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Research articles

ScienceAsia 42(2016): 33-39 |doi: 10.2306/scienceasia1513-1874.2016.42.033

The speed of reaction-diffusion fronts on fractals: testing the Campos-Méndez-Fort formula

Orapun Suwannasena, Michael A. Allenb,*, Julien Clinton Sprottc

ABSTRACT:     Campos, Méndez, and Fort (CMF) derived an approximate formula for the speed of reaction-diffusion fronts in fractal media. By way of a continuation of their earlier studies, we perform numerical simulations of reaction-diffusion equations with au(1−u)(1−αu) for 0≤α≤1 as the reaction term on various generalized Sierpiński carpets (including infinitely ramified and random ones). The CMF formula agrees well with the mean front speed as a function of a obtained from our simulations for the classic Sierpiński carpet, a randomized version of the carpet, and some finitely ramified carpets containing loops. In these cases the mean front speed also shows no significant dependence on α, as predicted by the CMF formula. However, the agreement is not so good in the case of the other carpets tested and this is probably a result of the mean distance of the front from the starting point against time behaving erratically in such cases. We also propose some nomenclature for generalized Sierpiński carpets and introduce a compact formulation of how to determine whether a point is on a generalized Sierpiński carpet lattice.

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a Mathematics Department, Faculty of Science, Mahidol University, Rama 6 Road, Bangkok 10400 Thailand
b Physics Department, Faculty of Science, Mahidol University, Rama 6 Road, Bangkok 10400 Thailand
c Department of Physics, University of Wisconsin, Madison, WI 53706 USA

* Corresponding author, E-mail: maa5652@gmail.com

Received 18 Sep 2015, Accepted 29 Feb 2016