Date of Completion

11-30-2018

Embargo Period

11-30-2018

Keywords

FitzHugh-Nagumo equations, traveling wave, standing pulse

Major Advisor

Yung Choi

Co-Major Advisor

Jeffrey Connors

Associate Advisor

Dmitriy Leykekhman

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

Algorithms are constructed to calculate standing pulse and traveling wave solu- tions for the FitzHugh-Nagumo equations in two dimensions. The algorithms are based on the application of a steepest descent method to some functionals. These algorithms are global in nature, in the sense that it does not require a good initial guess to guarantee convergence. Their numerical implementation involves construc- tion of asymptotic boundary conditions on truncated domains; asymptotic boundary conditions make the computation less expensive.

Our focus is on two special types of solutions for the FitzHugh-Nagumo equations: radially symmetric standing pulses in the whole space with Ω = R2 and traveling wave solutions in a strip Ω = R × [−L, L] for some L > 0. Using these algorithms we find multiple traveling pulse and front solutions for the same physical parameters. As an independent check, we test the traveling wave solutions from the steepest descent method using a parabolic solver, which reveals the stability of the solutions at the same time.

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