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Diffusion in a disk with inclusion: Evaluating Green's functions
1Department of Applied Mathematics, University of Leeds, Leeds, United Kingdom.
Plos One
|April 14, 2022
Summary
We derived exact Green's functions for diffusion in a two-dimensional circular domain with a removed inner region. This provides a new method to calculate the mean time for particles to reach boundaries.
Area of Science:
- Mathematical Physics
- Partial Differential Equations
- Stochastic Processes
Background:
- Calculating mean first passage times is crucial in diffusion processes.
- Green's functions offer a powerful tool for solving diffusion equations.
- Complex geometries, like domains with inclusions, pose significant analytical challenges.
Purpose of the Study:
- To derive exact Green's functions for diffusion in a 2D annulus.
- To provide closed-form expressions for mean first passage times.
- To analyze diffusion with mixed absorbing-reflecting boundary conditions.
Main Methods:
- Scaling and transformation to bipolar coordinates.
- Utilizing Green's functions in a scaled unit circle with a removed inner inclusion.
- Equivalence of series expansions and derivation of closed-form solutions.
Main Results:
- Exact Green's functions were obtained for the specified domain.
- Closed-form expressions, not series expansions, were derived.
- The mean time to reach the absorbing boundary was expressed as an integral of the Green's function.
Conclusions:
- The study provides an exact analytical solution for diffusion in a complex 2D domain.
- The derived Green's functions simplify calculations of mean first passage times.
- This method is applicable without restrictions on inclusion size or position.
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