Mathematical models describing diffusive transport of mass and energy are essential to our understanding of many problems in engineering, physics, biology and chemistry.
Classical analysis of mathematical models that describe diffusive transport focus on diffusion in simple geometries, such as lines, discs and spheres composed of homogeneous materials. In contrast, specific applications of diffusive transport theory in more complicated geometries are often explored computationally. This can include geometries with heterogeneous materials.
While computational approaches are necessary in certain circumstances, analytical insight is often preferred. This is because it can provide simple, easy-to-evaluate, closed-form mathematical expressions that explicitly highlight key relationships.
In this project we'll develop new techniques that give rise to simple mathematical expressions for the first passage time for diffusion in complicated geometries with heterogeneous material properties.
In this project you will:
- develop and implement stochastic random walk models of diffusion in
- classical geometries (e.g. sphere)
- more general non-radial geometries (e.g. oblate and prolate spheroids)
- derive and solve new partial differential equation models describing the first passage time in the non-radial geometries using perturbation methods
- compare stochastic estimates of the first passage time with the solutions of the new partial differential equations.
This project will result in new:
- algorithms and software that can numerically evaluate the first passage time in an arbitrary geometry with spatially varying material properties
- partial differential equation models that describe the first passage time in non-radial geometries, for both homogeneous and heterogeneous materials
- symbolic software (e.g. Maple) that can solve the systems of boundary value problems that construct the perturbation solution.
Skills and experience
This project requires good programming skills (e.g. MATLAB, Python).
Some experience in symbolic computing (e.g. Maple) would be an advantage. We will also highly regard good results in undergraduate subjects related to ordinary and partial differential equations.
You may be eligible to apply for a research scholarship.
Contact the supervisor for more information.