Applied Mathematics Commons^™

The Complementing Condition In Elasticity, Lavanya Ramanan

Masters Theses

We consider a boundary value problem of nonlinear elasticity on a domain [omega] in R³ [3-dimensional space] and compute the Complementing Condition for the linearized equations at a point X₀ [x zero] on boundary of omega. We assume a stored energy function depending on the first and third invariants of the deformation F and that the strong-ellipticity condition holds in [omega] . A surface traction boundary condition is imposed at X_0.
The Complementing Condition is calculated from a system of 3 second-order ordinary differential equations (0 less than and equal to t less than infinity) with boundary …

The Green's Function Method For Solutions Of Fourth Order Nonlinear Boundary Value Problem., Olga A. Teterina

Masters Theses

This thesis has demonstrated that Green’s functions have a wide range of applications with regard to boundary value problems. In particular, existence and uniqueness of solutions of a large class of fourth order boundary value problems has been established. In fact, given any fourth order ODE with homogeneous boundary conditions, as long as the corresponding Green’s function exists and f satisfies an appropriate Lipschitz condition, Theorem 2.1 guarantees such a solution under equally mild conditions. Similarly, Theorem 2.2 also guarantees such a solution under equally mild conditions. These theorems are contrasted with classical ODE existence theorems in that they get …

Comparison Of Methods For Estimating Stochastic Volatility, John Parnell Collins

Masters Theses

Understanding the ever changing stock market has long been of interest to both academic and financial institutions. The early attempts to model the dynamics treated the volatility or sensitivity of the price change to random effects as constant. However, in matching the model to real data it was realized that the volatility was actually a random variable, and thus began efforts to determine methods for estimating the stochastic volatility from experimental data.

In this thesis, we develop and compare three different computational statistical filtering methods for estimating the volatility: The Kalman Filter, the Gibbs Sampler, and the Particle Filter. These …

Immersed Finite Element Method For Interface Problems With Algebraic Multigrid Solver, Wenqiang Feng

Masters Theses

"This thesis is to discuss the bilinear and 2D linear immersed finite element (IFE) solutions generated from the algebraic multigrid solver for both stationary and moving interface problems. In contrast to the body-fitting mesh restriction of the traditional finite element methods or finite difference methods for interface problems, a number of numerical methods based on structured meshes independent of the interface have been developed. When these methods are applied to the real world applications, we often need to solve the corresponding large scale linear systems many times, which demands efficient solvers. The algebraic multigrid (AMG) method is a natural choice …

Validation Of Weak Form Thermal Analysis Algorithms Supporting Thermal Signature Generation, Elton Lewis Freeman

Masters Theses

Extremization of a weak form for the continuum energy conservation principle differential equation naturally implements fluid convection and radiation as flux Robin boundary conditions associated with unsteady heat transfer. Combining a spatial semi-discretization via finite element trial space basis functions with time-accurate integration generates a totally node-based algebraic statement for computing. Closure for gray body radiation is a newly derived node-based radiosity formulation generating piecewise discontinuous solutions, while that for natural-forced-mixed convection heat transfer is extracted from the literature. Algorithm performance, mathematically predicted by asymptotic convergence theory, is subsequently validated with data obtained in 24 hour diurnal field experiments for …

Decay Estimates For Nonlinear Wave Equations With Variable Coefficients, Michael Jacob Roberts

Masters Theses

We studied the long time behavior of solutions of nonlinear wave equations with variable coefficients and an absorption nonlinearity. Such an equation appears in models for traveling waves in a non-homogeneous gas with damping that changes with position. We established decay estimates of the energy of solutions. We found three different regimes of decay of solutions depending on the exponent of the absorption term. We show the existence of two critical exponents. For the exponents above the larger critical exponent, the decay of solutions of the nonlinear equation coincides with that of the corresponding linear problem. For exponents below the …

Latin Hypercube Sampling And Partial Rank Correlation Coefficient Analysis Applied To An Optimal Control Problem, Boloye Gomero

Masters Theses

Latin Hypercube Sampling/Partial Rank Correlation Coefficient (LHS/PRCC) sensitivity analysis is an efficient tool often employed in uncertainty analysis to explore the entire parameter space of a model. Despite the usefulness of LHS/PRCC sensitivity analysis in studying the sensitivity of a model to the parameter values used in the model, no study has been done that fully integrates Latin Hypercube sampling with optimal control analysis.

