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Updated: Sep 27, 2025

In Situ Monitoring of Diffusion of Guest Molecules in Porous Media Using Electron Paramagnetic Resonance Imaging
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Diffusion in a disk with inclusion: Evaluating Green's functions.

Remus Stana1, Grant Lythe1

  • 1Department of Applied Mathematics, University of Leeds, Leeds, United Kingdom.

Plos One
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PubMed
Summary
This summary is machine-generated.

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.

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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.