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Small-parameter expansion of linear Boltzmann or master operatorsAkama, Hachiro January 1964 (has links)
Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The differential-operator approximation to the linear Boltzmann operator (the Master equation operator) has been studied by several authors. In 1960 Siegel proposed a systematic approach called the CD expansion. He represents the approximating series in terms of creation and destruction operators for Hermite functions. In this thesis we study the physical meaning of a small parameter which usually exists in the CD expansion and which ensures the convergence of the series. We also establish the CD-expansion formalism for N-dimensional processes and initiate the study of the CD-expansion of the linear Boltzmann collision operator in the kinetic theory of gases.
In the case of one-dimensional processes; the models we study are the density fluctuations with a particle reservoir of finite volume, Alkemade's diode model, and Rayleigh disk. We find that the expansion parameter is the ratio of the average microscopic agitation interval to the macroscopic relaxation time. We further prove that this ratio is equal to the ratio of the average variance of the discontinuity of the random process determined by the linear. Boltzmann operator to the variance of the macroscopic observable at equilibrium. Since the CD expansion is an expansion with respect to the parameter of discontinuity, the expansion series reduces to the Fokker-Planck operator in the limiting case where the parameter becomes zero.
In the N-dimensional formalism, we use tensor Hermite polynomials and find a formalism valid for processes of any finite dimensionality. In extending the study to the kinetic theory of gases, we establish a method of obtaining derivate moments directly from the collision operator, and obtain a formula for the Hermite coefficients of derivate moments for an arbitrary force field.
We propose the CD hypothesis: The terms of the CD expansion are homogeneous and of successively increasing order in the parameter of discontinuity of the process. This hypothesis I holds for all the models we study in the one-dimension case. In three-dimensional collision processes it holds for an intermolecular force field obeying an inverse power law and for rigid spheres. Beyond these cases, the necessary and sufficient conditions for the hypothesis are rather complicated. A sufficient condition is, however, that the scattering cylinder (scattering cross section multiplied by the magnitude of the relative velocity) be a homogeneous function of the magnitude of the relative velocity.
In addition to the general results mentioned in the above, we obtain a number of particular results. Special cases of our density fluctuation model are the density fluctuations studied by van Kampen (1961), and Ehrenfest's urn model. We also introduce an isotropic Maxwellian particle which corresponds to s-wave scattering in wave mechanics, and which yields the CD expansion in a diagonal form. / 2999-01-01
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Series solution for the propagator of the linear Boltzmann equationKohlberg, Ira January 1965 (has links)
Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / This investigation is concerned with the evaluation and interpretation of the propagator, or conditional probability distribution function P(y,t| y0), of the Linear Boltzmann or Master equation from the "central limit viewpoint". We have obtained what is to our knowledge the first evaluation in series of the propagator for the typical kinetic-theoretical processes studied here, which are those underlying the problems usually studied in the approximation of the Fokker-Planck equation. We have been able to put the successive terms of this series in closed form; and have shown that the series can be interpreted as a generalized solution of the central limit problem of mathematical probability theory, the generalization consisting in the extension to a process in continuous time with non-independent increments. [TRUNCATED] / 2999-01-01
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Cost/Effectiveness Analysis of Obtaining Operational Estimates of Reference Evapotranspiration, Peninsular Florida, USAKittridge, Michael G. 20 July 2007 (has links)
The objective of this study is to conduct a cost/effectiveness analysis of the computation of reference evapotranspiration (ETo) in the peninsular of Florida. A meteorological station on the Fort Meade Mine in Polk County, Florida was used to provide data for the calculation of ETo. Five ETo equations were tested to determine the accuracy and cost/effectiveness to the fully measured ASCE Penman-Monteith (Full ASCE-PM) equation on daily, monthly, and annually time steps. The ETo equations ranged in amounts of parameters from the Full ASCE-PM to the Hargreaves. The energy terms accounted for approximately 90% of the total ETo flux. Solar radiation alone also accounted for approximately 90% of the total ETo flux. The highest cost-effectiveness ratios were equations that were able to accurately estimate values without relying on expensive meteorological equipment and/or omitted terms that had a lesser influence on the magnitude of ETo. The seasonal variability in the climate and consequently the emphasis of each meteorological parameter on ETo will create seasonal errors in the reduced sets of the ETo equations. Large seasonal errors were associated with temperature based ETo equations, while solar radiation based ETo equations accurately preserved the seasonal trends. At least in Florida, solar radiation is the key driving force in both the magnitude and the seasonality of ETo.
