Physical Sciences and Mathematics Commons^™

Investigating The Feasibility And Stability For Modeling Acoustic Wave Scattering Using A Time-Domain Boundary Integral Equation With Impedance Boundary Condition, Michelle E. Rodio

Mathematics & Statistics Theses & Dissertations

Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet and environmentally friendly aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for aircraft noise reduction and are characterized by a frequency-dependent impedance. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners

This work considers using either an impedance or an admittance (inverse …

An Inverse Eigenvalue Problem For The Schrödinger Equation On The Unit Ball Of R3, Maryam Ali Al Ghafli

Theses and Dissertations--Mathematics

The inverse eigenvalue problem for a given operator is to determine the coefficients by using knowledge of its eigenfunctions and eigenvalues. These are determined by the behavior of the solutions on the domain boundaries. In our problem, the Schrödinger operator acting on functions defined on the unit ball of $\mathbb{R}^3$ has a radial potential taken from $L^2_{\mathbb{R}}[0,1].$ Hence the set of the eigenvalues of this problem is the union of the eigenvalues of infinitely many Sturm-Liouville operators on $[0,1]$ with the Dirichlet boundary conditions. Each Sturm-Liouville operator corresponds to an angular momentum $l =0,1,2....$. In this research we focus on …

Theoretical Analysis Of Nonlinear Differential Equations, Emily Jean Weymier

Electronic Theses and Dissertations

Nonlinear differential equations arise as mathematical models of various phenomena. Here, various methods of solving and approximating linear and nonlinear differential equations are examined. Since analytical solutions to nonlinear differential equations are rare and difficult to determine, approximation methods have been developed. Initial and boundary value problems will be discussed. Several linear and nonlinear techniques to approximate or solve the linear or nonlinear problems are demonstrated. Regular and singular perturbation theory and Magnus expansions are our particular focus. Each section offers several examples to show how each technique is implemented along with the use of visuals to demonstrate the accuracy, …

Asymptotic Behavior Of Waves In A Nonuniform Medium, Nezam Iraniparast, Lan Nguyen, Mikhail Khenner

Applications and Applied Mathematics: An International Journal (AAM)

An incoming wave on an infinite string, that has uniform density except for one or two jump discontinuities, splits into transmitted and reflected waves. These waves can explicitly be described in terms of the incoming wave with changes in the amplitude and speed. But when a string or membrane has continuous inhomogeneity in a finite region the waves can only be approximated or described asymptotically. Here, we study the cases of monochromatic waves along a nonuniform density string and plane waves along a membrane with nonuniform density. In both cases the speed of the physical system is assumed to tend …

The Kerzman–Stein Operator For Piecewise Continuously Differentiable Regions, Michael Bolt, Andrew Raich

University Faculty Publications and Creative Works

The Kerzman–Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on a rectifiable curve. If the curve is continuously differentiable, the Kerzman–Stein operator is compact on the Hilbert space of square integrable functions; when there is a corner, the operator is noncompact. Here, we give a complete description of the spectrum for a finite symmetric wedge and we show how this reveals the essential spectrum for curves that are piecewise continuously differentiable. We also give an explicit construction for a smooth curve whose Kerzman–Stein operator has large norm.

Relationships Between Elements Of Leslie Matrices And Future Growth Of The Population, Lorisha Lynn Riley

Honors Program Projects

Leslie matrices have been used for years to model and predict the growth of animal populations. Recently, general rules have been applied that can relatively easily determine whether an animal population will grow or decline. My mentor, Dr. Justin Brown and I examine, more specifically, whether there are relationships between certain elements of a population and the dominant eigenvalue, which determines growth. Not only do we consider the general 3x3 Leslie matrix, but also we looked into modified versions for incomplete data and migration models of Leslie matrices. We successfully found several connections within these cases; however, there is much …

A Restarted Homotopy Method For The Nonsymmetric Eigenvalue Problem, Brandon Hutchison

Graduate Theses and Dissertations

The eigenvalues and eigenvectors of a Hessenberg matrix, H, are computed with a combination of homotopy increments and the Arnoldi method. Given a set, Ω, of approximate eigenvalues of H, there exists a unique vector f = f(H,Ω) in Rn where λ(H-e_{1ft)=Ω. A diagonalization of the homotopy H(t)=H−(1−t)e1ft at $t=0$ provides a prediction of the eigenvalues of H(t) at later times. These predictions define a new Ω that defines a new homotopy. The correction for each eigenvalue has an O(t2) error estimate, enabling variable step size and efficient convergence tests. Computations are done primarily in real arithmetic, and …}

Developing An Improved Shift-And-Invert Arnoldi Method, H. Saberi Najafi, M. Shams Solary

Applications and Applied Mathematics: An International Journal (AAM)

An algorithm has been developed for finding a number of eigenvalues close to a given shift and in interval [ Lb,Ub ] of a large unsymmetric matrix pair. The algorithm is based on the shift-andinvert Arnoldi with a block matrix method. The block matrix method is simple and it uses for obtaining the inverse matrix. This algorithm also accelerates the shift-and-invert Arnoldi Algorithm by selecting a suitable shift. We call this algorithm Block Shift-and-Invert or BSI. Numerical examples are presented and a comparison has been shown with the results obtained by Sptarn Algorithm in Matlab. The results show that the …

A Lower Estimate For The Norm Of The Kerzman-Stein Operator, Michael Bolt

University Faculty Publications and Creative Works

We establish an elementary lower estimate for the norm of the Kerzman-Stein operator for a smooth, bounded domain. The estimate involves the boundary length and logarithmic capacity. The estimate is tested on model domains for which the norm is known explicitly. It is shown that the estimate is sharp for an annulus and a strip, and is asymptotically sharp for an ellipse and a wedge. © 2007 Rocky Mountain Mathematics Consortium.