Qom University of TechnologyMathematics and Computational Sciences2717-27083320220901A numerical process of the mobile-immobile advection-dispersion model arising in solute transport11025490810.30511/mcs.2022.554606.1072ENYEsmaeelzade AghdamDepartment of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training UniversityBFarnamDepartment of Mathematics, Qom University of Technology, Qom, IranJournal Article20220526In 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.<br />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. <br />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.<br />The convergence order of the implicit numerical design for MI-ADM-TF is discussed in that is linear.<br />Moreover, the temporal-discretized structure of stability is also discussed theoretically in general in the article.<br />Eventually, two models are offered to demonstrate the quality and authenticity of the established process.http://mcs.qut.ac.ir/article_254908_22acd5f166739af21a07a0042fcac86d.pdf