# Category Archives: Applied Mathematics

## Wave Trapping Around Islands

Islands are subject to tides, seismic waves as well as wind generated waves. If any of these are trapped, that is the energy of the wave goes continuously around the island without escaping then there are practical consequences in terms of flood prediction as well as the detection of signals of say earthquakes or the

## Natural Ventilation

The study of how to ventilate a room without recourse to air conditioning has practical application to theatres and libraries where all but the quietest noise has to be avoided.   This project examines how classical fluid dynamics aids this practical application. You could be using complex analysis through mapping and solving partial differential equations in

## Wind Driven Circulation

In the ocean the main gyres responsible for how the sea circulates are generated by the climatic winds. In this project you will derive the equations that govern this process from first principles. These equations will then be solved in order to understand why there are strong currents on the western side of every world

## The Numerical Solution of the p-Laplace Equation in a Geometry with Sharp Corners

Certain types of fluids do not respond in a linear way to external forces – they are called non-Newtonian fluids. A simple model of a non-Newtonian fluid which is widely used in industrial applications (e.g. food processing) is the so-called power-law model. In very slow flows, the governing equation turns out to be the p-Laplace

## Perturbation Methods

Perturbation methods are used to investigate solutions to mathematical problems in which exact solutions can’t be found but in which there is a parameter whose size (usually very small) allows approximate solutions to be found. In this project the aim would be to look at a range of perturbation methods and the types of problems

## Non-linear Waves

This project concerns deriving equations that govern the propagation of waves on water, then solving them. These equations have two well-known solutions, the solitary wave (or soliton) and cnoidal waves. Some of the mathematics is quite involved but the satisfaction comes in being able to describe known observable wave phenomena. There is some opportunity for

## Waves

Wave motion is found everywhere: sound waves, water waves, electromagnetic waves, traffic waves and Mexican waves are just some examples. The mathematical description of wave motion can be fairly straightforward, as with the oscillations of a guitar string for example, but as more physical effects are allowed for the theory becomes more complicated. In this

## Chebychev Polynomials

In this project the aim is to study the mathematical properties of Chebychev polynomials and their role in numerical approximation. The concept of ‘good’ and ‘best’ numerical approximations and the Remes algorithm will be investigated and, if time permits, further applications of Chebychev polynomials could be considered. This project is mainly analytical but would involve

## Particle methods for fluid dynamics

Particle-based numerical methods have been used extensively by Hollywood in modelling dramatic events such as floods or sinking ships. This project would investigate smoothed particle hydrodynamics (SPH) models or, possibly, discrete vortex models (DVM) in modelling relatively simple flows. More SPH info can be found here. Supervisor: Dr David Graham

## Numerical methods for PDE’s

Partial differential equations describe how the three/four dimensional world works. Solving such equations is the key to understanding the behaviour of many physical, chemical, biological or financial systems. A large range of numerical methods is available to solve such problems (for example finite difference, finite element, finite volume, boundary element or pseudo-spectral methods to name

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