A physically based connection between fractional calculus and fractal geometry
Abstract
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the powerlaws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a powerlaw naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.
 Publication:

Annals of Physics
 Pub Date:
 November 2014
 DOI:
 10.1016/j.aop.2014.07.008
 arXiv:
 arXiv:1508.04652
 Bibcode:
 2014AnPhy.350..146B
 Keywords:

 Fractional calculus;
 Fractal geometry;
 Fractional differential equation;
 Transport process;
 Physics  Fluid Dynamics
 EPrint:
 23 pages, 6 figures