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Advisory Committee Chair

Rudi Weikard

Document Type

Dissertation

Date of Award

2018

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

In this dissertation we cover two main topics in Sturm-Liouville theory, namely comparison theory and spectral theory. First we give a brief introduction to the classical Sturm-Liouville equations, the development of comparison theorems, and the classical first order system of differential equations. Second we provide a flexible comparison theorem for two Sturm-Liouville equations with distributional coefficients, and we provide some applications of our result. Finally, we investigate spectral theory of first order system of differential equations whose coefficients are distributions of order zero. We define maximal and minimal relations and classify boundary conditions that produce self-adjoint restrictions of the maximal relation. Corresponding to each such restriction we show the existence of Green's function and generalized Fourier transform. We take a closer look at regular problems, and the case when we have a 2 by 2 system.

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