Physical Sciences and Mathematics Commons^™

Energy Methods For Reaction-Diffusion Problems, Xing Zhong

Dissertations

Nonlinear reaction-diffusion equations arise in many areas of applied sciences such as combustion modeling, population dynamics, chemical kinetics, etc. A fundamental problem in the studies of these equations is to understand the long time behavior of solutions of the associated Cauchy problem. These kinds of questions were originally studied in the context of combustion modeling.

For suitable nonlinearity and a monotone increasing one-parameter family of initial data starting with zero data, small values of the parameter lead to extinction, whereas large values of the parameter may lead to spreading, i.e., the solution converging locally uniformly to a positive spatially independent …

Instabilities In Newtonian Films And Nematic Liquid Crystal Droplets, Te-Sheng Lin

Dissertations

The instabilities of Newtonian films and nematic liquid crystal droplets within the framework of the long wave (lubrication) approximation are studied. For Newtonian films, it is found that, under destabilizing gravitational force, a contact line, modeled by a commonly used precursor film model, leads to free surface instabilities without any additional natural or imposed perturbations. In addition, there is a coupling between the surface instabilities and the transverse (fingering) instabilities which leads to complex behavior. All the observed phenomena are characterized by a single parameter D = (3Ca)^1/3 cot α where Ca is the capillary number and …

Mathematical Models Of Combustion At High Pressure, Daniel Fong

Dissertations

In this dissertation, we develop new mathematical theories of flame propagation that are valid at elevated, or extreme, pressures. Of particular interest is the regime of burning in which the pressure exceeds the critical pressure of the species undergoing chemical reaction. Fluids and flames are known to behave differently under these extreme conditions as opposed to atmospheric pressure. The focus of this dissertation is to investigate these differences by deriving reduced models that contain the unique features.

In the first part of this dissertation, we analyze the structure of laminar diffusion flames at high pressure in the limit of large …

The Effects Of Periodic And Non-Periodic Inputs On The Dynamics Of A Medial Entorhinal Cortex Layer Ii Stellate Cell Model, Dongwook Kim

Dissertations

Various neuron types exhibit sub-threshold and firing frequency resonance in which the sub-threshold membrane potential or firing frequency responses to periodic inputs peak at a preferred frequency (or frequencies). Previous experimental work has shown that medial entorhinal cortex layer II stellate cells (SCs) exhibit sub-threshold and firing frequency resonance in the theta frequency band (4 - 10 Hz). In this thesis we seek to understand the biophysical and dynamic mechanism underlying these phenomena and how they are related. We studied the effects of sinusoidal current and synaptic conductance inputs at various frequencies, with and without noise, on the supra-threshold dynamics …

Energy Propagation In Jammed Granular Matter, Xiaoni Fang

Dissertations

The systems built from dense granular materials are very important due to their relevance to a number of technological and other fields. However, they are difficult to study in particular due to a lack of accurate continuum description. In this work, studies on these systems are presented using discrete element simulations that model the granular particles as soft, elastic, and frictional disks which interact when in contact. These simulations are used for the purpose of analyzing a few granular systems with the main emphasis on understanding phenomena of energy and force propagation.

Analysis of energy propagation in a two-dimensional disordered …

Some Contributions To Modeling Usage Sensitive Warranty Servicing Strategies And Their Analyses, Rudrani Banerjee

Dissertations

Providing a warranty as a part of a product's sale is a common practice in industry. Parameters of such warranties (e.g., its duration limits, intensity of use) must be carefully specified to ensure their financial viability. A great deal of effort has been accordingly devoted in attempts to reduce the costs of warranties via appropriately designed strategies to service them. many such strategies, that aim to reduce the total expected costs of the warrantor or / and are appealing in other ways such as being more pragmatic to implement - have been suggested in the literature. Design, analysis and optimization …

Using Feed-Forward Networks To Infer The Activity Of Feedback Neuronal Networks, Xinxian Huang

Dissertations

The nervous system is one of the most important organ systems in a multicellular body. Animals, including human beings perceive, learn, think and deliver motion instructions through their nervous system. The basic structural units of the nervous system are individual neurons which constitute different neuronal networks with distinct functions. In each network, constituent neurons are coupled with different connection patterns, for example, some neurons send feed-forward information to the coupling neurons while others are mutually coupled. Because it is often difficult to analyze large interconnected feedback neuronal networks, it is important to derive techniques to reduce the complexity of the …

Confidence Bands For Survival Functions Under Semiparametric Random Censorship Models, Peixin Zhang

