A numerical process of the mobile-immobile advection-dispersion model arising in solute transport

In the present article‎, ‎to find the answer to the mobile-immobile advection-dispersion model of temporal fractional order $0 \beta \leq 1$ (MI-ADM-TF)‎, ‎which can be applied to model the solute forwarding in watershed catchment and flood‎, ‎the effective high-order numerical process is gonna be built‎. ‎To do this‎, ‎the temporal-fractional derivative of the MI-ADM-TF is discretized by using the linear interpolation‎, ‎and the temporal-first derivative by applying the first-order precision of the finite-difference method‎. ‎On the other hand‎, ‎After obtaining a semi-discrete form‎, ‎to obtain the full-discrete technique‎, ‎the space derivative is approximated utilizing a collocation approach based on the Legendre basis‎. ‎The convergence order of the implicit numerical design for MI-ADM-TF is discussed in that is linear‎. ‎Moreover‎, ‎the temporal-discretized structure of stability is also discussed theoretically in general in the article‎. ‎Eventually‎, ‎two models are offered to demonstrate the quality and authenticity of the established process‎.