Informal Applied and Numerical Analysis Seminars:
Autumn 2012

21 Sep 2012
Revenue Management of Airport Car Parks
Dmitry Yumashev
Revenue Management (RM) was first developed by US airlines for optimising
revenue from scheduled flights through dynamic adjustment of seat prices in response to
expected and actual demand. The setting of a carpark booking system at an airport is a
relatively new area of RM, and as such presents new and challenging problems.
Our analysis of the data from the Liverpool Airport longstay carpark reveals
strong behavioural patterns, which illustrate customers' preferences in terms of booking
in advance and staying for a certain number of days. These patterns are seasonal and tend
to repeat yearly with a very good accuracy, allowing us to make a reasonably good occupancy
forecast needed for RM.
We then show how RM techniques may be applied to the carpark problem in a very simplified
form, when customers have a choice between a more expensive premium carpark of a limited
capacity (which offers greater convenience) and a basic carpark with unlimited space (which
is cheaper but is less convenient in terms of the proximity to the terminal, etc). The model
allows to derive the optimal rule at which to accept customers into the premium car park
depending on their booking preferences (required duration of stay) in order to maximise the
revenue.

5 Oct 2012
Why Tennis Balls Are Hairy and Footballs Are Smooth
David Abrahams
A talk about aerodynamics in ball sports.

12 Oct 2012
Bringing Maths Together
Katie Steckles
A discussion about the problems presented at the MRSC.

19 Oct 2012
Image Reconstruction for Xray Computed Tomography
Will Thompson
Xray computed tomography (CT) is a technique for imaging slices through an object by measuring xray projections through it from a range of angles. The technique has been in widespread use in medical imaging since the mid 1970s, and is also now used in other diverse fields such as security screening, nondestructive testing and materials research.
The problem of image reconstruction, where we seek to recover the unknown structure of an object given only knowledge of a set of its xray projections, is an inverse problem, and is shown to be equivalent to finding a function f(x) given a set of its line integrals. In the two dimensional case, two alternative solution methods to this problem are presented.

26 Oct 2012
A Model of Earthquake Faulting With No Predictive Capabilities Whatsoever
Matthew Crooks
Plasticity is a term used to describe deformation that is permanent and remains even after all forces are removed. Granular materials behave in this way as can be seen in foot prints left in the sand on a beach. As such the constitutive equations for granular materials are a form of plasticity. These equations are often (but not always) hyperbolic and the implications of this are discussed within the context of the method of solution. The solution to problems with geometries not dissimilar to that of earthquakes are presented along with limitations and the route of future work within my PhD.

2 Nov 2012
Waves In Quasicrystals
Ruth Voisey
Understanding and predicting wave propagation and scattering by nonperiodic media is much more difficult than that for periodic composites, but is of great importance in many applications. No structure is exactly periodic, and so it is crucial to gain a better, more accurate understanding of nearly periodic, quasiperiodic and random structures. In terms of nondestructive testing one must know how the waves behave as they travel through nonperiodic structures in order to be aware of defects in the structure.
In this talk I will discuss part of my PhD research with quasiperiodic structures, where by quasiperiodic we mean nonperiodic in the sense that the structure is nonrepeating, but also nonrandom in the sense that the distribution is deterministic.
I will introduce quasicrystals, the Penrose tiling and how to model acoustic wave scattering through materials with similar properties.

9 Nov 2012
C1finite element implementation for the 4thorder problem with an effect of boundary approximation
Wassamon Phusakulkajorn
Finite element method is a numerical method to find an approximate solution to a partial differential equation. A shape function or an interpolation function plays an important role in accurately approximating the solution. To deal with the fourth order problem, the shape function has to be in the Hermite family; that is C1continuous.
In this talk, I will give an example of solving the fourth order problem with C1finite element. Two kinds of boundary domains will be considered: straight and curved boundary domains. You will see later on that linear interpolation of the domain geometry will dramatically reduce the accuracy and convergence rate of the solutions.

23 Nov 2012
Some results for stability of relative equilibria of point vortices on the hyperboloid.
Citlalitl Nava Gaxiola
Relative equilibria are trajectories that are invariant under the dynamics. For example, when we talk of a three body problem this would correspond simply to motions where the shape of the body ('triangle') does not change.
I would show what are the trajectories that we can expect for relative equilibria for this dynamical system, what are the possible classifications for two and three point vortices and, some results of its stability results.

30 Nov 2012
Bill Lionheart

7 Dec 2012
Acoustic wave scattering from macroscopic inclusions with arbitrary crosssection
Vicki Andrew
For wave scattering problems in which the wave number is of similar order to the characteristic length scale of the scatterer, homogenization techniques are generally redundant. Instead, we express the governing equation as a boundary integral equation of the second kind, where the unknown function appears both inside and outside the integral sign. A particularly useful numerical technique that can be used to solve integral equations of this kind is the boundary element method. This involves discretizing the boundary into segments and making some assumption about the variation in the shape of the element, and in the boundary function. We discuss different types of element; constant, linear and quadratic, and compare the convergence of each type.
We also briefly discuss how the method can be extended to solve for the reflection and transmission coefficients from a periodic array of cylinders with arbitrary cross section, and how the method can be extended to solve for 3D nonspherical scatterers.