Date of Award
Summer 2026
Embargo Period
8-14-2026
Document Type
Master's Thesis
Degree Name
Master of Arts (MA)
College/School/Department
Department of Mathematics and Statistics
Program
Mathematics
First Advisor
Ivana Alexandrova
Committee Members
Ivana Alexandrova, Joshua Isralowitz, Rongwei Yang
Keywords
matrix diffusion PDE, anisotropic diffusion, ellipsoids, principal semi-axis geometry, eigenstructure
Subject Categories
Analysis | Other Applied Mathematics | Partial Differential Equations
Abstract
We analyze the underlying geometry of the solution to an IVP for a matrix diffusion PDE. We first derive the fundamental solution to the PDE. We then determine the unique solution to the IVP. From there, we begin analyzing its underlying geometry. We first observe that the geometry exhibits an ellipsoidal nature. Furthermore, we observe that it is described by the principal semi-axis geometry of the ellipsoids associated with the solution. This conclusion follows from applying the Principal Axis Theorem to the associated ellipsoids to establish their principal semi-axis geometry, as governed by the eigenstructure of the matrix. This thesis thus draws from different areas - PDEs, real analysis, applied linear algebra, and eigenstructure-driven geometry - to provide a thorough analysis of the geometry of the solution.
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
Fox, Brandon M., "The Ellipsoidal Principal Semi-Axis Geometry of the Solution to an IVP for a Matrix Diffusion PDE" (2026). Electronic Theses & Dissertations (2024 - present). 543.
https://scholarsarchive.library.albany.edu/etd/543
Included in
Analysis Commons, Other Applied Mathematics Commons, Partial Differential Equations Commons