## Theses

#### Abstract

The study of the tear film of the eye is important for understanding the causes of dye eye syndrome, a disease causing damage to the ocular surface resulting in discomfort for many people. A popular mathematical model describing the evolution of the tear film thickness over time is a fourth-order nonlinear partial differential equation (PDE). This model has been formulated two different ways to facilitate numerical approximations. The first way is just one PDE including a fourth derivative along with the boundary and initial conditions that is solved for the tear film thickness. The second is a system of coupled second-order PDEs, with the analogous boundary and initial conditions, describing the tear film thickness and pressure. Typically, the second formulation is used when computing the solution as the first poses challenges. These challenges are generally attributed to approximating a scaled fourth derivative and specifying a third derivative for one of the boundary conditions.

In this thesis, I explore the computational differences between the two formulations of the tear film model in one spatial dimension. Both formulations are approximated by implementing a method of lines approach where spatial derivatives are approximated with second-order centered finite differences, and then the system of differential equations are integrated forward in time using a backward Euler method. The stability of each numerical method is proven analytically with von Neumann analysis, and the usefulness of each formulation is characterized by studying the condition number of each discretization for different parameters and boundary conditions. In particular, I examine the implications of the relationship between the lid function and boundary conditions. Lastly, a comparison is made with results from the two-dimensional equivalent model on a realistic blinking eye-shaped domain.

#### Library of Congress Subject Headings

Tears--Mathematical models; Fluid dynamics--Mathematics; Eye--Mathematical models

8-6-2018

Thesis

Graduate

#### Degree Name

Applied and Computational Mathematics (MS)

#### Department, Program, or Center

School of Mathematical Sciences (COS)

Kara L. Maki

#### Advisor/Committee Member

Nathaniel S. Barlow

#### Advisor/Committee Member

Matthew J. Hoffman

#### Campus

RIT – Main Campus

ACMTH-MS

COinS