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This paper considers a two-step fourth-order modified explicitEuler/Crank-Nicolson numerical method for solving the time-variable fractionalmobile-immobile advection-dispersion model subjects to suitable initial andboundary conditions. Both stability and error estimates of the new approach aredeeply analyzed in the $L^{\infty}(0,T;L^{2})$-norm. The theoretical studiesshow that the proposed technique is unconditionally stable with convergence oforder $O(k+h^{4})$, where $h$ and $k$ are space step and time step,respectively. This result indicate that the two-step fourth-order formulationis more efficient than a broad range of numerical schemes widely studied in theliterature for the considered problem. Numerical experiments are performed toverify the unconditional stability and convergence rate of the developedalgorithm.

Unconditional Stability Of A Two-Step Fourth-Order Modified Explicit  Euler/Crank-Nicolson Approach For Solving Time-Variable Fractional  Mobile-Immobile Advection-Dispersion Equation