Philosophiae Doctor - PhD / Singularly perturbed problems have been studied extensively over the past few years from different perspectives. The recent research has focussed on the problems whose solutions possess interior layers. These interior layers appear in the interior of the domain, location of which is difficult to determine a-priori and hence making it difficult to investigate these problems analytically. This explains the need for approximation methods to gain some insight into the behaviour of the solution of such problems. Keeping this in mind, in this thesis we would like to explore a special class of numerical methods, namely, fitted finite difference methods to determine reliable solutions. As far as the fitted finite difference methods are concerned, they are grouped into two categories: fitted mesh finite difference methods (FMFDMs) and the fitted operator finite difference methods (FOFDMs). The aim of this thesis is to focus on the former. To this end, we note that FMFDMs have extensively been used for singularly perturbed two-point boundary value problems (TPBVPs) whose solutions possess boundary layers. However, they are not fully explored for problems whose solutions have interior layers. Hence, in this thesis, we intend firstly to design robust FMFDMs for singularly perturbed TPBVPs whose solutions possess interior layers and to improve accuracy of these approximation methods via methods like Richardson extrapolation. Then we extend these two ideas to solve such singularly perturbed TPBVPs with variable diffusion coefficients. The overall approach is further extended to parabolic singularly perturbed problems having constant as well as variable diffusion coefficients. / 2023-08-31

The study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future discoveries. This dissertation develops methods for the accurate computation of high-order polynomial approximations of stable/unstable manifolds attached to long periodic orbits in discrete time dynamical systems. For this purpose a multiple shooting scheme is applied to invariance equations for the manifolds obtained using the Parameterization Method developed by Xavier Cabre, Ernest Fontich and Rafael De La Llave in [CFdlL03a, CFdlL03b, CFdlL05]. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2020. / FAU Electronic Theses and Dissertations Collection

This work addresses the design and real-time implementation of adaptive control strategies on the parabolic solar collector to enhance the production efficiency under varying working conditions. For example, the unpredictable variations of the solar irradiance and thermal losses, these factors can be a major problem in the control design. The control objective is to force the outlet temperature of the collector fluid, to track a predefined reference temperature regardless of the environmental changes. In this work, two control strategies have been designed and analyzed. First, an intelligent proportional-integral feedback control, which combines the proportionalintegral feedback control with an ultra-local model is proposed. This strategy uses a transfer function model that has been derived and identified from real-time data and used to test the controller performance. Second, an adaptive nonlinear control using Lyapunov stability theory combined with the phenomenological representation of the system is introduced. This strategy uses a bilinear model derived from the heat transfer equation. Both control strategies showed good performance in the simulations with respect to the convergence time and tracking accuracy. Besides, the conventional proportional-integral controller has been successfully implemented in the real system.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21539号 / 理博第4446号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 荒川 知幸, 教授 向井 茂, 准教授 加藤 周 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

HAWE Hydraulic SE, Munich, engineers and manufactures hydraulic drives (CSP-drives) for parabolic trough plants consisting of a compact power pack, directional and control valves, over-center valves, two cylinders and the fittings/hoses for connecting these components. Optional, but this is depending on the system and the control philosophy, also a hydralic accumulator. An optimized hydraulic drive for a parabolic trough field makes the power plant operator profit from savings at components, higher system efficiency, lower operational energy supply needs, less time spent on commissioning and first start-up, lower maintenance effort and increased life span of the drive and finally also savings on peripheral and safety devices. Many of shown proposals are even combining two or more of above mentioned advantages.

In this thesis, we study two stochastic reaction diffusion models - the diffusion limited reaction model of Smoluchowski, and a second approach popularized by Doi. Both models treat molecules as points undergoing Brownian motion. The former represents chemical reactions between two reactants through the use of reactive boundary conditions, with two molecules reacting instantly upon reaching the boundary of a properly embedded open set, termed the reaction region (or more generally some fixed lower dimensional sub-manifold). The Doi model uses reaction potentials, supported in the reaction region, whereby two molecules react with a fixed probability per unit time, λ, upon entering the reaction region. The problem considered is that of obtaining estimates for convergence rates, in λ, of the Doi model to the Smoluchowski model. This problem fits into the theory of singular perturbation or optimization, depending on which reactive boundary conditions one considers, and we solve it - at least for the bimolecular reaction with one stationary target.

Thomson scattering from free electrons in a high-intensity laser focus has been widely studied analytically, but not many measurements of this scattering have been made. We measure polarization-resolved nonlinear Thomson scattering from electrons in a high-intensity laser focus using a parabolic mirror. The weak scattering signal is distinguished from background noise using single-photon detectors and nanosecond time-resolution to distinguish a prompt scattering signal from noise photons. The azimuthal and longitudinal components of the fundamental, second, and third harmonics were measured. Our measurements reasonably match theoretical results, but also show some asymmetry in the emission patterns.

