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

摘要

Abstract In this paper, we propose a QSC-L1 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 L1 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 L2documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^2$$end{document} norms. Then, combined with the reduced basis technique, an efficient QSC-L1-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.