Thesis (M.S.)--University of North Carolina at Wilmington, 2006. / Includes bibliographical references (leaves: [73]-74)

The physicists and mathematicians have put a lot of efforts in the numerical analysis of various types of partial differential equations on unbounded domain. The time- dependent partial differential equations(PDEs) also have a wide range of applications in physics, geography and many other interdisciplines. This thesis is concerned with the numerical solutions of such kind of partial differential equations on unbounded spatial domain, especially the Korteweg-de Vries(KdV) equations. Since it is unable to solve the problem directly due to its unboundedness, the common way to surpass such difficulty is to introduce proper conditions on the truncated artificial boundaries and to approximate the problem on a bounded domain, which is also known as the Absorbing Boundary Conditions(ABCs). One of the main contributions of this thesis is to design accurate local absorbing boundary conditions for linearized KdV equations and to extend the method to non- linear KdV equations on unbounded domain. Pad´e approximation is the main tool to approximate the cubic root in the construction of local absorbing boundary conditions(LABCs) for a linearized KdV equation on unbounded domain. Besides, we also introduce the continued fraction method in the approximation of cubic root. To avoid the high-order derivatives in the absorbing boundary conditions, a sequence of auxiliary variables are applied accordingly. Then the original problem on unbounded domain is reduced to an approximated initial boundary value(IBV) problem defined on a finite domain. Based on previous work, we are able to extend the method to the design of efficient local absorbing boundary conditions for nonlinear KdV equations on unbounded domain. The unifying approach method is applied to this nonlinear case. The idea of the unifying approach method is to separate inward- and outward-going waves and to build suitable approximated linear operator with a “one-way operator”. Then we unite the approximated linear operator with the nonlinear subproblem and propose boundary conditions for the nonlinear subproblem along the artificial boundaries. The numerical simulations are given to demonstrate the effectiveness and accuracy of our local absorbing boundary conditions. Keywords: Korteweg-de Vries equation; Local absorbing boundary conditions; Pad´e approximation; Continued fraction method; Unifying approach.

Estudia las propiedades de controlabilidad para la ecuación Korteweg de Vries lineal e un intervalo limitado. Se establece un resultado, de controlabilidad nula para la ecuación lineal a través de la condijo de contorno tipo Durichlet. / Tesis

No description available.

We study the Korteweg-de Vries-Burgers equation. With a deep investigation into the spectral and smoothing properties of the linearized system, it is shown by applying Banach Contraction Principle and Gronwall's Inequality to the integral equation based on the variation of parameters formula and explicit representation of the operator semigroup associated with the linearized equation that, under appropriate assumption appropriate assumption on initial states w(x, 0), the nonlinear system is well-posed and its solutions decay exponentially to the mean value of the initial state in H1(O, 1) as t -> +". / Ph. D.

A lot of fluids are more complex than water: polymers, paints, gels, ketchup etc., because of big particles and their complicated microstructures, for instance, molecule entanglement. Due to this structure complexity, some material can display that it is still in yielded state when the imposed stress is released. This is referred to as thixotropy. This dissertation establishes mathematical analysis on a thixotropic yield stress fluid using a viscoelastic model under the limit that the ratio of retardation time versus relaxation time approaches zero. The differential equation model (the PEC model) describing the evolution of the conformation tensor is analyzed. We model the flow when simple shearing is imposed by prescribing a total stress. One part of this dissertation focuses on oscillatory shear stresses. In shear flow, different fluid states corresponding to yielded and unyielded phases occur. We use asymptotic analysis to study transition between these phases when slow oscillatory shearing is set up. Simulations will be used to illustrate and supplement the analysis. Another part of the dissertation focuses on planar Poiseuille flow. Since the flow is spatially inhomogeneous in this situation, shear bands are observed. The flow is driven by a homogeneous pressure gradient, leading to a variation of stress in the cross-stream direction. In this setting, the flow would yield in different time scales during the evolution. Formulas linking the yield locations, transition width, and yield time are obtained. When we introduce Korteweg stress in the transition, the yield location is shifted. An equal area rule is identified to fit the shifted locations. / Ph. D.

Tez (Yüksek Lisans) - Süleyman Demirel Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, 2007. / Kaynakça var.

Le but principal de cette thèse est d'obtenir des résultats d'existence et d'unicité pour des équations aux dérivées partielles dispersives avec conditions aux limites non homogènes. L'approche privilégiée est l'adaptation de techniques issues de la théorie classique des problèmes aux limites hyperboliques (que l'on rappelle au chapitre 1, en améliorant légèrement un résultat). On met en évidence au chapitre 3 une classe d'équations linéaires qu'on peut qualifier de dispersives satisfaisant des critères “minimaux”, et des résultats d'existence et d'unicité pour le problème aux limites associé à celles-ci sont obtenus au chapitre 4.Le fil rouge du mémoire est le modèle d'Euler-Korteweg, pour lequel on aborde l'analyse du problème aux limites sur une version linéarisée au chapitre 2. Toujours pour cette version linéarisée, on prouve un effet Kato-régularisant au chapitre 3. Enfin l'analyse numérique du modèle est abordée au chapitre 5. Pour cela, on commence par utiliser les résultats précédents pour décrire une manière simple d'obtenir les conditions aux limites dites transparentes dans le cadre des équations précédemment décrites puis on utilise ces conditions aux limites pour le modèle d'Euler-Korteweg semi-linéaire afin d'observer la stabilité/instabilité des solitons, ainsi qu'un phénomène d'explosion en temps fini. / The main aim of this thesis is to obtain well-posedness results for boundary value problems especially with non-homogeneous boundary conditions. The approach that we chose here is to adapt technics from the classical theory of hyperbolic boundary value problems (for which we give a brief survey in the first chapter, and a slight generalization). In chapter 3 we delimitate a class of linear dispersive equations, and we obtain well-posedness results for corresponding boundary value problems in chapter 4.The leading thread of this memoir is the Euler-Korteweg model. The boundary value problem for a linearized version is investigated in chapter 2, and the Kato-smoothing effect is proved (also for the linearized model) in chapter 3. Finally, the numerical analysis of the model is made in chapter 5. To begin with, we use the previous abstract results to show a simple way of deriving the so-called transparent boundary conditions for the equations outlined in chapter 3, and those conditions are then used to numerically solve the semi-linear Euler-Korteweg model. This allow us to observe the stability and instability of solitons, as well as a finite time blow up.

