TY  - JOUR
A1  - Awad, Emad
A1  - Metzler, Ralf
T1  - Closed-form multi-dimensional solutions and asymptotic behaviours for subdiffusive processes with crossovers: II. Accelerating case
JF  - Journal of physics : A, Mathematical and theoretical
N2  - 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.
KW  - multidimensional fractional diffusion equation
KW  - continuous time random
KW  - walks
KW  - crossover anomalous diffusion dynamics
KW  - non-Gaussian probability
KW  - density
Y1  - 2022
U6  - https://doi.org/10.1088/1751-8121/ac5a90
SN  - 1751-8113
SN  - 1751-8121
VL  - 55
IS  - 20
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  -