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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<sup>α</sup>''. 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 The classical physical laws such as [[Fick's laws of diffusion]], [[Darcy's law]] and [[Fourier's law]] are no longer applicable for such media, because they are based on [[Euclidean geometry]], which doesn't apply to media of non-[[integer]] [[fractal dimension]]s. The basic physical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<sup>β</sup>'', ''t<sup>α</sup>''). The elementary physical concepts such as velocity in a [[fractal spacetime]] (''x<sup>β</sup>'', ''t<sup>α</sup>'') can be redefined by: | |||
:<math> v' = \frac{dx'}{dt'}=\frac{dx^\beta}{dt^\alpha}\,,\quad \alpha,\beta>0</math>, | |||
where ''S<sup>α,β</sup>'' 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: | |||
:<math> | |||
\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</math>, | |||
A more general definition is given by | |||
:<math> | |||
\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</math>. | |||
[[Image:Fractal dervative.jpg|right|thumb|Fractal derivative for function ''f''(''t'') = ''t'', with derivative order is ''α'' ∈ <nowiki>(0,1]</nowiki>]] | |||
==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, | |||
:<math>\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)</math> | |||
:<math>u(x, 0)=\delta(x).</math> | |||
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 | |||
:<math>t'=t^\alpha\,,\quad x'=x^\beta. </math> | |||
then the equation (1) becomes the normal diffusion form equation, the solution of (1) has the stretched [[Gaussian distribution|Gaussian]] form: | |||
:<math>u(x,t)=\frac{1}{2\sqrt{\pi t^\alpha}} e^{-\frac{x^{2 \beta}}{4t^\alpha}}</math> | |||
The [[mean squared displacement]] of above fractal derivative diffusion equation has the [[asymptote]]: | |||
:<math>\left\langle x^2 (t) \right\rangle\propto t^{(3 \alpha-\alpha \beta)/2 \beta}.</math> | |||
==See also== | |||
* [[Fractional calculus]] | |||
* [[Fractional dynamics]] | |||
* [[Multifractal system]] | |||
==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. [http://www.sciencedirect.com/science/article/pii/S0898122109005525 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. [http://www.sciencedirect.com/science/article/pii/S0309170812002801 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== | |||
*[http://www.ismm.ac.cn/ismmlink/PLFD/index.html Power Law & Fractional Dynamics] | |||
[[Category:Fractals]] | |||
[[Category:Applied mathematics]] |
Latest revision as of 07:31, 23 May 2013
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 tα. 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 (xβ, tα). The elementary physical concepts such as velocity in a fractal spacetime (xβ, tα) can be redefined by:
where Sα,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:
A more general definition is given by
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,
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
then the equation (1) becomes the normal diffusion form equation, the solution of (1) has the stretched Gaussian form:
The mean squared displacement of above fractal derivative diffusion equation has the asymptote:
See also
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.