Applied partial differential equations 2

I am no longer teaching this unit! Please check the official online unit description.

I was teaching the second half of this unit (approximately weeks 18-23). For the first half of the unit, and some other resources such as maple animations, please see Diki Porter's page. Past exams are on Bb, in the "organisation" Maths Teaching.

Notes by chapter: 4 5 6 7

While not directly intended for this unit, you may find some of my old material for Methods 3 and (the discontinued) Applied Mathematics 2 helpful.

Some relevant research (well beyond the scope of this course):

The diffusion equation is useful in dynamical systems, for example situations where one variable is so erratic it is effectively random, and which controls the increments of another variable. Over long time scales the probability distribution (with respect to initial conditions) of the second variable can be described by a diffusion equation. Enforcing it to be positive corresponds to a hard wall (Neumann) boundary at zero, while allowing escape/transient behaviour leads to open (Dirichlet) boundary conditions. Deviations from the exactly known solutions to the diffusion equation also shed light on details of the dynamics. This approach is the subject of a recent preprint (ie a paper that has not yet been published in a peer-reviewed journal).

If a box is equally split into two regions in which a particle can diffuse at different rates, what fraction of time does it spend in each region? This apparently simple question has a rather subtle answer and is the subject of a recent preprint by Tupper and Yang.

The methods we used to study the modes of vibration of a drum can be applied to circular microdisks, which are used to construct lasers of a few microns in size. Under some approximations, the electromagnetic equations reduce to eigenvalues problems for the 2D Laplacian, matched at the boundary by conditions related to the refractive index of the material and the type of mode (transverse magnetic or electric). The solution inside is given in terms of J Bessel functions as the solution must be regular at the centre. The solution outside is given by Hankel functions, which are linear combinations of J and Y Bessel functions appropriate for outgoing waves at infinity. See this paper which was published in EPL 87 34003 (2009).