The dissertation consists of two parts.The first part is mainly to provide the algorithms and error estimates of the collocation Trefftz methods (CTMs) for seeking the solutions of partial differential equations. We consider several popular models of PDEs with singularities, including Poisson equations and the biharmonic equations. The second part is to present the collocation methods (CMs) and to give a unified framework of combinations of CMs with other numerical methods such as finite element method, etc. An interesting fact has been justified: The integration quadrature formulas only affect on the uniformly $V_h$-elliptic inequality, not on the solution accuracy. In CTMs and CMs, the Gaussian quadrature points will be chosen as the collocation points. Of course, the Newton-Cotes quadrature points can be applied as well. We need a suitable dense points to guarantee the uniformly $V_h$-elliptic inequality. In addition, the solution domain of problems may not be confined in polygons. We may also divide the domain into several small subdomains. For the smooth solutions of problems, the different degree polynomials can be chosen to approximate the solutions properly. However, different kinds of admissible functions may also be used in the methods given in this dissertation. Besides, a new unified framework of combinations of CMs with other methods will be analyzed. In this dissertation, the new analysis is more flexible towards the practical problems and is easy to fit into rather arbitrary domains. Thus is a great distinctive feature from that in the existing literatures of CTMs and CMs. Finally, a few numerical experiments for smooth and singularity problems are provided to display effectiveness of the methods proposed, and to support the analysis made.

Diss.--Math.--Chemnitz--Technische Universität, 1999. / Bibliogr. p. 103-106.

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abstract: The following research is a regulatory and emissions analysis of collocated sources of air pollution as they relate to the definition of "major, stationary, sources", if their emissions were amalgamated. The emitting sources chosen for this study are seven facilities located in a single, aggregate mining pit, along the Aqua Fria riverbed in Sun City, Arizona. The sources in question consist of Rock Crushing and Screening plants, Hot Mix Asphalt plants, and Concrete Batch plants. Generally, individual facilities with emissions of a criteria air pollutant over 100 tons per year or 70 tons per year for PM10 in the Maricopa County non-attainment area would be required to operate under a different permitting regime than those with emissions less than stated above. In addition, facility's that emit over 25 tons per year or 150 pounds per hour of NOx would trigger Maricopa County Best Available Control Technology (BACT) and would be required to install more stringent pollution controls. However, in order to circumvent the more stringent permitting requirements, some facilities have "collocated" in order to escape having their emissions calculated as single source, while operating as a single, production entity. The results of this study indicate that the sources analyzed do not collectively emit major source levels of emissions; however, they do trigger year and daily BACT for NOx. It was also discovered that lack of grid power contributes to the use of generators, which is the main source of emissions. Therefore, if grid electricity was introduced in outlying areas of Maricopa County, facilities could significantly reduce the use of generator power; thereby, reducing pollutants associated with generator use. / Dissertation/Thesis / M.S.Tech Technology 2011

This thesis is concerned with a particular numerical approach for solving the fractional partial differential equations (PDEs). In the last two decades, it has been observed that many practical systems are more accurately described by fractional differential equations (FDEs) rather than the traditional differential equation approaches. Consequently, it has become an important research area to study the theoretical and numerical aspects of various types of FDEs. This thesis will explore high order numerical methods for solving FDEs numerically. More precisely, spectral methods which exhibits exponential order of accuracy will be investigated. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss quadrature points. In this work, Hermite and modified rational functions are employed to serve as basis functions for solutions that decay exponentially and algebraically, respectively. The main emphasis of this thesis is to propose the spectral collocation method for FDEs posed in unbounded domains. Components of the differentiation matrix involving fractional Laplacian are derived which can then be computed recursively using the properties of confluent hypergeometric function or hypergeometric function. The first part of the thesis introduces preliminaries useful for other parts of the thesis. Review of the relevant definitions and properties of special functions such as Hermite functions, Bessel functions, hypergeometric functions, Gegenbauer polynomials, mapped Jacobi polynomials, modified rational functions are presented. Fractional Sobolev space is introduced and some lemmas on interpolation approximation in the fractional Sobolev space are also included. In the second part of the thesis, we present the spectral collocation method based on Hermite functions. Two bases are used, namely, the over-scaled Hermite function and generalized Hermite function, which are orthogonal functions on the whole line with appropriate weight functions. We will show that the fractional Laplacian of these two kinds of Hermite functions can be represented by confluent hypergeometric function. Behaviors of the condition numbers for the resulting spectral differentiation matrices with respect to the number of expansion terms are investigated. Moreover, approximation in two-dimensional space using the tensorized bases, application to multi-term problems and use of scaling to match different decay rate are also considered. Convergence analysis for generalized Hermite function are derived and numerical errors for two bases are analyzed. The third part of the thesis deals with the spectral collocation method based on modified rational functions. We first give a brief introduction for computation of the fractional Laplacian using modified rational functions, which is represented by hypergeometric functions. Then the differentiation matrix involving the fractional Laplace operator is given. Convergence analysis for modified Chebyshev rational functions and modified Legendre rational functions are derived and numerical experiments are carried out.

