Fractal derivative

In applied mathematics and mathematical analysis, the fractal derivative is a nonstandard type of derivative in which the variable such as t has been scaled according to t^α. The derivative is defined in fractal geometry.

Physical background

Porous media, aquifer, turbulence and other media usually exhibit fractal properties. The classical physical laws such as Fick's laws of diffusion, Darcy's law and Fourier's law are no longer applicable for such media, because they are based on Euclidean geometry, which doesn't apply to media of non-integer fractal dimensions. The basic physical concepts such as distance and velocity in fractal media are required to be redefined; the scales for space and time should be transformed according to (x^β, t^α). The elementary physical concepts such as velocity in a fractal spacetime (x^β, t^α) can be redefined by:

v' = \frac{dx'}{dt'}=\frac{dx^\beta}{dt^\alpha}\,,\quad \alpha,\beta>0

where S^α,β represents the fractal spacetime with scaling indices α and β. The traditional definition of velocity makes no sense in the non-differentiable fractal spacetime.

Definition

Based on above discussion, the concept of the fractal derivative of a function u(t) with respect to a fractal measure t has been introduced as follows:

\frac{\partial f(t)}{\partial t^\alpha}=\lim_{t_1 \rightarrow t}\frac{f(t_1)-f(t)}{t_1^\alpha-t^\alpha}\,, \quad \alpha>0

A more general definition is given by

\frac{\partial^\beta f(t)}{\partial t^\alpha}=\lim_{t_1 \rightarrow t}\frac{f^\beta (t_1)-f^\beta (t)}{t_1^\alpha-t^\alpha}\,, \quad\alpha>0, \beta>0

Fractal derivative for function f(t) = t, with derivative order is α ∈ (0,1]

Application in anomalous diffusion

As an alternative modeling approach to the classical Fick’s second law, the fractal derivative is used to derive a linear anomalous transport-diffusion equation underlying anomalous diffusion process,

\frac{d u (x,t)}{d t^\alpha}= D \frac{\partial }{\partial x^\beta} \left(\frac{\partial u(x,t)}{\partial x^\beta}\right), -\infty< x < +\infty\,, \quad (1)

u(x, 0)=\delta(x).

where 0 < α < 2, 0 < β < 1, and δ(x) is the Dirac Delta function.

In order to obtain the fundamental solution, we apply the transformation of variables

t'=t^\alpha\,,\quad x'=x^\beta.

then the equation (1) becomes the normal diffusion form equation, the solution of (1) has the stretched Gaussian form:

u(x,t)=\frac{1}{2\sqrt{\pi t^\alpha}} e^{-\frac{x^{2 \beta}}{4t^\alpha}}

The mean squared displacement of above fractal derivative diffusion equation has the asymptote:

\left\langle x^2 (t) \right\rangle\propto t^{(3 \alpha-\alpha \beta)/2 \beta}.

References

W. Chen. Time–space fabric underlying anomalous diffusion. Chaos, Solitons and Fractals 28 (2006), 923–929.
R. Kanno. Representation of random walk in fractal space-time, Physica A 248 (1998), 165-175.
W. Chen, H. G. Sun, X. Zhang, D. Korosak. Anomalous diffusion modeling by fractal and fractional derivatives. Computers and Mathematics with Applications，2010, 59 (5): 1754-1758.
H.G. Sun, M. M. Meerschaert, Y. Zhang, J. Zhu, W. Chen. A fractal Richards' equation to capture the non-Boltzmann scaling of water transport in unsaturated media. Advances in Water Resources, 2013, 52: 292-295.
J. H. Cushman, D. O'Malley and M. Park. Anomalous diffusion as modeled by a nonstationary extension of Brownian motion, Phys. Rev. E, 2009, 79, 032101.
F. Mainardi, A. Mura, and G. Pagnini. The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey. International Journal of Differential Equations, 2010, Article ID 104505, 29 pages, doi:10.1155/2010/104505.

External links

Power Law & Fractional Dynamics

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