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

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- {\alpha}u) for 0 < {\alpha} < 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 {\alpha}, 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.