Wave radiation is an important process in many geophysical flows. In particular, it is by wave radiation that flows may adjust to a state for which the dynamics is slow. Such a state is described as “balanced”, meaning there is an approximate balance between the Coriolis force and horizontal pressure gradients, and between buoyancy and vertical pressure gradients. In this thesis, wave radiation processes relevant to these enormously complex flows are studied through the use of some highly simplified models, and a parallel aim is to develop accurate numerical techniques for doing so. This thesis is divided into three main parts. 1. We consider accurate numerical boundary conditions for various equations which support wave radiation to infinity. Particular attention is given to discretely non-reflecting boundary conditions, which are derived directly from a discretised scheme. Such a boundary condition is studied in the case of the 1-d Klein-Gordon equation. The limitations concerning the practical implementation of this scheme are explored and some possible improvements are suggested. A stability analysis is developed which yields a simple stability criterion that is useful when tuning the boundary condition. The practical use of higher-order boundary conditions for the 2-d shallow water equations is also explored; the accuracy of such a method is assessed when combined with a particular interior scheme, and an analysis based on matrix pseudospectra reveals something of the stability of such a method. 2. Large-scale atmospheric and oceanic flows are examples of systems with a wide timescale separation, determined by a small parameter. In addition they both undergo constant random forcing. The five component Lorenz-Krishnamurthy system is a system with a timescale separation controlled by a small parameter, and we employ it as a model of the forced ocean by further adding a random forcing of the slow variables, and introduce wave radiation to infinity by the addition of a dispersive PDE. The dynamics are reduced by deriving balance relations, and numerical experiments are used to assess the effects of energy radiation by fast waves. 3. We study quasimodes, which demonstrate the existence of associated Landau poles of a system. In this thesis, we consider a simple model of wave radiation that exhibits quasimodes, that allows us to derive some explicit analytical results, as opposed to physically realistic geophysical fluid systems for which such results are often unavailable, necessitating recourse to numerical techniques. The growth rates obtained for this system, which is an extension of one considered by Lamb, are confirmed using numerical experiments.

In the first part of this thesis, we demonstrate theory and computations for finite-energy line defect solutions in an improvement of Ericksen-Leslie liquid crystal theory. Planar director fields are considered in two and three space dimensions, and we demonstrate straight as well as loop disclination solutions. The possibility of static balance of forces in the presence of a disclination and in the absence of ow and body forces is discussed. The work exploits an implicit conceptual connection between the Weingarten-Volterra characterization of possible jumps in certain potential fields and the Stokes-Helmholtz resolution of vector fields. The theoretical basis of our work is compared and contrasted with the theory of Volterra disclinations in elasticity. Physical reasoning precluding a gauge-invariant structure for the model is also presented. In part II of the thesis, the time-harmonic Schrodinger equation with periodic potential is considered. We derive the asymptotic form of the scattering wave function in the periodic space and investigate the possibility of its application as a DtN non-reflecting boundary condition. Moreover, we study the perfectly matched layer method for this problem and show that it is a reliable method, which converges rapidly to the exact solution, as the thickness of the absorbing layer increases. Moreover, we use the tight-binding method to numerically solve the Schrodinger equation for Graphene sheets, symmetry-adapted Carbon nanotubes and DNA molecules to demonstrate their electronic behavior in the presence of local defects. The results for Y-junction Carbon nanotubes depict very interesting properties and confirms the predictions for their application as new transistors.

These lecture notes are intended as a short introduction to diffusion processes on a domain with a reflecting boundary for graduate students, researchers in stochastic analysis and interested readers. Specific results on stochastic differential equations with reflecting boundaries such as existence and uniqueness, continuity and Markov properties, relation to partial differential equations and submartingale problems are given. An extensive list of references to current literature is included. This book has its origins in a mini-course the author gave at the University of Potsdam and at the Technical University of Berlin in Winter 2013.

Cette thèse est composée de deux parties indépendantes : la première partie traite des équations différentielles stochastiques dans le cadre de la G-espérance, tandis que la deuxième partie présente les résultats obtenus pour les équations différentielles stochastiques du seconde ordre. Dans un premier temps, on considère les intégrales stochastiques par rapport à un processus croissant, et on donne une extension de la formule d'Itô dans le cadre de la G-espérance. Ensuite, on étudie une classe d'équations différentielles stochastiques réfléchies unidimensionnelles dirigées par un G-mouvement brownien. Dans la suite, en utilisant une méthode de localisation, on prouve l'existence et l'unicité de solutions pour les équations différentielles stochastiques dirigées par un G-mouvement brownien, dont les coefficients sont localement lipschitziens. Enfin, dans le même cadre, on discute des problèmes de réflexion multidimensionnelle et on fournit quelques résultats de convergence. Dans un deuxième temps, on étudie une classe d'équations différentielles stochastiques rétrogrades du seconde ordre à croissance quadratique. Le but de ce travail est de généraliser le résultat obtenu par Possamaï et Zhou en 2012. On montre aussi l'existence et l'unicité des solutions pour ces équations, mais sous des hypothèses plus faibles. De plus, ce résultat théorique est appliqué aux problèmes de maximisation robuste de l'utilité du portefeuille en finance. / This thesis consists of two relatively independent parts : the first part concerns stochastic differential equations in the framework of the G-expectation, while the second part deals with a class of second order backward stochastic differential equations. In the first part, we first consider stochastic integrals with respect to an increasing process and give an extension of Itô's formula in the G-framework. Then, we study a class of scalar valued reflected stochastic differential equations driven by G-Brownian motion. Subsequently, we prove the existence and the uniqueness of solutions for some locally Lipschitz stochastic differential equations driven by G-Brownian motion. At the end of this part, we consider multidimensional reflected problems in the G-framework, and some convergence results are obtained. In the second part, we study the wellposedness of a class of second order backward stochastic differential equations (2BSDEs) under a quadratic growth condition on their coefficients. The aim of this part is to generalize a wellposedness result for quadratic 2BSDEs by Possamaï and Zhou in 2012. In this thesis, we work under some usual assumptions and deduce the existence and uniqueness theorem as well. Moreover, this theoretical result for quadratic 2BSDEs is applied to solve some robust utility maximization problems in finance.