In this thesis, we couple the optimal control numerical procedure to the LHS/PRCC procedure and perform a simultaneous examination of the effects of all the LHS parameter on the objective functional value. To test the effectiveness …

Lattice Residuability, Philip Theodore Thiem

Masters Theses

"Residuated lattices form the basis of certain kinds of logical interpretations. Also, complete commutative integral zero-bounded residuated lattices are used as a set of truth values for fuzzy logic values, Which are more general than the traditional bounded interval introduced by Zadeh. At times, it is important to know whether or not the lattice can be residuated in the first place. This thesis reviews the literature in lattice residuability and adds more observations. Specifically, (1) bounded chains and top-residuated lattices are show [sic] to be residuable, and (2) additional conditions necessary for residuability are established"--Abstract, page iii.

Transverse Waves In Simulated Liquid Rocket Engines With Arbitrary Headwall Injection, Charles Toufic Haddad

Masters Theses

This work introduces a closed-form analytical solution for the transverse vorticoacoustic wave in a circular cylinder with arbitrary headwall injection. This particular configuration mimics the conditions leading to the onset of traveling radial and tangential waves in a simple liquid rocket engine (LRE). Assuming a short cylindrical chamber with an injecting headwall, regular perturbations are used to linearize the problem’s mass, momentum, energy, ideal gas and isentropic relations. A Helmholtz decomposition is subsequently applied to the first-order disturbance equations, thus giving rise to a compressible, inviscid and acoustic set that is responsible for driving the unsteady motion and to an …

Development And Analysis Of Onboard Translunar Injection Targeting Algorithms, Phillippe Lyles Winters Reed

Masters Theses

Several targeting algorithms are developed and analyzed for possible future use onboard a spacecraft. Each targeter is designed to determine the appropriate propulsive burn for translunar injection to obtain desired orbital parameters upon arrival at the moon. Primary design objectives are to minimize the computational requirements for each algorithm but also to ensure reasonable accuracy, so that the algorithm’s errors do not force the craft to conduct large mid-course corrections. Several levels of accuracy for dynamical models are explored, the convergence range and speed of each algorithm are compared, and the possible benefits of the Broyden and trust-region targeters are …

A Time Series Approach To Electric Load Modelling, Matthias Benjamin Noller

Masters Theses

"With resources becoming more and more scarse [sic] as well as increasing competition caused by the liberalisation of the energy markets electric load modelling becomes ever more important for proper resource allocation.

This work tries to bridge the gap between long-term modelling done mainly via econometric approaches and short-term modelling in which time series models are more commonplace by focussing [sic] on pure time series modelling [sic] and exploring its limits in the process. Due to various seasonalities present in the data the approach chosen starts with a subdivision of the time axis in different time frames: A model for …

Analytical Computation Of Proper Orthogonal Decomposition Modes And N-Width Approximations For The Heat Equation With Boundary Control, Tasha N. Fernandez

Masters Theses

Model reduction is a powerful and ubiquitous tool used to reduce the complexity of a dynamical system while preserving the input-output behavior. It has been applied throughout many different disciplines, including controls, fluid and structural dynamics. Model reduction via proper orthogonal decomposition (POD) is utilized for of control of partial differential equations. In this thesis, the analytical expressions of POD modes are derived for the heat equation. The autocorrelation function of the latter is viewed as the kernel of a self adjoint compact operator, and the POD modes and corresponding eigenvalues are computed by solving homogeneous integral equations of the …

Propagation Of Periodic Waves Using Wave Confinement, Paula Cysneiros Sanematsu

Masters Theses

This thesis studies the behavior of the Eulerian scheme, with "Wave Confinement" (WC), when propagating periodic waves. WC is a recently developed method that was derived from the scheme "vorticity confinement" used in fluid mechanics, and it efficiently solves the linear wave equation. This new method is applicable for numerous simulations such as radio wave propagation, target detection, cell phone and satellite communications.

