A fast Fourier spectral exponential time-differencing method for solving the time-fractional mobile–immobile advection–dispersion equation

In this paper, we propose a fast and accurate numerical method for solving the time-fractional mobile–immobile advection–dispersion equation with Caputo derivative of order α for . The main advantage of this method is its order of accuracy, which is 3-α . We discretize the problem in space by the Fourier spectral method and time by the fourth-order exponential time-differencing Runge–Kutta method and a high-order approximation for Caputo derivative. We eliminate the first and second spatial derivatives of the well-known equation by the properties of the Fourier spectral method. We also prove the convergence of this method. Several numerical examples are provided to illustrate the efficiency and accuracy of the proposed method. We show that the temporal convergence rates for different values of α are more than two and conclude the effectiveness of the method for solving the well-known equation.