This work outlines the derivation and application of a class of stable and convergent finite difference methods to discretize weakly singular integrals which occur in Volterra integral equations. This derivation is motivated by classical discretizations that arise in Caputo fractional derivatives. We present a time-fractional diffusion equation as a case study to develop the finite difference scheme, where the Laplace transform is used to pose the problem equivalently as a Volterra integral equation, which is then discretized. A generalized scheme is presented to consider a much wider class of integral equations, which allows for the consideration of applications of the Fourier transform. This ultimately allows for a natural discretization of both time- and space-fractional diffusion and differential equations. Some natural physical applications are considered to fully utilize these schemes.

The novelty of these schemes is in its simplicity and efficiency when compared to classic methods of discretization, especially for Caputo fractional derivatives. Typical discretizations in the fractional derivative form over-assume regularity to discretize a full derivative term, which subsequently restricts the admissible solution space. Other considerations from discretizing the fractional derivative form include negatively impacting the rate of convergence from the remaining fractional integration term, which is recovered by the use of non-uniform mesh partitions to recover some of the order of convergence.

]]>In this dissertation, we first propose a sequential testing approach for detecting multiple change points in the Weibull accelerated failure time model, since this is sufficiently flexible to accommodate increasing, decreasing, or constant hazard rates and is also the only continuous distribution for which the accelerated failure time model can be reparametrized as a proportional hazards model. Our sequential testing procedure does not require the number of change points to be known; this information is instead inferred from the data. We conduct a simulation study to show that the method accurately detects change points and estimates the model. The numerical results along with a real data application demonstrate that our proposed method can detect change points in the hazard rate.

In survival analysis, most existing methods compare two treatment groups for the entirety of the study period. Some treatments may take a length of time to show effects in subjects. This has been called the time-lag effect in the literature, and in cases where time-lag effect is considerable, such methods may not be appropriate to detect significant differences between two groups. In the second part of this dissertation, we propose a novel non-parametric approach for estimating the point of treatment time-lag effect by using an empirical divergence measure. Theoretical properties of the estimator are studied. The results from the simulated data and real data example support our proposed method.

]]>This work considers using either an impedance or an admittance (inverse of impedance) boundary condition to allow for acoustic scattering problems to be modeled with geometries consisting of both unlined and lined surfaces. Three acoustic liner models are discussed: the *Extended Helmholtz Resonator Model*, the *Three-Parameter Impedance Model*, and the *Broadband Impedance Model*. In both the *Helmholtz* and *Three-Parameter* models, liner impedance is specified at a given frequency, whereas the *Broadband* model allows for the investigation of multiple frequencies simultaneously. The impedance and admittance boundary conditions for acoustic liners are derived for each model and coupled with a time-domain boundary integral equation. The scattering solution is obtained iteratively using a boundary element method with constant spatial and third-order temporal basis functions.

Time-domain boundary integral equations are unfortunately prone to numerical instabilities due to resonant frequencies resulting from non-trivial solutions in the interior domain. When reformulated with the Burton-Miller method, the instabilities are eliminated. Using a Burton-Miller reformulation, the stability of the boundary element method assuming a liner boundary condition is assessed using eigenvalue analysis. The stability of each liner model is discussed, and it is shown that the *Three-Parameter* and *Broadband *models are sufficient for modeling an acoustic liner on the surface of scattering bodies. The *Helmholtz* model demonstrates strict limitations for stability, whereas the *Three-Parameter* and *Broadband* models are stable for most cases.

Also included in this work is an assessment of the spatial accuracy of the time-domain boundary element method with respect to the surface element basis functions, as well as a performance study of the numerical algorithm.

]]>In the second part of this dissertation, we propose fast degenerate kernel by combining the practical truncation strategy in [25] with degenerate kernel method developed in [23]. Legendre piecewise orthogonal wavelets have been used to approximate the kernel which leads sparse structure in the linear system of the Fredholm equation and Jacobian matrix in Hammerstein equation. A fast degenerate kernel method takes place once the practical block truncation strategy implemented. Numerical examples are given throughout this dissertation.

]]>Interest in these classes of hypersingular integral equations is due to their occurrence in many physical applications. In particular, investigations into the scattering of acoustic waves by moving objects and the study of dynamic Griffith crack problems has necessitated a computationally efficient technique for solving such equations.

Fracture mechanic studies are performed using the aforementioned techniques. We focus our studies on problems addressing the Stress Intensity Factors (SIF) of a finite Griffith crack scattering an out of plane shear wave. In addition, we consider the problem of determining the SIF of two parallel Griffith cracks and two perpendicular Griffith cracks. It is shown that the method is very accurate and computationally efficient.