Os objetivos deste trabalho são: determinar o geóide do Estado de São Paulo (SP) pela técnica matemática Fast Collocation utilizando o modelo do geopotencial CG01c, derivado das novas missões gravimétricas de satélite CHAMP e GRACE, avaliar esse geóide através de estações com alturas geoidais conhecidas e do geóide calculado por Souza (2002), e apresentar uma nova abordagem que amplia a idéia sobre o sensoriamento remoto. O modelo do geopotencial CG01c, obtido com dados das missões CHAMP, GRACE e dados de superfície, e desenvolvido até o grau e ordem 360, foi utilizado para geração de anomalias gravimétricas, as quais foram subtraídas das anomalias gravimétricas terrestres, gerando as anomalias gravimétricas residuais. Essas anomalias residuais serviram de dados de entrada no programa FASTCOL para gerar as alturas geoidais residuais. A essas alturas geoidais residuais foi adicionado o modelo CG01c, representativo dos longos comprimentos de onda do campo de gravidade terrestre, produzindo-se o modelo geoidal chamado GEÓIDESP_FC_2007. Este geóide foi avaliado e posteriormente comparado com o Modelo Digital do Geóide Regular (MDGR). Ao final, argumentos foram apresentados para justificar o estudo do campo de gravidade da Terra no âmbito do sensoriamento remoto. / The objectives of this work are: to determine the geoid of the São Paulo (SP) State applying the mathematical technique so-called Fast Collocation using the CG01c geopotential model, from the new satellite gravity missions CHAMP and GRACE, to evaluate that geoid by stations with known geoidal heights and by the geoid determined by Souza (2002), and to show a new approach that enlarges the idea about the remote sensing. The CG01c geopotential model, obtained from missions CHAMP, GRACE and surface data, and developed up to degree and order 360, it was used to obtain the gravity anomalies, which were subtracted of the terrestrial gravity anomalies, yielding the residuals gravity anomalies. These residuals gravity anomalies it was input to the FASTCOL software to yield the residuals geoidal heights. The CG01c geopotential model, representative of the long wavelengths of the earth gravity field, was added to the residuals geoidal heights, yielding the geoid model so-called GEÓIDESP_FC_2007. This geoid was evaluated and later compared with the Modelo Digital do Geóide Regular (MDGR). At the end, arguments were presented to justify the study of the earth gravity field in the scope of the remote sensing

Os objetivos deste trabalho são: determinar o geóide do Estado de São Paulo (SP) pela técnica matemática Fast Collocation utilizando o modelo do geopotencial CG01c, derivado das novas missões gravimétricas de satélite CHAMP e GRACE, avaliar esse geóide através de estações com alturas geoidais conhecidas e do geóide calculado por Souza (2002), e apresentar uma nova abordagem que amplia a idéia sobre o sensoriamento remoto. O modelo do geopotencial CG01c, obtido com dados das missões CHAMP, GRACE e dados de superfície, e desenvolvido até o grau e ordem 360, foi utilizado para geração de anomalias gravimétricas, as quais foram subtraídas das anomalias gravimétricas terrestres, gerando as anomalias gravimétricas residuais. Essas anomalias residuais serviram de dados de entrada no programa FASTCOL para gerar as alturas geoidais residuais. A essas alturas geoidais residuais foi adicionado o modelo CG01c, representativo dos longos comprimentos de onda do campo de gravidade terrestre, produzindo-se o modelo geoidal chamado GEÓIDESP_FC_2007. Este geóide foi avaliado e posteriormente comparado com o Modelo Digital do Geóide Regular (MDGR). Ao final, argumentos foram apresentados para justificar o estudo do campo de gravidade da Terra no âmbito do sensoriamento remoto. / The objectives of this work are: to determine the geoid of the São Paulo (SP) State applying the mathematical technique so-called Fast Collocation using the CG01c geopotential model, from the new satellite gravity missions CHAMP and GRACE, to evaluate that geoid by stations with known geoidal heights and by the geoid determined by Souza (2002), and to show a new approach that enlarges the idea about the remote sensing. The CG01c geopotential model, obtained from missions CHAMP, GRACE and surface data, and developed up to degree and order 360, it was used to obtain the gravity anomalies, which were subtracted of the terrestrial gravity anomalies, yielding the residuals gravity anomalies. These residuals gravity anomalies it was input to the FASTCOL software to yield the residuals geoidal heights. The CG01c geopotential model, representative of the long wavelengths of the earth gravity field, was added to the residuals geoidal heights, yielding the geoid model so-called GEÓIDESP_FC_2007. This geoid was evaluated and later compared with the Modelo Digital do Geóide Regular (MDGR). At the end, arguments were presented to justify the study of the earth gravity field in the scope of the remote sensing