A convertible bond (CB) is a financial derivative, a so called hybrid security. It is an issued contract from a company or a government, which is paid for up-front. The contract yields a known amount at the specified maturity date, unless the holder chooses to convert it into an amount of the underlying asset. This kind of financial products can have complex features affecting the contract price and the optimal exercising situation. The partial differential equation (PDE) approach used for pricing financial derivatives makes it possible to describe convertible bonds with a physical model, a reversed diffusion described by a parabolic PDE. One can sometimes find both analytical and numerical solutions for this type of PDEs and interpret the solutions from a financial point of view, as they suggest predictable behaviour of the contract price.

Hospitals are one of the most energy consuming centers in which thermal energy is utilized for different medical equipments and others. Sterilizers, laundry and kitchens are the main thermal energy utilizing equipments. In addition, large amount of hot water is utilized mainly for showering and dish washing. The main sources of this thermal energy are fossil fuel for oil fired boilers and solar irradiation for solar thermal steam generation system. This project aims in analyzing the Technical performance of parabolic trough steam generation and oil fired boiler steam generation system for Black lion general specialized hospital which is located in Addis Ababa and to perform economic assessment on both systems so as to make comparison test. The result from technical feasibility study shows the parabolic trough can meet the steam demand of the hospital at the required time, more than 8hour per day, as the hospital currently require steam for different activities during the day time for 8hour per day. During cloudy day the conventional back up steam generation system will meet the daily demand for few days of the year. The economic assessment result shows that although the initial investment of concentrated solar steam generation is high as compared to convention steam generation system, the reverse is observed in operation and maintenance cost, resulting solar thermal steam generation break even (payback) to occur early, after 7 year the system let to operate over the conventional oil fired steam generation. In addition the levelized cost of energy for concentrated solar steam generation is found to be 58% higher than conventional steam generation.   Hence, the result shows that parabolic trough is found to be more economical for steam generation than oil fired boiler. If solar thermal steam generation (parabolic through) is implemented, the fuel consumption and operational cost of the boiler can be reduced appreciably.

Ce mémoire présente les travaux réalisés au cours de ma thèse dans le but d'étudier la contrôlabilité de quelques équations aux dérivées partielles. La première partie de cette thèse est consacrée à l'étude de la contrôlabilité de quelques équations cinétiques en différents régimes. Dans un régime collisionnel, nous étudions la contrôlabilité de l'équation de Kolmogorov, un modèle de type Fokker-Planck cinétique, posée dans l'espace de phases $R^d times R^d$. Nous obtenons la contrôlabilité à zéro de cette équation grâce à l'utilisation d'une inégalité spectrale associée à l'opérateur Laplacien dans tout l'espace. Dans un régime non-collisionnel, nous étudions la contrôlabilité de deux systèmes de couplage fluide-cinétique, les systèmes de Vlasov-Stokes et de Vlasov-Navier-Stokes, comportant des non-linéarités dues au terme de couplage. Dans ces cas, l'approche repose sur la méthode du retour.Dans la deuxième partie nous étudions la contrôlabilité d'une famille d'équations paraboliques dégénérées 1-D par la méthode de platitude, qui permet la constructions de contrôles explicites. La troisième partie porte sur le problème de la contrôlabilité de l'équation de Schrödinger par la forme du domaine, c'est-à-dire, en utilisant le domaine comme variable de contrôle. Nous obtenons un résultat de ce type dans le cas du disque unité bidimensionnel. Nos méthodes sont basées sur un résultat de contrôle exact local autour d'une certaine trajectoire, obtenu grâce au théorème d'inversion locale. / This memoir presents the results obtained during my PhD, whose goal is the study of the controllability of some Partial Differential Equations.The first part of this thesis is concerned with the study of the controllability of some kinetic equations undergoing different regimes. Under a collisional regime, we study the controllability of the Kolmogorov equation, a particular case of kinetic Fokker-Planck equation, in the phase space $R^d times R^d$. We obtain the null-controllability of this equation thanks to the use of a spectral inequality associated to the Laplace operator in the whole space. Under a non-collisional regime, we study the controllability of two fluid-kinetic models, the Vlasov-Stokes system and the Vlasov-Navier-Stokes system, which exhibe nonlinearities due to the coupling terms. In those cases, the strategy relies on the Return method.In the second part, we study the controllability of a family of 1-D degenerate parabolic equations by the flatness method, which allows the construction of explicit controls.The third part is focused on the problem of the controllability of the Schrödinger equation via domain deformations, i.e., using the domain as a control. We obtain a result of this kind in the case of the two-dimensional unit disk, for radial data. Our methods are based on a local exact controllability result around a certain trajectory, obtained thanks to the Inverse Mapping theorem.