Modeling diffusive transport with a fractional derivative without singular kernel
In this paper we present an alternative representation of the diffusion equation and the diffusion advection equation using the fractional calculus approach, the spatial time derivatives are approximated using the fractional definition recently introduced by Caputo and Fabrizio in the range b,g(0;2] for the space and time domain respectively. In this representation two auxiliary parameters sx and st are introduced, these parameters related to equation results in a fractal space time geometry provide an entire new family of solutions for the diffusion processes. The numerical results showed different behaviors when compared with classical model solutions. In the range b,g(0;1) the concentration exhibits the non Markovian Levy flights and the subdiffusion phenomena; when b=g=1 the classical case is recovered; when b,g(1;2] the concentration exhibits the Markovian Levy flights and the superdiffusion phenomena; finally when b=g=2 the concentration is anomalous dispersive and we found ballistic diffusion.