Suggested Project Topics

Here are a few ideas for project topics for final year, higher diploma, and Masters students. The only common thread is that they combine some mathematics with computation.

1. Solving fractional order differential equations with Chebfun

Cauchy formula for repeated integration is \[ f^{(-n)}(x) = \frac{1}{(n-1)!}\int_a^x (x-t)^{-1}f(t) dt. \] where \[ f^{(-n)}(x) = \int_a^x \int_a^{\sigma_1} \cdots \int_a^{\sigma_{n-1}} f(\sigma_n) d\sigma_n \dots d\sigma_2 d \sigma_1. \] Remarkably, the formula holds for any real $n$, leading to the concepts of fractional integrals and derivatives. There is a lot of research interest in the numerical solution of fractional order differential equations. In this project we'll investigate how the Matlab toolbox, Chebfun, can be used to solve these problems.

A student who undertakes this project would investigate the mathematics that underpins the system, while developing new programmes and examples to solve problems with chebfuns.

An excellent reference for this project is Approximation Theory and Approximation Practice by Nick Trefethen, available from the library at 511.4 TRE.

2. Solving the Navier-Stokes equations with FEniCS

The FEniCS Project is a collection of free software for the (somewhat) automated solution of differential equations by finite element methods. Through a relatively simple Python interface, users and developers can quickly describe complex geometries and differential equations, and compute accurate approximations.

This project would focus on modelling fluid flow on realistic domains (like Lough Corrib, or Galway Bay).

3. Other topics