Os objetivos deste trabalho são: determinar o geóide do Estado de São Paulo (SP) pela técnica matemática Fast Collocation utilizando o modelo do geopotencial CG01c, derivado das novas missões gravimétricas de satélite CHAMP e GRACE, avaliar esse geóide através de estações com alturas geoidais conhecidas e do geóide calculado por Souza (2002), e apresentar uma nova abordagem que amplia a idéia sobre o sensoriamento remoto. O modelo do geopotencial CG01c, obtido com dados das missões CHAMP, GRACE e dados de superfície, e desenvolvido até o grau e ordem 360, foi utilizado para geração de anomalias gravimétricas, as quais foram subtraídas das anomalias gravimétricas terrestres, gerando as anomalias gravimétricas residuais. Essas anomalias residuais serviram de dados de entrada no programa FASTCOL para gerar as alturas geoidais residuais. A essas alturas geoidais residuais foi adicionado o modelo CG01c, representativo dos longos comprimentos de onda do campo de gravidade terrestre, produzindo-se o modelo geoidal chamado GEÓIDESP_FC_2007. Este geóide foi avaliado e posteriormente comparado com o Modelo Digital do Geóide Regular (MDGR). Ao final, argumentos foram apresentados para justificar o estudo do campo de gravidade da Terra no âmbito do sensoriamento remoto. / The objectives of this work are: to determine the geoid of the São Paulo (SP) State applying the mathematical technique so-called Fast Collocation using the CG01c geopotential model, from the new satellite gravity missions CHAMP and GRACE, to evaluate that geoid by stations with known geoidal heights and by the geoid determined by Souza (2002), and to show a new approach that enlarges the idea about the remote sensing. The CG01c geopotential model, obtained from missions CHAMP, GRACE and surface data, and developed up to degree and order 360, it was used to obtain the gravity anomalies, which were subtracted of the terrestrial gravity anomalies, yielding the residuals gravity anomalies. These residuals gravity anomalies it was input to the FASTCOL software to yield the residuals geoidal heights. The CG01c geopotential model, representative of the long wavelengths of the earth gravity field, was added to the residuals geoidal heights, yielding the geoid model so-called GEÓIDESP_FC_2007. This geoid was evaluated and later compared with the Modelo Digital do Geóide Regular (MDGR). At the end, arguments were presented to justify the study of the earth gravity field in the scope of the remote sensing

<div>In recent decades the revolutionary possibilities of low-thrust electric propulsion have been demonstrated by the success of missions such as Dawn and Hayabusa 1 and 2. The efficiency of low-thrust engines reduces the propellant mass required to achieve mission objectives and this benefit is frequently worth the additional time of flight incurred, particularly for robotic spacecraft. However, low-thrust trajectory design poses a challenging optimal control problem. At each instant in time, spacecraft control parameters that minimize an objective, typically propellant consumption or time of flight, must be determined. The characteristics of low-thrust optimal solutions are often unintuitive, making it difficult to develop an <i>a priori</i> estimate for the state and control history of a spacecraft that can be used to initialize an optimization algorithm. This investigation seeks to develop a low-thrust trajectory design framework to address this challenge by combining the existing techniques of orbit chaining and direct collocation. Together, these two methods offer a novel approach for low-thrust trajectory design that is intuitive, flexible, and robust.</div><div><br></div><div>This investigation presents a framework for the construction of orbit chains and the convergence of these initial guesses to optimal low-thrust solutions via direct collocation. The general procedure is first demonstrated with simple trajectory design problems which show how dynamical structures, such as periodic orbits and invariant manifolds, are employed to assemble orbits chains. Following this, two practical mission design problems demonstrate the applicability of this framework to real world scenarios. An orbit chain and direct collocation approach is utilized to develop low-thrust transfers for the planned Gateway spacecraft between a variety of lunar and libration point orbits (LPOs). Additionally, the proposed framework is applied to create a systematic method for the construction of transfers for the Lunar IceCube spacecraft from deployment to insertion upon its destination orbit near the Moon. Three and four-body dynamical models are leveraged for preliminary trajectory design in the first and second mission design applications, respectively, before transfers are transitioned to an ephemeris model for validation. Together, these realistic sample applications, along with the early examples, demonstrate that orbit chaining and direct collocation constitute an intuitive, flexible, and robust framework for low-thrust trajectory design. </div>

Formulaic language (FL) has long been a research topic investigated by various schools of researchers. Most of the previous researches focused on English as Second Language. The applicability of the past research results to other languages, especially Chinese as a second language, is still uncertain.The present study attempts to investigate the classroom teaching of Chinese formulaic language and its effects on students writing performance. Two sections of 3rd year Chinese L2 learners at Brigham Young University were chosen to participate the research, serving as the treatment group and the control group respectively. Both groups were required to write on the same topics at different times in a semester. 80 responses from four topics were then chosen to analyze the effects of using Chinese FL.Results show that among the four types of Chinese FL, both collocations and transitions are used frequently, while the other two types of Chinese FL are less used by participants from the experimental group. Except the first topic, the adoption of Chinese FL does contribute to more Chinese characters in the writing responses of the participants from the experimental group, which reveal learners confidence in Chinese writing after the FL training. Also, both the remarkably higher average scores for each topic and the decreasing error rates demonstrate that the adoption of Chinese FL teaching positively influences participants writing performance. Lastly, the contributions of the four different types of Chinese FL to participants better writing performance are also discussed.