Open AccessDiffusion in a disk with inclusion: Evaluating Green’s functions
Author(s) -
Remus Stana,
Grant Lythe
Publication year - 2022
Publication title -
plos one
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.99
H-Index - 332
ISSN - 1932-6203
DOI - 10.1371/journal.pone.0265935
Subject(s) - radius , mathematical analysis , unit circle , domain (mathematical analysis) , boundary (topology) , series (stratigraphy) , physics , equivalence (formal languages) , function (biology) , position (finance) , mathematics , thermal diffusivity , diffusion , geometry , quantum mechanics , pure mathematics , paleontology , computer security , finance , evolutionary biology , computer science , economics , biology
We give exact Green’s functions in two space dimensions. We work in a scaled domain that is a circle of unit radius with a smaller circular “inclusion”, of radius a , removed, without restriction on the size or position of the inclusion. We consider the two cases where one of the two boundaries is absorbing and the other is reflecting. Given a particle with diffusivity D , in a circle with radius R , the mean time to reach the absorbing boundary is a function of the initial condition, given by the integral of Green’s function over the domain. We scale to a circle of unit radius, then transform to bipolar coordinates. We show the equivalence of two different series expansions, and obtain closed expressions that are not series expansions.