Date of Award

Spring 5-2017

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Committee Chair

Dr. James Lambers

Committee Chair Department

Mathematics

Committee Member 2

Dr. Ching-Shyang Chen

Committee Member 2 Department

Mathematics

Committee Member 3

Dr. Haiyan Tian

Committee Member 3 Department

Mathematics

Committee Member 4

Dr. Huiqing Zhu

Committee Member 4 Department

Mathematics

Abstract

As a result of stiff systems of ODEs, difficulties arise when using time stepping methods for PDEs. Krylov subspace spectral (KSS) methods get around the difficulties caused by stiffness by computing each component of the solution independently. In this dissertation, we extend the KSS method to a circular domain using polar coordinates. In addition to using these coordinates, we will approximate the solution using Legendre polynomials instead of Fourier basis functions. We will also compare KSS methods on a time-independent PDE to other iterative methods. Then we will shift our focus to three families of orthogonal polynomials on the interval (−1,1), with weight function ω(x) ≡ 1. These families of polynomials satisfy the boundary conditions (1) p(1) = 0, (2) p(−1) = p(1) = 0, and (3) p(1) = p′(1) = 0. The first two boundary conditions two arise naturally from PDEs defined on a disk with Dirichlet boundary conditions and the requirement of regularity in Cartesian coordinates. The third boundary condition includes both Dirichlet and Neumann boundary conditions for a higher-order PDE. The families of orthogonal polynomials are obtained by orthogo- nalizing short linear combinations of Legendre polynomials that satisfy the same boundary conditions. Then, the three-term recurrence relations are derived. Finally, it is shown that from these recurrence relations, one can efficiently compute the corresponding recurrences for generalized Jacobi polynomials (GJPs) that satisfy the same boundary conditions.

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