An Efficient QSC Approximation of Variable-Order Time-Fractional Mobile-Immobile Diffusion Equations with Variably Diffusive Coefficients

In this paper, we propose a QSC- L 1 method to solve the two-dimensional variable-order time-fractional mobile-immobile diffusion (TF-MID) equations with variably diffusive coefficients, in which the quadratic spline collocation (QSC) method is employed for the spatial discretization, and the classical L 1 formula is used for the temporal discretization. We show that the method is unconditionally stable and convergent with first-order in time and second-order in space with respect to some discrete and continuous L^2 norms. Then, combined with the reduced basis technique, an efficient QSC- L 1-RB method is proposed to improve the computational efficiency. Numerical examples are attached to verify the convergence orders, and also the method is applied to identify parameters of the variable-order TF-MID equations. Numerical results confirm the contributions of the reduced basis technique, even when the observation data is contaminated by some levels of random noise.