Consider the release into the air of a contaminated particle that could be harmful to nearby wildlife and agriculture. To understand the effect of this particle on the environment, it becomes...Show moreConsider the release into the air of a contaminated particle that could be harmful to nearby wildlife and agriculture. To understand the effect of this particle on the environment, it becomes important to find out whether it hits the ground and how long it takes to do so. Mathematically speaking, we are searching for the the exit probability and the expected exit time. Observations show that the particle doesn’t just move along with the wind. It also performs some random motion which can cause it to move against the wind. This has to be taken into account when choosing a model to describe the movement of the particle. In this thesis we look at the situation described above and also at other practical examples. Our main focus will be on finding the exit probability and the expected exit time. We start with an introduction to stochastic differential equations (SDE’s), because the models we consider will be in that form. Connected with each SDE is a partial differential equation called the Fokker-Planck (FP) equation. This FP-equation will then lead us to our first means to obtain the expected exit time. Next, we introduce the numerical simulation of SDE’s and this will provide us with a second way to obtain the expected exit time. Finally, four examples are chosen from non-mathematical research areas. Both approaches to finding the expected exit time will be applied and the results will be compared. The differential equations that arise in the second approach are approximated using singular perturbations. All results can be verified with the MATLAB code from the appendix. The examples we look at are a population model from biology, a membrane voltage model from neurology, a particle movement model from physics and a model for groundwater pollution from hydrology.Show less
We present a variant of Newton’s Method for computing travelling wave solutions to bistable lattice differential equations. We prove that the method converges to a solution, obtain existence and...Show moreWe present a variant of Newton’s Method for computing travelling wave solutions to bistable lattice differential equations. We prove that the method converges to a solution, obtain existence and uniqueness of solutions to such equations with a small second order term and study the limiting behaviour of such solutions as this second order term tends to zero. The robustness of the algorithm will be discussed using numerical examples. These results will also be used to illustrate phenomena like propagation failure, which are encountered when studying lattice differential equations. We finish by outlining some properties of higher dimensional systems, including a period two bifurcation.Show less
