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Diffusion on middle-ξ Cantor sets
In this paper, we study C^ζ -calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the C^ζ -calculus on the generalized Cantor sets known as middle-ξ Cantor sets. We have suggested a calculus on the middle-ξ Cantor sets for different values of ξ with 0 < ξ < 1. Differential equations on the middle-ξ Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given
Diffusion on middle- Cantor sets
In this paper, we study -calculus on generalized Cantor sets,
which have self-similar properties and fractional dimensions that exceed their
topological dimensions. Functions with fractal support are not differentiable
or integrable in terms of standard calculus, so we must involve local
fractional derivatives. We have generalized the -calculus on the
generalized Cantor sets known as middle- Cantor sets. We have suggested a
calculus on the middle- Cantor sets for different values of with
. Differential equations on the middle- Cantor sets have been
solved, and we have presented the results using illustrative examples. The
conditions for super-, normal, and sub-diffusion on fractal sets are given.Comment: 15 pages, 11 figure
Anomalous diffusion on a fractal mesh
An exact analytical analysis of anomalous diffusion on a fractal mesh is
presented. The fractal mesh structure is a direct product of two fractal sets
which belong to a main branch of backbones and side branch of fingers. The
fractal sets of both backbones and fingers are constructed on the entire
(infinite) and axises. To this end we suggested a special algorithm of
this special construction. The transport properties of the fractal mesh is
studied, in particular, subdiffusion along the backbones is obtained with the
dispersion relation , where the transport
exponent is determined by the fractal dimensions of both backbone and
fingers. Superdiffusion with has been observed as well when the
environment is controlled by means of a memory kernel
Fourier spectra of measures associated with algorithmically random Brownian motion
In this paper we study the behaviour at infinity of the Fourier transform of
Radon measures supported by the images of fractal sets under an algorithmically
random Brownian motion. We show that, under some computability conditions on
these sets, the Fourier transform of the associated measures have, relative to
the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity.
The argument relies heavily on a direct characterisation, due to Asarin and
Pokrovskii, of algorithmically random Brownian motion in terms of the prefix
free Kolmogorov complexity of finite binary sequences. The study also
necessitates a closer look at the potential theory over fractals from a
computable point of view.Comment: 24 page
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