In acoustics, we first consider the moving wing problem. For this problem we wish to find the sound produced by the interaction of a moving wing with a known incident sound source. Although this problem is relatively simple, it is a good precursor to the two-dimensional, finite, moving duct problem.

The bulk of the research is focused on solving the two-dimensional, finite, moving duct problem. Here we look at sound propagation and radiation from a finite, two-dimensional, moving duct with a variety of inlet configurations. In particular, we conduct studies on the redirection of sound by a so-called scarf inlet design. In said designs, we are able to demonstrate the ability to redirect sound away from sensitive areas.

]]>First, a detailed explanation of the modeling process is given and the full set of rate equations is obtained. The model is then simplified and certain qualitative properties of the solution are obtained.

In the second part the equilibrium solutions are obtained and a local stability analysis is performed. The system of rate equations is solved numerically and the effects, on the solution, of varying physical parameters is discussed.

Finally, the third part addresses the oscillatory behavior of the system by "tracking" the eigenvalues of the linearized system. A comparison is made between the frequency of oscillations in the linear and nonlinear system. Pertinent physical processes--back transfer, Q-switching, and up-conversion--are then examined.

The laser system consists of thulium and holmium ions in a YAG crystal operated in a Fabrey-Perot cavity. All computer programs were written in FORTRAN and currently run on either an IBM-PC or a DEC VAX 11/750.

]]>In utilizing a NURB a designer may desire that it pass through a set of data points {x_{i}} This interpolation problem is solved by the assigning of weights to each data point. Up to now little has been known regarding the relationship between these assigned weights and the weights of the corresponding interpolating NURB. In this thesis this relationship is explored. Sufficient conditions are developed to produce interpolating NURBS which have positive weights. Applications to the problems of degree reduction and curve fairing are presented. Both theoretical and computational results are presented.

The analysis of the model is conducted in two parts. First, by formally taking an average over the spatial variable, the system of partial differential equations is reduced to a system of ordinary differential equations describing the temporal behavior of the spatially-averaged dynamic quantities. Several qualitative properties of the solutions of this system are proved and stability of the solutions under various operating conditions is investigated. The rate equations are solved numerically and the effects on the solutions of changes in the physical parameters are discussed.

The second part of this study is concerned with the qualitative and numerical analysis of the spatial and temporal model of a Ti: Sapphire ring laser. Several qualitative properties of the solution are established. The system of partial differential equations is solved numerically by integration along the characteristic lines using an implicit integration scheme developed for this problem. The computed solutions are compared to those obtained by using a stable finite difference approximation. The results of the comparison demonstrate that the implicit integration scheme is viable as well as efficient for numerically solving the system of partial differential equations and can be considered a useful analytical tool for studying the dynamics of this type of laser system.

All computer codes are written in FORTRAN and currently run on a DEC VAX 11/750.

]]>Topological considerations in structuring the grid generation mapping are discussed. In particular, this thesis examines the concept of the degree of a mapping and how it can be used to determine what requirements are necessary if a mapping is to produce a suitable grid.

The grid generation algorithm uses a mapping composed of bicubic B-splines. Boundary coefficients are chosen so that the splines produce Schoenberg's variation diminishing spline approximation to the boundary. Interior coefficients are initially chosen to give a variation diminishing approximation to the transfinite bilinear interpolant of the function mapping the boundary of the unit square onto the boundary of the grid.

The practicality of optimizing the grid by minimizing a functional involving the Jacobian of the grid generation mapping at each interior grid point and the dot product of vectors tangent to the grid lines is investigated.

Grids generated by using the algorithm are presented.

]]>Finally, an application of truncated distributions is presented. The fit of returns on common stocks to the normal, Cauchy, truncated normal, and truncated Cauchy distributions is compared via the Kolmogorov-Smirnov statistic. The results show that a truncated distribution is a better fitting model in virtually all cases.

]]>Such an interpolant is found by posing and solving a minimization problem. The solution is a piecewise cubic polynomial. We actually solve this problem indirectly by using the Peano kernel theorem to recast this problem into an equivalent minimization problem having the second derivative of the interpolant as the solution.

We are then led to solve a nonlinear system of equations. We show that with Newton's method we have an exceptionally attractive and efficient method for solving this nonlinear system of equations.

We display examples of such interpolants as well as convergence results obtained by using Newton's method. We list a FORTRAN program to compute these shape-preserving interpolants.

Next we consider the problem of computing the interpolant of minimal norm from a convex cone in a normed dual space. This is an extension of de Boor's work on minimal norm unconstrained interpolation.

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