Ce doctorat porte sur trois domaines de la théorie du contrôle : le contrôle par le bord d'une équation de Korteweg-de Vries, le contrôle de trois équations de la chaleur couplées par des termes cubiques et la stabilisation en temps fini de trois systèmes classiques de dimension finie. Pour l'équation de Korteweg-de Vries, on démontre d'abord une inégalité de Carleman en utilisant un poids exponentiel bien choisi, puis on en déduit la contrôlabilité à 0 de l'équation. Pour le système de trois équations de la chaleur couplées par des termes cubiques, on montre la contrôlabilité à 0 globale alors que le linéarisé autour de 0 n'est pas contrôlable. On applique la méthode du retour pour obtenir la contrôlabilité locale : on construit des trajectoires du système de contrôle allant de 0 à 0 et ayant un linéarisé contrôlable. Puis un changement d'échelle permet d'obtenir un résultat global. Enfin, concernant les trois systèmes de dimension finie, il s'agit de systèmes contrôlables mais à linéarisés non contrôlables et qui ne sont pas stabilisables à l'aide de feedbacks stationnaires (continus). On construit des feedbacks explicites dépendant du temps conduisant à une stabilisation en temps fini. Pour cela on s'occupe de différentes parties du systèmes pendant différents intervalles de temps. / This doctoral thesis focuses on three fields of Control Theory: the control on the edge of the Korteweg-de Vries equation, the control of three heat equations coupled by cubic terms, and the stabilisation in finite time of three classic systems of finite dimension. For the KdV equation, we first demonstrate a Carleman inequality using a well-chosen exponential weight, then we deduce the controllability at zero of the equation. For the system of three heat equations coupled by cubic terms, we show the global controllability at zero even though the linearized system around zero is not controllable. We apply the return method to obtain local controllability: we build control system trajectories going from zero to zero and whose linearised systems are controllable. Then a scale change allows us to obtain a global result. Finally, concerning the three systems of finite dimension, these systems are controllable systems but the linearised systems are not controllable and are not stabilised with means of continuous stationary feedback. We construct an explicit time-dependent feedback leading to a stabilisation in finite time. For this we deal with different parts of systems during different intervals of time.

La cavitation est la transformation d'un liquide en vapeur qui est causée par une chute de pression en dessous de la pression de saturation vapeur. Ce phénomène se manifeste le plus souvent dans les turbomachines qui sont en interaction avec des liquides. On peut citer les pompes hydrauliques, les injecteurs, les inducteurs ou encore les hélices de bateaux. Vue les effets néfastes qu'elle engendre : bruit, vibrations, détérioration du métal et baisse des performances (chute des rendements et pertes de charges), sa prise en compte est indispensable dans le design des turbomachines. Cette thèse a pour objectif de modéliser ce phénomène de manière à reproduire la nucléation, la convection et l'implosion des bulles de cavitation. Nous nous basons sur un modèle à interface diffuse (le modèle d'équilibre homogène) sur lequel nous greffons un modèle de tension de surface basé sur les équations de Navier Stokes & Korteweg compressibles. Nous réalisons en somme une étude sur l'influence de la tension de surface sur le phénomène de collapse. Nous utilisons un code de volumes finis dont la discrétisation spatiale est assurée par méthode des moindres carrés mobiles. Combinée à un solveur de Riemann de type SLAU, le modèle numérique permet d'outre passer les difficultés liés à la nature du phénomène de cavitation qui sont principalement les forts gradients qui subsistent à travers l'interface liquide-vapeur. L'autre point traité dans la thèse est la détermination d'un coefficient capillaire numérique qui correspond à une tension de surface réelle en fonction de l'épaisseur de l'interface artificiellement élargie pour un maillage donné. / Cavitation is the transformation of a liquid into vapor which is caused by a pressure drop below the vapor saturation pressure. This phenomenon usually occurs in turbine engines that interact with liquids like: hydraulic pumps, injectors, inductors or boat propellers. View its negative effects: noise, vibrations, damage to the metal and decreased performance, it should be included in the design of turbomachinery The main objective of this thesis is to model this phenomenon so as to reproduce the nucleation, convection and the implosion of cavitation bubbles. We rely on a diffuse interface model (the homogeneous equilibrium model) on which we graft a surface tension model based on compressible Navier Stokes & Korteweg equations. We study the influence of surface tension on the bubble collapse. We used a finite volume approach whose spatial discretization is made by moving least squared method. Coupled with a Riemann solver called SLAU, the numerical model can go further difficulties related to the nature of the cavitation phenomenon which is mainly the strong gradients that remain through the liquid-vapor interface. Another issue addressed in this thesis is the determination of a numerical capillary coefficient which corresponds to a real surface tension in function of the thickness of the artificially extended interface for a given mesh.