Dissertations

In medical reports point estimates and pointwise confidence intervals of parameters are usually displayed. When the parameter is a survival function, however, the approach of joining the upper end points of individual interval estimates obtained at several points and likewise for the lower end points would not produce bands that include the entire survival curve with a given confidence. Simultaneous confidence bands, which allow confidence statements to be valid for the entire survival curve,would be more meaningful

This dissertation focuses on a novel method of developing one-sample confidence bands for survival functions from right censored data. The approach is model- …

High-Order Adaptive Methods For Computing Invariant Manifolds Of Maps, Jacek K. Wrobel

Dissertations

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the …

Markovian And Stochastic Differential Equation Based Approaches To Computer Virus Propagation Dynamics And Some Models For Survival Distributions, Lianzhe Xu

Dissertations

This dissertation is divided in two Parts. The first Part explores probabilistic modeling of propagation of computer 'malware' (generally referred to as 'virus') across a network of computers, and investigates modeling improvements achieved by introducing a random latency period during which an infected computer in the network is unable to infect others. In the second Part, two approaches for modeling life distributions in univariate and bivariate setups are developed.

In Part I, homogeneous and non-homogeneous stochastic susceptible-exposed-infectious- recovered (SEIR) models are specifically explored for the propagation of computer virus over the Internet by borrowing ideas from mathematical epidemiology. Large computer …

Sequential Bayesian Filtering For Spatial Arrival Time Estimation, Rashi Jain

Dissertations

Locating and tracking a source in an ocean environment as well as estimating environmental parameters of a sound propagation medium is of utmost importance in underwater acoustics. Matched field processing is often the method of choice for the estimation of such parameters. This approach, based on full field calculations, is computationally intensive and sensitive to assumptions on the structure of the environment. As an alternative, methods that use only select features of the acoustic field for source localization and environmental inversion have been proposed. The focus here is on inversion using arrival times of identified paths within recorded time-series. After …

Numerical And Asymptotic Modeling Of Evolving Nonlinear Ocean Surface Wave Fields, Matt Malej

Dissertations

The main focus of this dissertation is the asymptotic and numerical modeling of nonlinear ocean surface wave fields. In particular, a development of an accurate numerical model for the evolution of nonlinear ocean waves, including extreme waves known as Rogue/Freak waves. Due to their elusive and destructive nature, the media often portrays Rogue waves as unimaginatively huge and unpredictable monsters of the sea. To address these concerns, derivations of asymptotically reduced models, based on the small wave steepness assumption, are presented and their corresponding numerical simulations via a Fourier pseudo-spectral method are discussed. The simulations are initialized with a well-known …

Uniform Heating Of Thin Ceramic Slabs In A Multimode Microwave Cavity, Shuchi Agrawal

Dissertations

Two-dimensional reaction diffusion equations, which contain a functional and an inhomogeneous source term, are good models for describing microwave heating of thin ceramic slabs and cylinders in a multi mode, highly resonant cavity. A thin ceramic slab situated in a TEN03 rectangular cavity modeled in the small Biot number limit and a thin silicon wafer situated in a TM101 cylindrical cavity modeled in the small fineness ratio limit are studied to gain insight into the dynamics of the heating process. The evolution of temperature is governed by a two-dimensional reaction diffusion equation and a spatially non-homogeneous reaction term. Numerical methods …

Asymptotic And Numerical Analysis Of Time-Dependent Wave Propagation In Dispersive Dielectric Media That Exhibit Fractional Relaxation, Matthew Frank Causley

Dissertations

This dissertation addresses electromagnetic pulse propagation through anomalously dispersive dielectric media. The Havriliak-Negami (H-N) and Cole-Cole (C-C) models capture the non-exponential nature of such dielectric relaxation phenomena, which is manifest in a variety of dispersive dielectric media. In the C-C model, the dielectric polarization is coupled to the time-dependent Maxwell's equations by a fractional differential equation involving the electric field. In the H-N case, a more general pseudo-fractional differential operator describes the polarization.

The development and analysis of a robust method for implementing such models is presented, with an emphasis on algorithmic efficiency. Separate numerical schemes are presented for C-C …

Pattern Formation In Oscillatory Systems, Hui Wu

Dissertations

Synchronization is a kind of ordinary phenomenon in nature, the study of it includes many mathematical branches. Phase space is one of the most powerful inventions of modern mathematical science. There are two variables, the position and velocity, that can describe the 2-dimensional phase space system. For example, the state of pendulum may be specified by its position and its velocity, so its phase space is 2-dimensional. The state of the system at a given time has a unique corresponding point in the phase space. In order to describe the motion of an oscillator, we can talk about its motion …

Computational Methods For Two-Phase Flow With Soluble Surfactant, Kuan Xu

Dissertations

A mathematical model is formulated and solved for the two-phase flow of a viscous drop or inviscid bubble in an immiscible, viscous surrounding fluid in the zero Reynold's number or Stokes flow limit. A surfactant that is present on the interface is also soluble in the exterior fluid, and the drop is deformed by an imposed linear flow. The geometry is two-dimensional and Cartesian.

