In this thesis, we consider four different stochastic partial differential equations. Firstly, we study stochastic Helmholtz equation driven by an additive white noise, in a bounded convex domain with smooth boundary of Rd (d = 2, 3). And then with the help of the perfectly matched layers technique, we also consider the stochastic scattering problem of Helmholtz type. The second part of this thesis is to investigate the time harmonic case for stochastic Maxwell's equations driven by an color noise in a simple medium, and then we expand the results to the stochastic Maxwell's equations in case of dispersive media in Rd (d = 2, 3). Thirdly, we study stochastic parabolic partial differential equation driven by space-time color noise, where the domain O is a bounded domain in R2 with boundary ∂O of class C2+alpha for 0 < alpha < 1/2. In the last part, we discuss the stochastic wave equation (SWE) driven by nonlinear noise in 1D case, where the noise 626x6t W(x, t) is the space-time mixed second-order derivative of the Brownian sheet. / Many physical and engineering phenomena are modeled by partial differential equations which often contain some levels of uncertainty. The advantage of modeling using so-called stochastic partial differential equations (SPDEs) is that SPDEs are able to more fully capture interesting phenomena; it also means that the corresponding numerical analysis of the model will require new tools to model the systems, produce the solutions, and analyze the information stored within the solutions. / One of the goals of this thesis is to derive error estimates for numerical solutions of the above four kinds SPDEs. The difficulty in the error analysis in finite element methods and general numerical approximations for a SPDE is the lack of regularity of its solution. To overcome such a difficulty, we follow the approach of [4] by first discretizing the noise and then applying standard finite element methods and discontinuous Galerkin methods to the stochastic Helmholtz equation and Maxwell equations with discretized noise; standard finite element method to the stochastic parabolic equation with discretized color noise; Galerkin method to the stochastic wave equation with discretized white noise, and we obtain error estimates are comparable to the error estimates of finite difference schemes. / We shall focus on some SPDEs where randomness only affects the right-hand sides of the equations. To solve the four types of SPDEs using, for example, the Monte Carlo method, one needs many solvers for the deterministic problem with multiple right-hand sides. We present several efficient deterministic solvers such as flexible CG method and block flexible GMRES method, which are absolutely essential in computing statistical quantities. / Zhang, Kai. / Adviser: Zou Jun. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3552. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 144-155). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Neste trabalho, estudamos a existˆencia e multiplicidade de solu¸c˜oes de certos problemas p-sublineares envolvendo o operador p-laplaciano usando teoria de Morse. / The purpose of this text is to provide a didactic exposition of the paper “Solutions of p-sublinear p-Laplacian equation via Morse theory” by Yuxia Guo and Jiaquan Liu [8]. This paper addresses the existence and multiplicity of solutions for the problem where is a smooth, bounded domain of RN, p is the p-Laplacian operator and f satisfies certain conditions, in particular f is p-sublinear at 0. Morse theory is used to infer the existence of critical points of a functional associated to this problem. In Chapter 2, we introduce the necessary Morse theoretic concepts, assuming basic knowledge of singular homology theory. In Chapter 3, we introduce basic properties of the p-Laplacian operator, assuming knowledge of Sobolev spaces, including imbedding and compactness results. Finally, in Chapter 4, we follow Guo and Liu’s paper itself.

Orientador: Rodney Carlos Bassanezi / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-09-11T21:18:32Z (GMT). No. of bitstreams: 1 Leite_JeffersonCruzdosSantos_D.pdf: 39583926 bytes, checksum: ac69c5a564ed32a9d1eb58ac0e71c1fd (MD5) Previous issue date: 2011 / Resumo: Neste trabalho definiremos solução fuzzy para problemas que envolvam difusão e exploraremos algumas propriedades importantes como unicidade e estabilidade dessas soluções. Basicamente estamos interessados em considerar algumas características importantes desses problemas difusivos como incertos, para isso, usaremos o conceito de numero fuzzy. Termos como coeficiente de difusão e condição inicial serão considerados como incertos e através da extensão de Zadeh aplicado a solução da equação determinística associada ao problema teremos a solução fuzzy. Serão obtidas também soluções via base de regras, utilizando sistemas dinâmicos pfuzzy, garantindo assim, uma maneira eficiente e prática de obtermos, boas respostas para os problemas, sem necessariamente termos as soluções explícitas. Aplicações desses resultados também serão apresentados / Abstract: This work will define fuzzy solution for problems involving di_usion and explore some important properties such as uniqueness and stability of these solutions. Basically we are interested in considering some important features of these diffusion problems as uncertain and, we use the concept of fuzzy numbers for this. Terms such as diffusion coefficient and initial condition are considered as uncertain and by the extension of Zadeh's solution applied to deterministic equation associated with the problem we have the fuzzy solution. Solutions for rule-base situations are also obtained, using p-fuzzy dynamic systems, thus guaranteeing an, efficient and practical way of obtaining adequate answers to the problems, not necessarily under the explicit solutions. Applications of these results will also be discussed / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

