Date of Completion

7-26-2018

Embargo Period

7-26-2018

Major Advisor

Yung-Sze Choi

Co-Major Advisor

Jeffrey Connors

Associate Advisor

Dmitriy Leykekhman

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

ABSTRACT

Algorithms are proposed to calculate traveling pulses and fronts in both directions

for the FitzHugh- Nagumo equations in one dimensional spatial domain. The rst

algorithm is based on the application of the steepest descent method to a certain

functional on some admissible sets. These sets are dierent for pulses and for fronts.

This approach is global in nature, so that an initial guess for the wave prole and the

speed can be quite dierent from the correct solution. The second algorithm is the

pseudo arc length continuation method, which solves the governing equations directly.

The two algorithms are complementary. Continuation makes the computation of a

bifurcation diagram more ecient, but it requires a good initial guess. This is supplied

by the steepest descent algorithm. Also, the two algorithms serve as an independent

check for one another.

Depending on the physical parameter values, we observe the existence of single,

multiple (stable and unstable) or no traveling pulses and fronts, within the corresponding

admissible set. At suitable parameter values, we found as many as ve

traveling wave solutions; two distinct pulses and two fronts moving to the right, and

one front moving to the left. The computed wave proles are tested numerically using

a parabolic solver and, for stable solutions, the speed and shape are maintained very

well for a large number of time steps.

Additional Files
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