The vast majority of theses in this collection are open access and freely available. There are a small number of theses that have access restricted to the WWU campus. For off-campus access to a thesis labeled "Campus Only Access," please log in here with your WWU universal ID, or talk to your librarian about requesting the restricted thesis through interlibrary loan.

Date of Award

Spring 2025

Document Type

Masters Thesis

Department or Program Affiliation

Mathematics

Degree Name

Master of Science (MS)

Department

Mathematics

First Advisor

McDowall, Stephen R.

Second Advisor

Barnard, Richard Charles

Third Advisor

Glimm, Tilmann

Abstract

In this thesis I extend the work of McDowall [McD1] (which extends the work of [CS]), where in his paper, he showed the solvability of the stationary linear transport equation on a compact Riemannian manifold with boundary. We will solve this equation in the more general setting of a compact Finsler manifold with boundary. We define what a Finsler manifold is and characterize its geodesics as solutions to Euler-Lagrange equations, as well as explore the symplectic structure of the tangent bundle to a Finsler manifold. The boundary value problem is then posed, and an appropriate function space is defined so that boundary conditions are well posed. With this in hand, a trace theorem is proven and estimates established. Finally, we construct an operator in order to reformulate our equation as an operator equation. From previous estimates, we are then able to show this operator has operator norm less than one so that a Neumann series for an inverse exists and a solution is written down explicitly.

Type

Text

Keywords

stationary transport, Finsler, manifold, optical tomography

Publisher

Western Washington University

OCLC Number

1523220523

Subject – LCSH

Manifolds (Mathematics); Finsler spaces; Optical tomography; Boundary value problems

Format

application/pdf

Genre/Form

masters theses

Language

English

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Download

Included in

Mathematics Commons

Share

COinS