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Equations d'évolution sur certains groupes hyperboliques / Evolution equation on some hyperbolic groupsJamal Eddine, Alaa 06 December 2013 (has links)
Cette thèse porte sur l’étude d’équations d’évolution sur certains groupes hyperboliques, en particulier, nous étudions l’équation de la chaleur, l’équation de Schrödinger et l’équation des ondes modifiée, d’abord sur les arbres homogènes, ensuite sur des graphes symétriques. Sur les arbres homogènes, nous montrons que, sous une hypothèse d’invariance de jauge, on a existence globale des solutions de l’équation de Schrödinger ainsi qu’un phénomène de ’scattering’ pour des données arbitraires dans l’espace des fonctions de carré intégrable sans restriction sur le degré de la non-linéarité, contrairement au cas euclidien ou au cas hyperbolique. Nous généralisons ensuite ce résultat sur les graphes symétriques de degré (k − 1)(r − 1) sous la condition k < r. Un de nos principaux résultats sur les graphes symétriques est l’estimation du noyau de la chaleur associé au laplacien combinatoire. Pour finir, nous établissons une expression explicite des solutions de l’équation des ondes modifiée sur les graphes symétriques. / This thesis focuses on the study of evolution equations on certain hyperbolic groups, in particular, we study the heat equation, the Schrödinger equation and the modified wave equation first on homogeneous trees then on symmetric graphs. In the homogeneous trees case, we show that under a gauge invariance condition, we have global existence of solutions of the Schrödinger equation and scattering for arbitrary data in the space of square integrable functions without any restriction on the degree of the nonlinearity, in contrast to the euclidean and hyperbolic space cases. We then generalize this result on symmetric graphs of degree (k − 1)(r − 1) under the condition k < r . One of our main results on symmetric graphs is the estimate of the heat kernel associated to the combinatorial laplacian. Finally, we establish an explicit expression of solutions of the modified wave equation on symmetric graphs.
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Inverse Scattering For The Zero-Energy Novikov-Veselov EquationMusic, Michael 01 January 2016 (has links)
For certain initial data, we solve the Novikov-Veselov equation by the inverse scat- tering method. This is a (2+1)-dimensional completely integrable system that gen- eralizes the (1+1)-dimensional Korteweg-de-Vries equation. The method used is the inverse scattering method. To study the direct and inverse scattering maps, we prove existence and uniqueness properties of exponentially growing solutions of the two- dimensional Schrodinger equation. For conductivity-type potentials, this was done by Nachman in his work on the inverse conductivity problem. Our work expands the set of potentials for which the analysis holds, completes the study of the inverse scattering map, and show that the inverse scattering method yields global in time solutions to the Novikov-Veselov equation. This is the first proof that the inverse scattering method yields classical solutions to the Novikov-Veselov equation for the class of potentials considered here.
