Time-integral iteration method for two-dimensional anomalous transport

A methodology is developed to describe time-dependent phenomena associated with nonlocal transport in complex, two-dimensional geometries. It is an extension of the "iterative method" introduced previously to solve steady-state transport problems [Maggs and Morales, Phys. Rev. E 99, 013307 (2019)], and it is based on the "jumping particle" concepts associated with the continuous-time random walk (CTRW) model. The method presented explicitly evaluates the time integral contained in the CTRW master equation. A modified version of the Mittag-Leffler function is used for the waiting-time probability distributions to incorporate memory effects. Calculations of the propagation of "anomalous transport waves" in various systems, with and without memory, illustrate the technique.