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. •Fractional calculus is applied to the diffusion and the diffusion–advection equation.•The Caputo–Fabrizio fractional derivative is applied.•The generalization of the equations in space–time exhibits anomalous behavior.•To keep the dimensionality an auxiliary parameter σ is introduced.•The numerical solutions are obtained using the numerical Laplace transform algorithm.