@article{AwadMetzler2020,
  author    = {Awad, Emad and Metzler, Ralf},
  title     = {Crossover dynamics from superdiffusion to subdiffusion},
  series = {Fractional calculus and applied analysis : an international journal for theory and applications},
  volume    = {23},
  journal   = {Fractional calculus and applied analysis : an international journal for theory and applications},
  number    = {1},
  publisher = {De Gruyter},
  address   = {Berlin ; Boston},
  issn      = {1311-0454},
  doi       = {10.1515/fca-2020-0003},
  pages     = {55 -- 102},
  year      = {2020},
  abstract  = {The Cattaneo or telegrapher's equation describes the crossover from initial ballistic to normal diffusion. Here we study and survey time-fractional generalisations of this equation that are shown to produce the crossover of the mean squared displacement from superdiffusion to subdiffusion. Conditional solutions are derived in terms of Fox H-functions and the dth-order moments as well as the diffusive flux of the different models are derived. Moreover, the concept of the distribution-like is proposed as an alternative to the probability density function.},
  language  = {en}
}
@article{AwadMetzler2022,
  author    = {Awad, Emad and Metzler, Ralf},
  title     = {Closed-form multi-dimensional solutions and asymptotic behaviours for subdiffusive processes with crossovers: II. Accelerating case},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {55},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {20},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8121/ac5a90},
  pages     = {29},
  year      = {2022},
  abstract  = {Anomalous diffusion with a power-law time dependence vertical bar R vertical bar(2)(t) similar or equal to t(alpha i) of the mean squared displacement occurs quite ubiquitously in numerous complex systems. Often, this anomalous diffusion is characterised by crossovers between regimes with different anomalous diffusion exponents alpha(i). Here we consider the case when such a crossover occurs from a first regime with alpha(1) to a second regime with alpha(2) such that alpha(2) > alpha(1), i.e., accelerating anomalous diffusion. A widely used framework to describe such crossovers in a one-dimensional setting is the bi-fractional diffusion equation of the so-called modified type, involving two time-fractional derivatives defined in the Riemann-Liouville sense. We here generalise this bi-fractional diffusion equation to higher dimensions and derive its multidimensional propagator (Green's function) for the general case when also a space fractional derivative is present, taking into consideration long-ranged jumps (Levy flights). We derive the asymptotic behaviours for this propagator in both the short- and long-time as well the short- and long-distance regimes. Finally, we also calculate the mean squared displacement, skewness and kurtosis in all dimensions, demonstrating that in the general case the non-Gaussian shape of the probability density function changes.},
  language  = {en}
}