An efficient numerical treatment of variable-order time fractional advection–diffusion equation

The variable-order fractional advection–diffusion equation is essential for the modeling of the anomalous diffusion behavior of contaminant transport in complex media of variable nature. Consequently, the solution of the variable-order advection–diffusion model is highly important as it can provide a reference value for environmental protection. The present work is concerned with the numerical solution of the variable-order time fractional advection–diffusion model using the fractional explicit decoupled group method (FEDGM). To the best of the authors’ knowledge, this is the first application of this approach to variable-order fractional advection–diffusion equations. The proposed scheme is based on a finite difference technique in the time domain and skewed difference approximations in the space domain. This approach allows one to reduce the computational work effectively as it comprises only half of the nodal points in the iteration process, compared to the standard finite difference schemes. A Crank–Nicolson difference scheme (CNDS) is also suggested for comparison purposes. The stability and convergence of the presented methods are investigated and theoretically proven. Several numerical simulations were performed, and the obtained results were matched with the exact solutions for validation. The results showed that the FEDGM can improve computational efficiency while maintaining good accuracy in solving the variable-order advection–diffusion model, as compared to the CNDS.