Date of Award

Summer 6-27-2023

Degree Type

Masters Thesis

Degree Name

Master of Science (MS)

School

Mathematics and Natural Sciences

Committee Chair

Dr. James Lambers

Committee Chair School

Mathematics and Natural Sciences

Committee Member 2

Dr. Bernd Schroeder

Committee Member 2 School

Mathematics and Natural Sciences

Committee Member 3

Dr. Huiqing Zhu

Committee Member 3 School

Mathematics and Natural Sciences

Abstract

In this thesis we shall perform the comparisons of a Krylov Subspace Spectral method with Forward Euler, Backward Euler and Crank-Nicolson to solve the Cable Equation. The Cable Equation measures action potentials in axons in a mammalian brain treated as an ideal cable in the first part of the study. We shall subject this problem to the further assumption of a non-ideal cable. Assume a non-uniform cross section area along the longitudinal axis. At the present time, the effects of torsion, curvature and material capacitance are ignored. There is particular interest to generalize the application of the PDEs including and other than Cable Equation to the study of Neurodegenerative diseases like multiple sclerosis, Alzheimer’s, Parkinsons etc. The ultimate goal would be to be able to study a broad application of numerical methods to understand features of the human brain and its functions without involving medically invasive procedures. ii

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