  The generalized diffusion equations with fractional order derivatives have
shown be quite efficient to describe the diffusion in complex systems, with the
advantage of producing exact expressions for the underlying diffusive
properties. Recently, researchers have proposed different fractional-time
operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently
from the well-known Riemann-Liouville operator, are defined by non-singular
memory kernels. Here we proposed to use these new operators to generalize the
usual diffusion equation. By analyzing the corresponding fractional diffusion
equations within the continuous time random walk framework, we obtained waiting
time distributions characterized by exponential, stretched exponential, and
power-law functions, as well as a crossover between two behaviors. For the mean
square displacement, we found crossovers between usual and confined diffusion,
and between usual and sub-diffusion. We obtained the exact expressions for the
probability distributions, where non-Gaussian and stationary distributions
emerged. This former feature is remarkable because the fractional diffusion
equation is solved without external forces and subjected to the free diffusion
boundary conditions. We have further shown that these new fractional diffusion
equations are related to diffusive processes with stochastic resetting, and to
fractional diffusion equations with derivatives of distributed order. Thus, our
results show that these new operators are a simple and efficient way for
incorporating different structural aspects into the system, opening new
possibilities for modeling and investigating anomalous diffusive processes.
