Numerical method for the estimation of the fractional parameters in the fractional mobile/immobile advection–diffusion model

In this paper, we mainly investigate the numerical identification of the fractional parameters in the fractional mobile/immobile advection–diffusion model. For the direct problem, two efficient numerical schemes with second-order spatial accuracy and fourth-order spatial accuracy are derived, respectively. The unconditional stability and convergence of the numerical schemes are analysed rigorously by the Fourier analysis method. For the inverse problem, we first propose a novel numerical scheme to compute the fractional sensitivity matrix, and then employ the nonlinear Levenberg–Marquardt iterative method to estimate the fractional parameters. Finally, numerical examples are given to illustrate the effectiveness and accuracy of the proposed numerical schemes, and the numerical identification of the fractional parameters is also presented in tabular form, which demonstrate the effectiveness of the proposed numerical method for the identification of the fractional parameters in the fractional mobile/immobile advection–diffusion model.