Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel

In this paper, using the fractional operators with Mittag-Leffler kernel in Caputo and Riemann-Liouville sense the space-time fractional diffusion equation is modified, the fractional equation will. be examined separately; with fractional spatial derivative and fractional temporal derivative. For the study cases, the order considered is 0 < beta, gamma <= 1 respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation, these parameters related to equation results in a fractal space-time geometry provide a new family of solutions for the diffusive processes. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, viscoelastic materials, material heterogeneities and media with different scales. (C) 2016 Elsevier B.V. All rights reserved.