Faculty/School

Faculty of Science

School of Mathematical Sciences

Topic status

We're looking for students to study this topic.

Research centre

Supervisors

Professor Scott McCue
Position
Professor
Division / Faculty
Faculty of Science

Overview

Burgers' equation is a nonlinear second-order pde (partial differential equation) that acts as a model in fluid mechanics and other systems such as traffic flow.  This pde is special because we can solve it exactly using a Cole-Hopf transformation, as described in MXB325, whereas most second-order nonlinear pdes do not have exact solutions.  Therefore much is known about solutions to Burgers' equation.

This project involves revisiting Burgers' equation but allowing the spatial variable to be a complex number, instead of a real number.  This is a fascinating mathematical exercise that allows us to i) learn new mathematics about complexified pdes, and ii) learn more about nonlinear pde models more generally.

Research engagement

  • reviewing results from MXB325
  • simulating solutions of Burgers' equation
  • reading and interpreting aspects of a recent paper by supervisor Prof Scott McCue
  • visualising solutions in the complex plane
  • applying various approaches to track complex singularities as they evolve in time

Research activities

  • revisit Burgers' equation and explore solutions for initial conditions that lead to travelling waves
  • reformulate the problem in terms of a complex variable, and study these solutions in the complex plane
  • explore a number of open questions about how the complexified Burgers' equation behaves
  • all of this work will be undertaken under the supervision of Prof Scott McCue

Outcomes

  • new insight into nonlinear pde models
  • new techniques for studying nonlinear pdes
  • new understanding of Burgers' equation

Skills and experience

  • preferably a strong background in MXB322 Partial Differential Equations, MXB325 Modelling with Differential Equations 2 and MXB326 Computational Methods 2 (or equivalent third-year undergraduate courses)
  • preferably some elementary training in complex variable theory

Start date

1 November, 2024

End date

28 February, 2025

Location

Gardens Point campus (note the timing is negotiable)

Keywords

Contact

Scott McCue, scott.mccue@qut.edu.au