Moments for Tempered Fractional Advection-Diffusion Equations

This paper develops moment formulas for exponentially tempered, fractional advection-diffusion equations (TFADEs) that transition from anomalous to asymptotic diffusion limits over time. Exact analytical expressions or series representations for spatial moments up to the fourth order are derived by integral transform or asymptotic expansion approach. A fully Lagrangian solver, cross verified by an implicit Eulerian approach, is also developed to calculate numerically the complete evolution of moments for the TFADEs with complex initial and boundary conditions. Moment analysis identifies the diffusion equation that attracts the tempered anomalous diffusion in the long time limit. Fitting of moments measured at two end members of alluvial systems checks the applicability of moment analysis in understanding real diffusion.