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ASYMPTOTIC ANALYSIS FOR THE MEAN FIRST PASSAGE TIME IN FINITE OR SPATIALLY PERIODIC 2D DOMAINS WITH A CLUSTER OF SMALL TRAPS

Part of: Representations of solutions Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  01 March 2021

S. IYANIWURA  and
M. J. WARD Open the ORCID record for M. J. WARD [Opens in a new window]
S. IYANIWURA
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada; [email protected].
M. J. WARD*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada; [email protected].
*
[email protected]
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Abstract

A hybrid asymptotic-numerical method is developed to approximate the mean first passage time (MFPT) and the splitting probability for a Brownian particle in a bounded two-dimensional (2D) domain that contains absorbing disks, referred to as “traps”, of asymptotically small radii. In contrast to previous studies that required traps to be spatially well separated, we show how to readily incorporate the effect of a cluster of closely spaced traps by adapting a recently formulated least-squares approach in order to numerically solve certain local problems for the Laplacian near the cluster. We also provide new asymptotic formulae for the MFPT in 2D spatially periodic domains where a trap cluster is centred at the lattice points of an oblique Bravais lattice. Over all such lattices with fixed area for the primitive cell, and for each specific trap set, the average MFPT is smallest for a hexagonal lattice of traps.

Keywords

trap clusterlogarithmic capacitancemean first passage timesplitting probabilityGreen’s function

MSC classification

Primary: 35B25: Singular perturbations
Secondary: 35C20: Asymptotic expansions 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 35J08: Green's functions
Type
Research Article
Information
The ANZIAM Journal , Volume 63 , Issue 1 , January 2021 , pp. 1 - 22
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Copyright
© Australian Mathematical Society 2021

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