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Local orthogonal mappings and operator formulation for varying cross-sectional ducts.Ahmed, Naveed, Ahmed, Waqas January 2010 (has links)
<p>A method is developed for solving the two dimensional Helmholtz equation in a ductwith varying cross-section region bounded by a curved top and flat bottom, having oneregion inside. To compute the propagation of sound waves in a curved duct with a curvedinternal interface is difficult problem. One method is to transform the wave equation intoa solvable form and making the curved interface plane. To this end a local orthogonaltransformation is developed for the varying cross-sectional duct having one medium inside.This transformation is first used to make the curved top of the waveguide flat andto transform the Helmholtz equation into an initial value problem. Later on the local orthogonaltransformation is developed for a waveguide having two media inside with flattop, a flat bottom and a curved interface. This local orthogonal transformation is used toflatten the interface and also to transform the Helmholtz equation into a simple, solvableordinary differential equation. In this paper we present operator formulation for the partwith flat bottom and curved top including a curved interface. In the ordinary differentialequation with operators in coefficients, obtained after the transformation, all the operationsrelated to the transverse variable are treated as operators while the derivative withrespect to the range variable is kept.</p>
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Regularity and approximation of a hyperbolic-elliptic coupled problemKruse, Carola January 2010 (has links)
In this thesis, we investigate the regularity and approximation of a hyperbolic-elliptic coupled problem. In particular, we consider the Poisson and the transport equation where both are assigned nonhomogeneous Dirichlet boundary conditions. The coupling of the two problems is executed as follows. The right hand side function of the Poisson equation is the solution ρ of the transport equation whereas the gradient field E = −∇u, with u being solution of the Poisson problem, is the convective field for the transport equation. The analysis is done throughout on a nonconvex, not simply connected domain that is supposed to be homeomorph to an annular domain. In the first part of this thesis, we will focus on the existence and uniqueness of a classical solution to this highly nonlinear problem using the framework of Hölder continuous functions. Herein, we distinguish between a time dependent and time independent formulation. In both cases, we investigate the streamline functions defined by the convective field E. These are used in the time dependent case to derive an operator equation whose fixed point is the streamline function to the gradient of the classical solution u. In the time independent setting, we formulate explicitly the solution operators L for the Poisson and T for the transport equation and show with a fixed point argument the existence and uniqueness of a classical solution (u,ρ). The second part of this thesis deals with the approximation of the coupled problem in Sobolev spaces. First, we show that the nonlinear transport equation can be formulated equivalently as variational inequality and analyse its Galerkin finite element discretization. Due to the nonlinearity of the coupled problem, it is necessary to use iterative solvers. We will introduce the staggered algorithm which is an iterative method solving alternating the Poisson and transport equation until convergence is obtained. Assuming that LοT is a contraction in the Sobolev space H1(Ω), we will investigate the convergence of the discrete staggered algorithm and obtain an error estimate. Subsequently, we present numerical results in two and three dimensions. Beside the staggered algorithm, we will introduce other iterative solvers that are based on linearizing the coupled problem by Newton’s method. We illustrate that all iterative solvers converge satisfactorily to the solution (u, ρ).
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Sturmian theory and its applicationsLawson, R. D. Unknown Date (has links)
No description available.
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The Nonisospectral and variable coefficient Korteweg-de Vries equation.January 1992 (has links)
by Li Kam Shun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaf 65). / Chapter CHAPTER 1 --- Soliton Solutions of the Nonisospectral and Variable Coefficient Korteweg-de Vries Equation / Chapter §1.1 --- Introduction --- p.4 / Chapter §1.2 --- Inverse Scattering --- p.6 / Chapter §1.3 --- N-Soliton Solution --- p.11 / Chapter §1.4 --- One-Soliton Solutions --- p.15 / Chapter §1.5 --- Two-Soliton Solutions --- p.18 / Chapter §1.6 --- Oscillating and Asymptotically Standing Solitons --- p.23 / Chapter CHAPTER 2 --- Asymptotic Behaviour of Nonsoliton Solutions of the Nonisospectral and Variable Coefficient Korteweg-de Vries Equation / Chapter §2.1 --- Introduction --- p.31 / Chapter §2.2 --- Main Results --- p.36 / Chapter §2.3 --- Lemmas --- p.39 / Chapter §2.4 --- Proof of the Main Results --- p.59 / References --- p.65
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On a shallow water equation.January 2001 (has links)
Zhou Yong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 51-53). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Chapter 1 --- Introduction --- p.2 / Chapter 2 --- Preliminaries --- p.10 / Chapter 3 --- Periodic Case --- p.22 / Chapter 4 --- Non-periodic Case --- p.35 / Bibliography --- p.51
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