progressive approach introduces two new image processing paradigms, i.e., concurrent image deblurring and denoising with coherence enhancement, and joint image fusion and denoising, defined within a Partial Differential Equation -variational theoretical setting.The first image processing paradigm represents an intermediary step in validating and testing the concept of compound image restoration and enhancement, while the second proposition, i.e., the joint fusion-denoising model fully illustrates the advantages of using concurrent image processing as opposed to sequential approaches.Both propositions are theoretically formalized and experimentally analyzed and compared with the similar existing methodology, proving thus their validity and emphasizing their characteristics and advantages when considered an alternative to a sequential image processing chain.

Dans la première partie de cette thèse on étudie les systèmes abcd qui ont été dérivés par J.L. Bona, M. Chen et J.-C. Saut en 2002. Ces systèmes sont des modèles approximant le problème d'ondes hydrodynamiques dans le régime de Boussinesq, à savoir, des vagues de faible amplitude et de grande longueur d'onde. Dans les deux premiers chapitres on considère le problème d'existence en temps long à savoir la construction de solutions pour les systèmes abcd qui ont leur temps d'existence minoré par $1/varepsilon$ où $varepsilon$ est le rapport entre une amplitude typique du vague et la profondeur du canal. Dans un premier temps on considère des données initiales appartenant aux espaces de Sobolev qui sont inclus dans l'espace des fonctions continues qui s'annulent à l'infini. D'un point de vue physique cette situatuion correspond à des vagues sont localisées en espace. Le point clé est la construction d'une fonctionnelle non linéaire d'énergie qui contrôle certaines normes de Sobolev sur un intervalle de temps long. Pour y arriver, on travaille avec des équations localisées en fréquence. Cette approche nous permet d'obtenir des résultats d'existence en temps long en demandant moins de régularité sur les données initiales. Un deuxième avantage de notre méthode est que l'on peut traiter d'une manière unifiée presque tous les cas correspondant aux différentes valeurs des paramètres abcd. Dans le deuxième chapitre on montre des résultats d'existence en temps long pour le cas des données ayant un comportement non trivial à l'infini.Ce type des données est relevant pour l'étude de la propagation des mascarets. L'idée qui est à la base de ces résultats est de considérer un découpage convenable de la donnée initiale en hautes et basses fréquences. Dans le troisième chapitre on emploie des schémas de volumes finis afin de construire des solutions numériques. On utilise ensuite nos schémas pour étudier l'interaction d'ondes progressives.La deuxième partie de ce manuscrit est consacrée à l'étude des problèmes de régularité optimale pour le système de Navier-Stokes qui régi l'évolution d'un fluide incompressible, inhomogène et pour le système Navier-Stokes-Korteweg utilisé pour prendre en compte les effets de capillarité. Plus précisément, on montre que ces systèmes sont bien-posés dans leurs espaces critiques, à savoir, les espaces quiont la même invariance par changement d'échelle que les systèmes eux-mêmes. Pour pouvoir démontrer ce type de résultats on a besoin d'établir de nouvelles estimations pour un problème de type Stokes avec des coefficients variables / The first part of the present thesis deals with the so -called abcd systems which were derived by J.L. Bona, M. Chen and J.-C. Saut back in 2002. These systems are approximation models for the waterwaves problem in the Boussinesq regime, that is, waves of small amplitude and long wavelength. In the first two chapters we address the long time existence problem which consists in constructing solutions for the Cauchy problem associated to the abcd systems and prove that the maximal time of existence is bounded from below by some physically relevant quantity. First, we consider the case of initial data belonging to some Sobolev spaces imbedded in the space of continuous functions which vanish at infinity. Physically, this corresponds to spatially localized waves. The key ingredient is to construct a nonlinear energy functional which controls appropriate Sobolev norms on the desired time scales. This is accomplished by working with spectrally localized equations. The two important features of our method is that we require lower regularity levels in order to develop a long time existence theory and we may treat in an uni ed manner most of the cases corresponding to the di erent values of the parameters. In the second chapter, we prove the long time existence results for the case of data thatdoes not necessarily vanish at in nity. This is especially useful if one has in mind bore propagation. One of the key ideas of the proof is to consider a well-adapted high-low frequency decomposition of the initial data. In the third chapter, we propose infinite volume schemes in order to construct numerical solutions. We use these schemes in order to study traveling waves interaction.The second part of this manuscript, is devoted to the study of optimal regularity issues for the incompressible inhomogeneous Navier-Stokes system and the Navier-Stokes-Korteweg system used in order to take in account capillarity effects. More precisely, we prove that these systems are well-posed in their truly critical spaces i.e. the spaces that have the same scale invariance as the system itself. Inorder to achieve this we derive new estimates for a Stoke-like problem with time independent variable coefficients

Do mechanického modelu kochley zavádzame implicitné numerické metódy. Tes- tujeme konkrétne štyri metódy: implicitný Euler, Crank-Nicolson, BDF druhého a tretieho rádu na lineárnej a nelineárnej verzii modelu. Nelineárny model obsahuje funkciu so saturujúcou vlastnosťou. Aplikácia implicitných metód na nelineárny model vedie na sústavu nelineárnych rovníc. Predstavujeme dva spôsoby, ako túto sústavu numericky riešiť. Prvý z nich zahrňuje nelinearitu do pravej strany novovzniknutej lineárnej sústavy. Druhý robí linearizáciu nelineárnej funkcie. V práci porovnávame oba spôsoby z hľadiska efektivity a sledujeme ich konvergenciu k referenčnému riešeniu. Pre hodnotu tolerancie, ktorú používame na určenie numerickej konvergencie, je prvý spôsob efektívnejší. V úplne nelineárnom režime druhý spôsob zlyháva, pretože nekon- verguje k referenčnému riešeniu. Výsledkom porovnania implicitných metód je, že Crank-Nicolsonova metóda s prvým spôsobom riešenia nelineárnej sústavy je pre účely nášho modelu najlepšia. Použitie tejto metódy v mechanickom modeli nám umožňuje vytvoriť ľubovoľne presné prepojenie medzi mechanickým a elektrickým modelom kochley, rešpektujúc fyziológiu človeka. 1

We develop expansion methods as a new computational approach towards high-dimensional partial differential equations (PDEs), particularly of such type as arising in the valuation of financial derivatives. The proposed methods are extended from [41] and use principal component analysis (PCA) of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. They enable calculation of highly accurate approximate solutions with computational complexity polynomial in the number of dimensions for PDEs with a low number of dominant principal components. For the case of PDEs with constant coefficients, we show existence of expansion solutions and prove theoretical error bounds. We give a precise characterisation of when our methods can be applied and construct specific examples of a first and second order version. We provide numerical results showing that the empirically observed convergence speeds are in agreement with the theoretical predictions. For the case of PDEs with varying coefficients, we give a heuristic motivation using the Parametrix approach and empirically test the methods' accuracy for a range of variable parameter stock models. We demonstrate the applicability of our expansion methods to real-world securities pricing problems by considering path-dependent and early-exercise options in the LIBOR market model. Using the example of Bermudan swaptions and Ratchet floors, which are considered difficult benchmark problems, we give a careful analysis of the numerical accuracy and computational complexity. We are able to demonstrate that for problems with medium to high dimensionality, around 60-100, and moderate time horizons, the presented PDE methods deliver results comparable in accuracy to benchmark state-of-the-art Monte Carlo methods in similar or (significantly) faster run time.