Fractal derivative
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In applied mathematics and mathematical analysis, the fractal derivative is a nonNewtonian type of derivative in which the variable such as t has been scaled according to t^{α}. The derivative is defined in fractal geometry.
Contents
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 The classical physical laws such as Fick's laws of diffusion, Darcy's law and Fourier's law are no longer applicable for such media, because they are based on Euclidean geometry, which doesn't apply to media of noninteger fractal dimensions.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:
 ,
where S^{α,β} represents the fractal spacetime with scaling indices α and β. The traditional definition of velocity makes no sense in the nondifferentiable 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 transportdiffusion 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
 Chen, W. (2006). "Time–space fabric underlying anomalous diffusion". Chaos, Solitons and Fractals. 28 (4): 923–929. arXiv:mathph/0505023. doi:10.1016/j.chaos.2005.08.199.
 Kanno, R. (1998). "Representation of random walk in fractal spacetime". Physica A. 248 (1–2): 165–175. doi:10.1016/S03784371(97)004226.
 Chen, W.; Sun, H.G.; Zhang, X.; Korosak, D. (2010). "Anomalous diffusion modeling by fractal and fractional derivatives". Computers and Mathematics with Applications. 59 (5): 1754–8. doi:10.1016/j.camwa.2009.08.020.
 Sun, H.G.; Meerschaert, M.M.; Zhang, Y.; Zhu, J.; Chen, W. (2013). "A fractal Richards' equation to capture the nonBoltzmann scaling of water transport in unsaturated media". Advances in Water Resources. 52 (52): 292–5. doi:10.1016/j.advwatres.2012.11.005. PMC 3686513.
 Cushman, J.H.; O'Malley, D.; Park, M. (2009). "Anomalous diffusion as modeled by a nonstationary extension of Brownian motion". Phys. Rev. E. 79 (3): 032101. doi:10.1103/PhysRevE.79.032101.
 Mainardi, F.; Mura, A.; Pagnini, G. (2010). "The MWright Function in TimeFractional Diffusion Processes: A Tutorial Survey". International Journal of Differential Equations. 2010: 104505. arXiv:1004.2950. Bibcode:2010arXiv1004.2950M. doi:10.1155/2010/104505.
External links
 Power Law & Fractional Dynamics
 NonNewtonian calculus website