Date of Award

8-2011

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Vasilios Alexiades

Committee Members

Ohannes Karakashian, Xiaobing Feng, Jack Buchanan

Abstract

Electrical propagation in excitable tissue, such as nerve fibers and heart muscle, is described by a nonlinear diffusion-reaction parabolic partial differential equation for the transmembrane voltage $V(x,t)$, known as the cable equation. This equation involves a highly nonlinear source term, representing the total ionic current across the membrane, governed by a Hodgkin-Huxley type ionic model, and requires the solution of a system of ordinary differential equations. Thus, the model consists of a PDE (in 1-, 2- or 3-dimensions) coupled to a system of ODEs, and it is very expensive to solve, especially in 2 and 3 dimensions.

In order to solve this equation numerically, we develop an algorithm, extended from the Parareal Algorithm, to efficiently incorporate space-parallelized solvers into the framework of the Parareal algorithm, to achieve time-and-space parallelization. Numerical results and comparison of the performance of several serial, space-parallelized and time-and-space-parallelized time-stepping numerical schemes in one-dimension and in two-dimensions are also presented.

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