Applied Mathematics Commons^™

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 …