Propagation Failure in Discrete Inhomogeneous Medium Using a Caricature of the Cubic

Spatially discrete Nagumo equations have widespread physical applications, including modeling electrical impulses traveling through a demyelinated axon, an environment typical in multiple scle- rosis. We construct steady-state, single front solutions by employing a piecewise linear reaction term. Using a combination of Jacobi-Operator theory and the Sherman-Morrison formula we de- rive exact solutions in the cases of homogeneous and inhomogeneous diffusion. Solutions exist only under certain conditions outlined in their construction. The range of parameter values that satisfy these conditions constitutes the interval of propagation failure, determining under what circumstances a front becomes pinned in the media. Our exact solutions represent a very specific solution to the spatially discrete Nagumo equation. For example, we only consider inhomogeneous media with one defect present. We created an original script in MATLAB which algorithmically solves more general cases of the equation, including the case for multiple defects. The algorithmic solutions are then compared to known exact solutions to determine their validity.

Author dc:creator

• Lydon, Elizabeth

Contributors dc:contributor

• Moore, Brian

• Reaction diffusion equation; propagation failure; discrete nagumo
• Mathematics
• Dissertations, Academic -- Sciences; Sciences -- Dissertations, Academic

Language dc:language

English

Identifier

CFE0005831

OAI identifier oai:identifier

oai:stars.library.ucf.edu:etd-2227