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.
| Identifer | oai:union.ndltd.org:ucf.edu/oai:stars.library.ucf.edu:etd-2227 |
| Date | 01 January 2015 |
| Creators | Lydon, Elizabeth |
| Publisher | STARS |
| Source Sets | University of Central Florida |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Electronic Theses and Dissertations |
Page generated in 0.0027 seconds