Mathematics Ph.D. Dissertations

Title

Finite Elements and Practical Error Analysis of Huxley and EFK Equations

Date of Award

2008

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics/Applied Math and Scientific Computation

First Advisor

Tong Sun (Advisor)

Second Advisor

So-Hsiang Chou (Committee Member)

Third Advisor

Steven Seubert (Committee Member)

Fourth Advisor

Hassan Rajaei (Committee Member)

Abstract

In this dissertation, long time error estimates are obtained using non-traditional methods for the Hodgkin-Huxley equation

ut + uxx = u(1-u)(u-a) for a ∈ (0,1/2)

ut + γ u xxxx - uxx = u-u3

Traditional methods for analyzing exact error propagation depends on the stability of the numerical method employed. Whereas, in this dissertation the analysis of the exact error propagation uses evolving attractors and only depends on the stability of the dynamical system. The use of the smoothing indicator yields a posteriori estimates on the numerical error instead of a priori estimates.

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