The dissolved surfactant is considered in the physically realistic limit of large bulk Péclet number. That is, it convects and diffuses as a passive scalar in the bulk flow where the ratio of its convection to …

A Numerical Study On The Propagation And Interaction Of Strongly Nonlinear Solitary Waves, Qiyi Zhou

Dissertations

We study numerically a strongly nonlinear long wave model for surface gravity waves propagating in both one and two horizontal dimensions. This model often referred to as the Su-Gardner or Green-Naghdi equations can be derived from the Euler equations under the assumption that the ratio between the characteristic wavelength and water depth is small, but no assumption on the wave amplitude is required. We first generalize the model to describe large amplitude one-dimensional solitary waves in the presence of background shear of constant vorticity. After computing the solitary wave solution of the strongly nonlinear model, the interaction between two solitary …

Modeling With Bivariate Geometric Distributions, Jing Li

Dissertations

This dissertation studied systems with several components which were subject to different types of failures. Systems with two components having frequency counts in the domain of positive integers, and the survival time of each component following geometric or mixture geometric distribution can be classified into this category. Examples of such systems include twin engines of an airplane and the paired organs in a human body. It was found that such a system, using conditional arguments, can be characterized as multivariate geometric distributions. It was proved that these characterizations of the geometric models can be achieved using conditional probabilities, conditional failure …

Perturbed Spherical Objects In Acoustic And Fluid Flow Fields, Manmeet Kaur

Dissertations

In this study, the time averaged acoustic radiation force and drag on a small, nearly spherical object suspended freely in a stationary sound wave field in a compressible, low viscosity fluid is to be calculated. This problem has been solved for a spherical object, and it has many important engineering applications related to segregation and separation processes for particles in fluids such as water. Small but significant errors have occurred in the predicted behavior of the particles using the existing approximate solutions based on perfect spheres. The classical approach has been extended in this research to objects that deviate slightly …

Nonlinear Evolution Of Annular Layers And Liquid Threads In Electric Fields, Qiming Wang

Dissertations

The nonlinear dynamics of viscous perfectly conducting liquid jets or threads under the action of a radial electric field are studied theoretically and numerically here. The field is generated by a potential difference between the jet surface and a concentrically placed electrode of given radius. A long-wave nonlinear model that is used to predict the dynamics of the system and in particular to address the effect of the radial electric field on jet breakup is developed, Two canonical regimes are identified that depend on the size of the gap between the outer electrode and the unperturbed jet surface. For relatively …

Modeling And Quasi-Monte Carlo Simulation Of Risk In Credit Portfolios, Bo Ren

Dissertations

Credit risk is the risk of losing contractually obligated cash flows promised by a counterparty such as a corporation, financial institution, or government due to default on its debt obligations. The need for accurate pricing and hedging of complex credit derivatives and for active management of large credit portfolios calls for an accurate assessment of the risk inherent in the underlying credit portfolios. An important challenge for modeling a credit portfolio is to capture the correlations within the credit portfolio. For very large and homogeneous portfolios, analytic and semi-analytic approaches can be used to derive limiting distributions. However, for portfolios …

Reduced Order Models For Fluid-Structure Interaction Systems By Mixed Finite Element Formulation, Ye Yang

Dissertations

In this work, mixed finite element formulations are introduced for acoustoelastic fluid- structure interaction (FSI) systems. For acoustic fluid, in addition to displacement- pressure (u/p) mixed formulation, a three-field formulation, namely, displacement-pressure-vorticity moment formulation (u - p -Λ) is employed to eliminate some zero frequencies. This formulation is introduced in order to compute the coupled frequencies without the contamination of nonphysical spurious non-zero frequencies. Furthermore, gravitational forces are introduced to include the coupled sloshing mode. In addition, u/p mixed formulation is the first time employed in solid. The numerical examples will demonstrate that the mixed formulations are capable of predicted …

Self Similar Flows In Finite Or Infinite Two Dimensional Geometries, Leonardo Xavier Espin Estevez

Dissertations

This study is concerned with several problems related to self-similar flows in pulsating channels. Exact or similarity solutions of the Navier-Stokes equations are of practical and theoretical importance in fluid mechanics. The assumption of self-similarity of the solutions is a very attractive one from both a theoretical and a practical point of view. It allows us to greatly simplify the Navier-Stokes equations into a single nonlinear one-dimensional partial differential equation (or ordinary differential equation in the case of steady flow) whose solutions are also exact solutions of the Navier-Stokes equations in the sense that no approximations are required in order …

