Many problems in physics, biology and economics depend on the duration of time required for a diffusing particle to cross a boundary. As a result, calculating the distribution of the first passage time, and the mean first passage time is an active area of research. Exact results for the mean first passage time for diffusion on simple geometries, such as discs and spheres are well known, whereas computational methods are often used to study the first passage time on realistic geometries where closedform solutions of the governing elliptic boundary value problems are not available. Recent work at QUT has demonstrated how to take classical results for the mean exit time on discs and ellipses and then construct new results for the mean exit time on irregular shapes that are small perturbations of the classical shape. In this project we will develop and validate new approximate solutions for the mean exit time for diffusion on a range of shapes that are not small perturbations of classical shapes. We will achieve this by taking advantage of the fact that the governing equation is linear and use superposition to generate approximate solutions for shapes that are large deviations from classical shapes, such as the shape defined by overlapping two discs or two ellipses. The new approximations will be validated using stochastic simulation and numerical solutions of the governing boundary value problem using a boundary integral approximation.
Queensland University of Technology
Josh Roebuck is a fourth-year student undertaking a double degree in engineering (honours) and mathematics at QUT. He is originally from Bundaberg and moved to Brisbane to pursue this pathway as it is an area he has always had a passion for. Maths has always been something Josh has loved learning and applying, and he is aspiring to make his own contributions in the field. Since moving, he has also been working as a high-school tutor to assist his younger peers with similar interests to him, and to continue inspiring others to undertake this pathway. Outside of uni and work, Josh enjoys travelling, writing and music, and incorporates these into his daily life to stay motivated and fulfilled.