A fractional differential equation is studied and its application for describing diffusion on random fractal structures is considered. It represents the simplest generalization of the fractional diffusion equation valid in Euclidean systems. The solution of the fractional equation in one dimension is discussed, and compared with exact results for the fractional Brownian motion and the one-dimensional version of the 'standard' diffusion equation on fractals. In higher dimensions, it correctly describes the asymptotic scaling behaviour of the probability density function on random fractals, as obtained recently by using scaling arguments and exact enumeration calculations for the infinite percolation cluster at criticality.
Giona, M., Roman, H. (1992). Fractional diffusion equation on fractals: One-dimensional case and asymptotic behaviour. JOURNAL OF PHYSICS. A, MATHEMATICAL AND GENERAL, 25(8), 2093-2105 [10.1088/0305-4470/25/8/023].
Fractional diffusion equation on fractals: One-dimensional case and asymptotic behaviour
Roman H. E.
1992
Abstract
A fractional differential equation is studied and its application for describing diffusion on random fractal structures is considered. It represents the simplest generalization of the fractional diffusion equation valid in Euclidean systems. The solution of the fractional equation in one dimension is discussed, and compared with exact results for the fractional Brownian motion and the one-dimensional version of the 'standard' diffusion equation on fractals. In higher dimensions, it correctly describes the asymptotic scaling behaviour of the probability density function on random fractals, as obtained recently by using scaling arguments and exact enumeration calculations for the infinite percolation cluster at criticality.
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