TURK 2025

In this study, we investigate numerical solutions of the variable-order fractional mobile–immobile advection–dispersion model in the Caputo sense using the collocation finite element method. The fractional-order time derivative and the classical-order space derivative are discretized using the \L1 algorithm and the Crank–Nicolson approach, respectively. An approximate solution is then constructed with the help of trigonometric cubic B-splines and time-dependent parameters, converting the discretized equation into a linear system of algebraic equations. Solving this system yields the numerical results and the associated errors in the \L_2 and \L_\infty norms. Additionally, comparison tables are presented to evaluate the results against those obtained with other methods. The efficiency and accuracy of the proposed method are demonstrated through some examples. These results indicate that the proposed method is both efficient and accurate.
[1] M. M. Meerschaert , C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations. Appl Numer Math 2006;56:80-90 .
[2] S. B. Yuste, L. Acedo, An explicit finite difference method and a new von neu- mann-type stability analysis for fractional diffusion equations, SIAM J Numer Anal 2005;42:1862-74.
[3] Q. Yang , I. Turner , F. Liu , Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation. ANZIAM J 20 08;50:80 0-14 .
[4] H. Zhang et al., A novel numerical method for the time variable fractional order mobile--immobile advection--dispersion model, Computers \& Mathematics with Applications 66.5 (2013): 693-701.
[5] Z. Liu, L. Xiaoli, A Crank--Nicolson difference scheme for the time variable fractional mobile--immobile advection—dispersion equation, Journal of Applied Mathematics and Computing 56.1 (2018): 391-410.
[6] M. Yang, L. Lijie, W. Leilei , Local Discontinuous Galerkin Method for the Variable-Order Fractional Mobile-Immobile Advection-Dispersion Equation, Computational Mathematics and Mathematical Physics 65.2 (2025): 308-319.