Numerical treatment for after-effected multi-term time-space fractional advection–diffusion equations

Anomalous transport and the concept of delayed advection and diffusion can be combined for the description of transport phenomena in heterogeneous environments. A class of multi-term time-space fractional advection–diffusion model with after-effect that is general enough to express the above possible mechanisms of transient dispersion is numerically investigated in this work. To that end, a fully implicit difference method is first constructed for the numerical solution of those equations. This scheme is built on the idea of separating the current state and the prehistory function such that the prehistory function is approximated by means of an appropriate interpolation–extrapolation operator. A discrete form of the fractional Grönwall inequality is used and employed to provide an optimal error estimate for the proposed scheme. The existence and uniqueness of the numerical solution, the order of approximation error for the constructed scheme, the stability and the order of convergence are mathematically investigated in this work. Finally, some numerical experiments with fixed, variable and distributed delays with respect to time are tested to clarify the efficiency of the proposed scheme and to ensure the coincide between theoretical and experimental results.