Solutions of variable order mobile–immobile advection–dispersion models, their existence and uniqueness

Abstract This study employs a numerical approach, specifically the two-step Adomian decomposition method (TSADM), which provides an analytical solution to the variable fractional-order multi-dimensional mobile–immobile advection–dispersion equation. We employ three widely recognized and efficient numerical schemes: the Adomian decomposition method (ADM), the modified ADM, and TSADM. A comparison of the obtained results demonstrates the superior efficiency of TSADM among these approaches, as it requires less computational effort while maintaining high accuracy. Additionally, the theoretical aspects are a crucial part of this work. We investigate the existence and uniqueness of solutions using Schaefer’s fixed point theorem within a Banach space. The stability analysis is also discussed, and we provide results on the positive, maximal, and minimal solutions for the variable fractional-order multi-dimensional mobile-immobile advection–dispersion equation.