Exact and approximate solutions of diffusion on evolving domains

Overview

Classical applications of mathematical analysis involve solving partial differential equation models on fixed domains, e.g. 0 < x < L.  Applications in biology, however, involve studying diffusive transport on rapidly evolving domains, e.g. 0 < x < L(t), where L(t) represents the length of the evolving tissue. While many problems have been addressed for the case where L(t) increases, less attention has been paid to cases where we consider diffusion on an oscillating domain.

In this project we will construct exact solutions of diffusive transport problems on oscillatory domains (lines, discs, spheres) and then analyse certain limits using asymptotic methods to provide additional insight into the underlying physical phenomena.

Research activities

• Using transforms to construct exact solutions of PDE models of diffusion, or systems of PDE models of multicomponent. diffusion, on oscillatory domains in 1,2 and 3 dimensions.
• Use numerical methods to check the accuracy of the solutions.
• Use perturbation methods to examine various limits (slow, fast oscillations).

Outcomes

• New exact expressions for the solution of PDE models on growing domains.
• New numerical algorithms to check the exact solutions.
• Publish the outcomes in the applied mathematics literature.

Skills and experience

The student should have:

• strong programming skills (e.g. MATLAB, Julia or equivalent)
• completed undergraduate subjects in ordinary differential equations (MXB225) and partial differential equations (MXB325).

Keywords

• partial differential equation
• applied mathematics
• computational mathematics

Contact

Contact Professor Matthew Simpson for more information.