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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 σ x and σ t 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. [ABSTRACT FROM AUTHOR]