McDonald, NR;
Harris, Samuel J;
(2024)
Exact and numerical solutions of a free boundary problem with a reciprocal
growth law.
IMA Journal of Applied Mathematics
, 89
(2)
pp. 374-386.
10.1093/imamat/hxae014.
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Abstract
A two-dimensional free boundary problem is formulated in which the normal velocity of the boundary is proportional to the inverse of the gradient of a harmonic function T. The field T is defined in a simply connected region which includes the point at infinity where it has a logarithmic singularity. The growth problem in which the boundary expands outward is formulated both in terms of the Schwarz function of the boundary and a Polubarinova-Galin equation for the conformal map of the region from the exterior of the unit disk. An expanding free boundary is shown to be stable and explicit solutions for growing ellipses and a class of polynomial lemniscates are derived. Numerical solution of the Polubarinova-Galin equation is used to compute the evolution of the boundary having other initial shapes.
| Type: | Article |
|---|---|
| Title: | Exact and numerical solutions of a free boundary problem with a reciprocal growth law |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1093/imamat/hxae014 |
| Publisher version: | https://doi.org/10.1093/imamat/hxae014 |
| Language: | English |
| Additional information: | © The Author(s) 2024. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. |
| Keywords: | Free boundary; Schwarz function; wildfire; lemniscate growth |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10191351 |
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