Modeling Diffusion and Capture

Start: 04/12/2017 - 4:15pm
End  : 04/12/2017 - 5:15pm


Alan Lindsay (Notre Dame)


Diffusion is a fundamental transport mechanism whereby spatial paths are determined from probabilistic distributions. In examples such as the pollination of a flower or immune response to infection, the arrival of a single particle can initiate a cascade of events. The movement of these particles is driven by random motions, yet these systems largely function in an ordered and predictable way. This process, and others like it, can be modeled as a problem for the arrival time of a diffusing molecule to hit a small absorbing target.  In this talk I will discuss the mathematical models of this phenomenon and introduce the governing equations which are PDEs of parabolic and elliptic type with a mix of Dirichlet (Absorbing) and Neumann (Reflecting) boundary conditions. A particular feature of cellular problems is that the absorbing set has a large number of very small sites. I will present a new homogenized theory which replaces the heterogeneous configuration of boundary conditions with a uniform Robin type condition. To verify this limit, I will describe a novel spectral boundary element method for the exterior mixed Neumann-Dirichlet boundary value problem. The numerical formulation reduces the problem to a linear integral equation supported on each of the absorbing sites. Real biological systems feature thousands of absorbing sites and our numerical method can rapidly resolve this realistic limit to high accuracy.

Shanhan B460, Harvey Mudd College