Physical Sciences and Mathematics Commons^™

Diagonalization Of 1-D Schrodinger Operators With Piecewise Constant Potentials, Sarah Wright

Master's Theses

In today's world our lives are very layered. My research is meant to adapt current inefficient numerical methods to more accurately model the complex situations we encounter. This project focuses on a specific equation that is used to model sound speed in the ocean. As depth increases, the sound speed changes. This means the variable related to the sound speed is not constant. We will modify this variable so that it is piecewise constant. The specific operator in this equation also makes current time-stepping methods not practical. The method used here will apply an eigenfunction expansion technique used in previous …

Methods In Modeling Wildlife Disease From Model Selection To Parameterization With Multi-Scale Data, Ian Mcgahan

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The effects of emerging wildlife diseases are global and profound, resulting in loss of human life, economic and agricultural impacts, declines in wildlife populations, and ecological disturbance. The spread of wildlife diseases can be viewed as the result of two simultaneous processes: spatial spread of wildlife populations and disease spread through a population. For many diseases these processes happen at different timescales, which is reflected in available data. These data come in two flavors: high-frequency, high-resolution telemetry data (e.g. GPS collar) and low-frequency, low-resolution presence-absence disease data. The multi-scale nature of these data makes analysis of such systems challenging. Mathematical …

Hadamard Well-Posedness For Two Nonlinear Structure Acoustic Models, Andrew Becklin

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation focuses on the Hadamard well-posedness of two nonlinear structure acoustic models, each consisting of a semilinear wave equation defined on a smooth bounded domain $\Omega\subset\mathbb{R}^3$ strongly coupled with a Berger plate equation acting only on a flat portion of the boundary of $\Omega$. In each case, the PDE is of the following form: \begin{align*} \begin{cases} u_{tt}-\Delta u +g_1(u_t)=f(u) &\text{ in } \Omega \times (0,T),\\[1mm] w_{tt}+\Delta^2w+g_2(w_t)+u_t|_{\Gamma}=h(w)&\text{ in }\Gamma\times(0,T),\\[1mm] u=0&\text{ on }\Gamma_0\times(0,T),\\[1mm] \partial_\nu u=w_t&\text{ on }\Gamma\times(0,T),\\[1mm] w=\partial_{\nu_\Gamma}w=0&\text{ on }\partial\Gamma\times(0,T),\\[1mm] (u(0),u_t(0))=(u_0,u_1),\hspace{5mm}(w(0),w_t(0))=(w_0,w_1), \end{cases} \end{align*} where the initial data reside in the finite energy space, i.e., $$(u_0, u_1)\in H^1_{\Gamma_0}(\Omega) \times L^2(\Omega) \, \text{ …

Mathematical Modeling Of Microemulsification Processes, Numerical Simulations And Applications To Drug Delivery, Ogochukwu Nneka Ifeacho

Open Access Theses & Dissertations

Microemulsion systems are a great pharmaceutical tool for the delivery of formulations containing multiple hydrophilic and hydrophobic ingredients of varying physicochemical properties. These systems are gaining popularity because of its long shelf life, improved drug solubilisation capacity, easy preparation and improvement of bioavailability. Despite the advantages associated with the use of microemulsion systems in pharmaceutical industries, the major challenge impeding their use has been and continues to be the lack of understanding of these systems.

Microemulsions can be mathematically modeled by an initial boundary value problem involving a sixth order nonlinear time dependent equation. In this Thesis, we present a …

The Direct Scattering Map For The Intermediate Long Wave Equation, Joel Klipfel

Theses and Dissertations--Mathematics

In the early 1980's, Kodama, Ablowitz and Satsuma, together with Santini, Ablowitz and Fokas, developed the formal inverse scattering theory of the Intermediate Long Wave (ILW) equation and explored its connections with the Benjamin-Ono (BO) and KdV equations. The ILW equation\begin{align*} u_t + \frac{1}{\delta} u_x + 2 u u_x + Tu_{xx} = 0, \end{align*} models the behavior of long internal gravitational waves in stratified fluids of depth $0< \delta < \infty$, where $T$ is a singular operator which depends on the depth $\delta$. In the limit $\delta \to 0$, the ILW reduces to the Korteweg de Vries (KdV) equation, and in the limit $\delta \to \infty$, the ILW (at least formally) reduces to the Benjamin-Ono (BO) equation.

While the KdV equation is very well understood, a rigorous analysis of inverse scattering for the ILW equation remains to be accomplished. There is currently no rigorous proof that the Inverse Scattering …