A novel numerical method for the time variable fractional order mobile–immobile advection–dispersion model

Evolution equations containing fractional derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations. Recently, researchers have found that many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the mobile–immobile advection–dispersion model with the Coimbra variable time fractional derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint. A novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated. As for the convergence of the numerical method, we only consider a special case, i.e., the time fractional derivative is independent of the time variable tt. The case where the time fractional derivative depends on both the time variable tt and the space variable xx will be considered in a future work. Finally, numerical examples are provided to show that the implicit difference approximation is computationally efficient.