23,303 research outputs found
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
Heterogeneous diffusion in comb and fractal grid structures
We give an exact analytical results for diffusion with a power-law position
dependent diffusion coefficient along the main channel (backbone) on a comb and
grid comb structures. For the mean square displacement along the backbone of
the comb we obtain behavior , where
is the power-law exponent of the position dependent diffusion
coefficient . Depending on the value of we
observe different regimes, from anomalous subdiffusion, superdiffusion, and
hyperdiffusion. For the case of the fractal grid we observe the mean square
displacement, which depends on the fractal dimension of the structure of the
backbones, i.e., , where
is the fractal dimension of the backbones structure. The reduced
probability distribution functions for both cases are obtained by help of the
Fox -functions
Simultaneous denoising and enhancement of signals by a fractal conservation law
In this paper, a new filtering method is presented for simultaneous noise
reduction and enhancement of signals using a fractal scalar conservation law
which is simply the forward heat equation modified by a fractional
anti-diffusive term of lower order. This kind of equation has been first
introduced by physicists to describe morphodynamics of sand dunes. To evaluate
the performance of this new filter, we perform a number of numerical tests on
various signals. Numerical simulations are based on finite difference schemes
or Fast and Fourier Transform. We used two well-known measuring metrics in
signal processing for the comparison. The results indicate that the proposed
method outperforms the well-known Savitzky-Golay filter in signal denoising.
Interesting multi-scale properties w.r.t. signal frequencies are exhibited
allowing to control both denoising and contrast enhancement
A note on fractional derivative modeling of broadband frequency-dependent absorption: Model III
By far, the fractional derivative model is mainly related to the modelling of
complicated solid viscoelastic material. In this study, we try to build the
fractional derivative PDE model for broadband ultrasound propagation through
human tissues
Non-perturbative calculations for the effective potential of the symmetric and non-Hermitian field theoretic model
We investigate the effective potential of the symmetric
field theory, perturbatively as well as non-perturbatively. For the
perturbative calculations, we first use normal ordering to obtain the first
order effective potential from which the predicted vacuum condensate vanishes
exponentially as in agreement with previous calculations. For the
higher orders, we employed the invariance of the bare parameters under the
change of the mass scale to fix the transformed form totally equivalent to
the original theory. The form so obtained up to is new and shows that all
the 1PI amplitudes are perurbative for both and regions. For
the intermediate region, we modified the fractal self-similar resummation
method to have a unique resummation formula for all values. This unique
formula is necessary because the effective potential is the generating
functional for all the 1PI amplitudes which can be obtained via and thus we can obtain an analytic calculation for the 1PI
amplitudes. Again, the resummed from of the effective potential is new and
interpolates the effective potential between the perturbative regions.
Moreover, the resummed effective potential agrees in spirit of previous
calculation concerning bound states.Comment: 20 page
Extending the D'Alembert Solution to Space-Time Modified Riemann-Liouville Fractional Wave Equations
In the realm of complexity, it is argued that adequate modeling of
TeV-physics demands an approach based on fractal operators and fractional
calculus (FC). Non-local theories and memory effects are connected to
complexity and the FC. The non-differentiable nature of the microscopic
dynamics may be connected with time scales. Based on the Modified
Riemann-Liouville definition of fractional derivatives, we have worked out
explicit solutions to a fractional wave equation with suitable initial
conditions to carefully understand the time evolution of classical fields with
a fractional dynamics. First, by considering space-time partial fractional
derivatives of the same order in time and space, a generalized fractional
D'Alembertian is introduced and by means of a transformation of variables to
light-cone coordinates, an explicit analytical solution is obtained. To address
the situation of different orders in the time and space derivatives, we adopt
different approaches, as it will become clear throughout the paper. Aspects
connected to Lorentz symmetry are analyzed in both approaches.Comment: 8 page
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