O objetivo deste projeto é o estudo de técnicas numéricas robustas para aproximação da solução de leis de conservação hiperbólicas escalares unidimensionais e bidimensionais e de sistemas de leis de conservação hiperbólicas. Para alcançar tal objetivo, estudamos esquemas conservativos com propriedades especiais, tais como, esquemas upwind, TVD, Godunov, limitante de fluxo e limitante de inclinação. A solução de um sistema de leis de conservação pode exibir descontinuidades do tipo choque, rarefação ou de contato. Assim, o desenvolvimento de técnicas numéricas capazes de reproduzir e tratar esses comportamentos é desejável. Além de representar corretamente a descontinuidade os esquemas numéricos têm ainda uma tarefa mais árdua; aquela de escolher a solução singular correta, a chamada solução entrópica. Os métodos de Godunov, limitantes de fluxo e limitantes de inclinação são técnicas numéricas que possuem as características apropriadas para aproximar a solução entrópica de uma lei de conservação. / The aim of this work is the study of robust numerical techniques for approximating the solution of scalar and systems of hyperbolic conservation laws. To achieve this, we studied conservative schemes with special properties, such as, schemes upwind, TVD, Godunov, flux limiters and slope limiters. The solution of a system of conservation laws can present discontinuities, like shocks, rarefaction or contact. Therefore, the development of numerical techniques capable of reproducing such featurs are highly desirable. Furthermore, besides resolving singularities, it is required that the numerical method chooses the correct weak solution, that is, the entropic solution. Godunov, flux limiters and slope limiters are techniques that show the appropriate behaviour when applied to conservation laws.

Neste trabalho consideramos sistemas de equações diferenciais parciais lineares de primeira ordem, com coeficientes analíticos, definidos em variedades analíticas reais, no caso particular em que seu coposto é igual a um. Demonstramos que esse tipo de sistema admite integrais primeiras locais, e buscamos caracterizar sua hipoelipticidade analítica local e global em termos de propriedades topológicas das mesmas. Também provamos a Fórmula de Aproximação de Baouendi-Trèves / In this work we consider systems of first-order linear partial differential equations, with analytic coefficients, defined on real-analytic manifolds, in the special case in which the corank is equal to one. We prove that this type of systems admits local first integrals, and we seek to characterize their local and global analytic hypoellipticity in terms of topological properties of these first integrals. We also prove the Baouendi-Trèves Approximation Formula

The radiative transfer equation (RTE) models the transport of radiation through a participating medium. In particular, it captures how radiation is scattered, emitted, and absorbed as it interacts with the medium. This process arises in numerous application areas, including: neutron transport in nuclear reactors, radiation therapy in cancer treatment planning, and the investigation of forming galaxies in astrophysics. As a result, there is great interest in the solution of the RTE in many different fields. We consider the energy dependent form of the RTE and allow media containing regions of negligible absorption. This particular case is not often considered due to the additional dimension and stability issues which arise by allowing vanishing absorption. In this thesis, we establish the existence and uniqueness of the underlying boundary value problem. We then proceed to develop a stable numerical algorithm for solving the RTE. Alongside the construction of the method, we derive corresponding error estimates. To show the validity of the algorithm in practice, we apply the algorithm to four different example problems. We also use these examples to validate our theoretical results.

A Partial Differential Equation (PDE) based model combining surface electromigration and wetting is developed for the analysis of the morphological instability of mono-crystalline metal films in a high temperature environment typical to operational conditions of microelectronic interconnects. The atomic mobility and surface energy of such films are anisotropic, and the model accounts for these material properties. The goal of modeling is to describe and understand the time-evolution of the shape of film surface. I will present the formulation of a nonlinear parabolic PDE problem for the height function h(x,t) of the film in the horizontal electric field, followed by the results of the linear stability analyses and computations of fully nonlinear evolution equation.

In 2014, Astala, Päivärinta, Reyes, and Siltanen conducted numerical experiments reconstructing a piecewise continuous conductivity. The algorithm of the shortcut method is based on the reconstruction algorithm due to Nachman, which assumes a priori that the conductivity is Hölder continuous. In this dissertation, we prove that, in the presence of infinite-precision data, this shortcut procedure accurately recovers the scattering transform of an essentially bounded conductivity, provided it is constant in a neighborhood of the boundary. In this setting, Nachman’s integral equations have a meaning and are still uniquely solvable. To regularize the reconstruction, Astala et al. employ a high frequency cutoff of the scattering transform. We show that such scattering transforms correspond to Beltrami coefficients that are not compactly supported, but exhibit certain decay at infinity. For this class of Beltrami coefficients, we establish that the complex geometric optics solutions to the Beltrami equation exist and exhibit the same subexponential decay as described in the 2006 work of Astala and Päivärinta. This is a first step toward extending the inverse scattering map of Astala and Päivärinta to non-compactly supported conductivities.

Spectral methods for partial differential equations with boundary conditions in complex domains are developed with the help of a fictitious domain approach. For rectangular embedding, spectral-Galerkin formulations with special trial and test functions are presented and discussed, as well as the well-posedness and the error analysis. For circular and annular embedding, dimension reduction is applied and a sequence of 1-D problems with artificial boundary values are solved. Applications of our methods include the fractional Laplace problem and the Helmholtz equations. In numerical examples, our methods show good performance on the boundary value problems in both smooth and polygonal complex domains, and the L2 errors decay exponentially for smooth solutions. For singular problems, high-order convergence rates can also be obtained.

This work is a study of the relationship between Brownian motion and elementary, linear partial differential equations. In the text, I have shown that Brownian motion is a Markov process, and that Brownian motion itself, and certain Stochastic processes involving Brownian motion are also martingales. In particular, Dynkin's formula for Brownian motion was shown. Using Dynkin's formula and Brownian motion, I then constructed solutions for the classical Dirichlet problem and the heat equation, given by Δu=0 and ut= 1/2Δu+g, respectively. I have shown that the bounded solution is unique if Brownian motion will always exit the domain of the function once it has started at a point in the domain. The heat equation also has a unique bounded solution.