The WC scheme adds a nonlinear term to the discrete wave equation that adds stability with negative and positive diffusion, conserves integral quantities such as total amplitude and wave speed, and it allows wave propagation over long …

Nonlinear Acoustics Of Piston-Driven Gas-Column Oscillations, Andrew William Wilson

Masters Theses

The piston-driven oscillator is traditionally modeled by directly applying boundary conditions to the acoustic wave equations; with better models re-deriving the wave equations but retaining nonlinear and viscous effects. These better models are required as the acoustic solution exhibits singularity near the natural frequencies of the cavity, with an unbounded (and therefore unphysical) solution. Recently, a technique has been developed to model general pressure oscillations in propulsion systems and combustion devices. Here, it is shown that this technique applies equally well to the piston-driven gas-column oscillator; and that the piston experiment provides strong evidence for the validity of the general …

Closed-Form Solutions To Discrete-Time Portfolio Optimization Problems, Mathias Christian Goeggel

Masters Theses

"In this work, we study some discrete time portfolio optimization problems. After a brief introduction of the corresponding continuous time models, we introduce the discrete time financial market model. The change in asset prices is modeled in contrast to the continuous time market by stochastic difference equations. We provide solutions for these stochastic difference equations. Then we introduce the discrete time risk-measure and the portfolio optimization problems. We provide closed form solutions to the discrete time problems. The main contribution of this thesis are the closed form solutions to the discrete time portfolio models. For simulation purposes the discrete time …

The Analogue Of The Iterated Logarithm For Quantum Difference Equations, Karl Friedrich Ulrich

Masters Theses

"In this thesis, we consider oscillation and nonoscillation of q-difference equations, i.e., equations that arise while studying q-calculus. In particular, we prove an extension of Kneser’s theorem on q-calculus to cases in which no conclusion can be drawn by applying Kneser’s theorem. In order to accomplish this, we establish a change of variables which yields, when applied iteratively, a sequence of comparison functions. We use these comparison functions to establish our main result. Finally, we consider an analogue result for time scales which are unbounded from above"--Abstract, page iii.

Dynamic Equations With Piecewise Continuous Argument, Christian Keller

Masters Theses

"We extend the theory of differential equations with piecewise continuous argument to general time scales. Linear and quasi-linear systems of functional dynamic equations with alternating retarding and advanced argument will be investigated and conditions for globally asymptotic stability of those systems will be stated and proven. Furthermore, oscillation criteria for linear first-order equations with piecewise continuous argument will be established"--Abstract, page iii.

Some Properties Of Hereditarily Indecomposable Chainable Continua, Thomas John Kacvinsky

Masters Theses

"In 1920, B. Knaster and C. Kuratowski raised the question of whether each homogeneous plane continuum is a simple closed curve. In 1921, S. Mazurkiewicz raised the question of whether each subcontinuum of Euclidean n-space which is homeomorphic to each of its subcontinua is necessarily an arc. In that same year, B. Knaster and C. Kuratowski raised the question of whether there exists a nondegenerate hereditarily indecomposable continuum.

The third question was answered in the affirmative in 1922 by B. Knaster, when he constructed a nondegenerate hereditarily indecomposable subcontinuum of the plane.

The second question was answered in 1947 by …

A Numerical Method For The Solution Of The Schrödinger Equation By A Trial Wavefunction Improvement Formula, Chun-Sheng Ko

Masters Theses

A numerical method and corresponding computer algorithm for solving the one-dimensional radial Schrödinger equation to any desired accuracy is developed. The method uses a finite difference scheme in which an initial trial wavefunction is digitalized over a lattice covering the region of integration. The values of a rough solution are then altered at each lattice point by a simple improvement formula decreasing the value of the variational energy until the desired minimum is reached. The accuracy of these solutions depends only on the grid size. This method is characterized and tested with a harmonic oscillator potential. Practical evaluations and applications …

Evaluation And Improvement Of The Wu-Koh Model For The Prediction Of Multiple-Cell Cooling Tower Plumes, Michael Joseph Wastag

Masters Theses

This paper evaluates and improves the performance of the Wu-Koh mathematical model for the prediction of plume dispersion from multiple natural-draft (NDCT) and mechanical-draft (MDCT) cooling towers. The Wu-Koh multiple-tower model was chosen for study due to its advanced treatment of plume merging.

