Modeling diffusive transport with a fractional derivative without singular kernel

J.F. Gómez-Aguilar, M.G. López-López, V.M. Alvarado-Martínez, J. Reyes-Reyes and M. Adam-Medina

Physica A: Statistical Mechanics and its Applications, 2016, vol. 447, issue C, 467-481

Abstract: 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 β,γ∈(0;2] for the space and time domain respectively. In this representation two auxiliary parameters σx and σt 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 β,γ∈(0;1), the concentration exhibits the non-Markovian Lévy flights and the subdiffusion phenomena; when β=γ=1 the classical case is recovered; when β,γ∈(1;2] the concentration exhibits the Markovian Lévy flights and the superdiffusion phenomena; finally when β=γ=2 the concentration is anomalous dispersive and we found ballistic diffusion.

Keywords: Fractional calculus; Non-local transport processes; Non-Fickian diffusion; Caputo–Fabrizio fractional derivative; Dissipative dynamics; Anomalous diffusion (search for similar items in EconPapers)
Date: 2016
References: View complete reference list from CitEc
Citations: View citations in EconPapers (16)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:447:y:2016:i:c:p:467-481

DOI: 10.1016/j.physa.2015.12.066

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