Discreet Dynamical Population Models : Higher Dimensional Pioneer-Climax Models, Yogesh Joshi

Dissertations

There are many population models in the literature for both continuous and discrete systems. This investigation begins with a general discrete model that subsumes almost all of the discrete population models currently in use. Some results related to the existence of fixed points are proved. Before launching into a mathematical analysis of the primary discrete dynamical model investigated in this dissertation, the basic elements of the model - pioneer and climax species - are described and discussed from an ecological as well as a dynamical systems perspective. An attempt is made to explain why the chosen hierarchical form of the …

Numerical Detection Of Complex Singularities In Two And Three Dimensions, Kamyar Malakuti

Dissertations

Singularities often occur in solutions to partial differential equations; important examples include the formation of shock fronts in hyperbolic equations and self-focusing type blow up in nonlinear parabolic equations. Information about formation and structure of singularities can have significant role in interfacial fluid dynamics such as Kelvin-Helmholtz instability, Rayleigh-Taylor instability, and Hele-Shaw flow. In this thesis, we present a new method for the numerical analysis of complex singularities in solutions to partial differential equations. In the method, we analyze the decay of Fourier coefficients using a numerical form fit to ascertain the nature of singularities in two and three-dimensional functions. …

Loss Of Synchrony In An Inhibitory Network Of Type-I Oscillators, Myongkeun Oh

Dissertations

Synchronization of excitable cells coupled by reciprocal inhibition is a topic of significant interest due to the important role that inhibitory synaptic interaction plays in the generation and regulation of coherent rhythmic activity in a variety of neural systems. While recent work revealed the synchronizing influence of inhibitory coupling on the dynamics of many networks, it is known that strong coupling can destabilize phase-locked firing. Here we examine the loss of synchrony caused by an increase in inhibitory coupling in networks of type-I Morris-Lecar model oscillators, which is characterized by a period-doubling cascade and leads to mode-locked states with alternation …

On The Rolling Motion Of Viscous Fluid On A Rigid Surface, Xinli Wang

Dissertations

This thesis considers two closely related problems. First, the influence of insoluble surfactant at a moving contact line is considered. This work is mostly motivated by the air entrainment during the coating process where there is a three-phase contact point (e.g., air, liquid and solid). For moving contact line problems, when the fluid is assumed to be an incompressible Newtonian fluid and a no-slip boundary condition is enforced at the solid boundary, the non-integrable stress singularity arises at the contact line, which is physically unrealistic. The contact angle of 180° is considered as a special case in which the singularity …

A Mathematical And Computational Exploration Of The Effect Of The A-Current In Determining The Activity Phase Of Follower Neurons, Yu Zhang

Dissertations

Bursting oscillations are prevalent in neurons of central pattern generators (CPGs) that produce rhythmic motor activity, and the activity phase plays an important role in determining the normal or dysfunctional network output. The activity phase is the delay time-with respect to some reference time in each cycle and normalized by the oscillation cycle period-of the onset of action potentials by a neuron. This dissertation investigates how the A-current, in conjunction with other intrinsic properties, sets the activity phase of a neuron driven by inhibition.

This dissertation is divided into two major components. In the first component, methods of dynamical systems …

The Role Of Short Term Synamptic Plasticity In Temporal Coding Of Neuronal Networks, Lakshmi Chandrasekaran

Dissertations

Short term synaptic plasticity is a phenomenon which is commonly found in the central nervous system. It could contribute to functions of signal processing namely, temporal integration and coincidence detection by modulating the input synaptic strength. This dissertation has two parts. First we study the effects of short term synaptic plasticity in enhancing coincidence detecting ability of neurons in the avian auditory brainstem. Coincidence detection means a target neuron has a higher firing rate when it receives simultaneous inputs from different neurons as opposed to inputs with large phase delays. This property is used by birds in sound localization. When …

Roles Of Gap Junctions In Neuronal Networks, Joon Ha

Dissertations

This dissertation studies the roles of gap junctions in the dynamics of neuronal networks in three distinct problems. First, we study the circumstances under which a network of excitable cells coupled by gap junctions exhibits sustained activity. We investigate how network connectivity and refractory length affect the sustainment of activity in an abstract network. Second, we build a mathematical model for gap junctionally coupled cables to understand the voltage response along the cables as a function of cable diameter. For the coupled cables, as cable diameter increases, the electrotonic distance decreases, which cause the voltage to attenuate less, but the …