Our evaluation of the Wu-Koh model was carried out by comparing model predictions of field and laboratory data. Comparisons of model predictions of visible plume outlines were made to single- and multiple-tower visible plume data. The sites of the single-tower data were Lunen (300 MW_e), Chalk Point (630 MW_e) and Paradise (1100 MW …

An Investigation Of The Plume Theories Of Briggs And Hanna, George Kyle Cooper

Masters Theses

Many models have been developed for or applied to the prediction of mechanical draft cooling tower plumes. However, due to a serious lack of data up to this date, little has been done to verify the predictive capability of these mathematical models. This thesis attempts to rectify this situation somewhat by undertaking a thorough investigation of the plume theories of G. A. Briggs and S. R. Hanna. A study of their theoretical foundations and development, practical formulations, and ability to predict the height and length of the visible portion of mechanical-draft cooling tower plumes is undertaken. Detailed derivations of their …

Comparative Studies Of The Computational Analysis Of One Dimensional Gas Flow, Johnny Ziebarth

Masters Theses

No abstract provided.

Imbedding Cayley Graphs, Brian L. Garman

Masters Theses

No abstract provided.

Models For Molecular Vibration, Allan Bruce Capps

Masters Theses

“The purpose of the investigation is to determine if a classical model can be used to characterize molecular vibrational and librational (restricted rotation) frequencies. In general, the model treats a molecule as asymmetrical and rigid and simulates the intermolecular forces by springs along the bonds. Molecules constrained to a plane and molecules free to move in three dimensions are analyzed. The frequencies of water molecules are investigated, in particular. The model and analytic components are found to function well. Within the limits set for the model, the water molecule simulation is not successful as the motion becomes anharmonic for energy …

Numerical Study Of Slow Motion Of A Smoke Filament, Tapan Sen

Masters Theses

No abstract provided.

Tschebyscheff Fitting With Polynomials And Nonlinear Functions, George F. Luffel

Masters Theses

"It is the purpose of this study to survey the properties of the Tschebyscheff polynomials with particular reference to how they are employed as approximants and interpolants. The survey is extended to include the process known as "Tschebyscheff Approximation" or "Tschebyscheff Fitting" of a function by functions other than polynomials. One such fitting technique which will be of particular interest in this study is that of fitting f(x) by the function ab^x + c where a, b, c are real and b ≠ 1. In addition to a survey of the properties of the Tschebyscheff polynomials and of Tschebyscheff …

Latent Class Analysis And Information Retrieval, George Loyd Jensen

Masters Theses

"Information retrieval may be defined roughly as a procedure to either locate or physically retrieve a document or documents containing information on a given topic with a high degree of reliability. Information retrieval is one of the newest fields in computing science. Being so new there are many unexplored areas. The research that has been done has not been standardized beyond the point of the effort which has been slanted toward solving the "Library Problem" ... Several different information retrieval methods have been suggested and some research done on them to try and solve the "Library Problem". Of these methods …

Linear And Quadratic Programming With More Than One Objective Function, William John Lodholz

Masters Theses

"A computational procedure is presented for determining optimal solutions to the linear and quadratic programming problem when there is more than one objective function subject to linear constraints. In general a unique solution does not exist and a set of "best" or "efficient" points is determined and presented in graphical or tabular form. To solve the mathematical programming problems the simplex method is used for linear objective functions and Wolfe's method is used for quadratic objective functions"--Abstract, page ii.

A Study Of Certain Conservative Sets For Parameters In The Linear Statistical Model, Roger Alan Chapin

Masters Theses

"In the case of the linear statistical model, it has been shown that under certain conditions the confidence intervals obtained by considering the parameters one at a time are conservative when used as a joint confidence region, using the product of confidences as the confidence. However, nothing had been known of how conservative they are. This research provides accurate estimates of the true confidence for these cases.

Also, it has not yet been proved that they are indeed conservative for all cases. It is thought that they are, and the results of this research support this conjecture"--Abstract, page ii.

Comparison Of Methods To Select A Probability Model, Howard Lyndal Colburn

Masters Theses

"A comparison is made between the estimates of the parameters in a gamma distribution obtained by the method of moments with those obtained by a numerical approximation to the maximum likelihood estimates. The estimates obtained by the numerical approximation had a smaller mean squared error from the true value than the estimates obtained by the method of moments. Modifications to tests of fit are made in order to develop methods to select a distribution from a set of possible distributions for a population with an unknown distribution. These selection methods are compared in their ability to make correct selections. Although …