A QSC method for fractional subdiffusion equations with fractional boundary conditions and its application in parameters identification

摘要

A quadratic spline collocation (QSC) method combined with L1 time discretization, named QSC-L1, is proposed to solve fractional subdiffusion equations with artificial boundary conditions. A novel norm-based stability and convergence analysis is carefully discussed, which shows that the QSC-L1 method is unconditionally stable in a discrete space-time norm, and has a convergence order O(τ~(2-α) + h~2), where τ and h are the temporal and spatial step sizes, respectively. Then, based on fast evaluation of the Caputo fractional derivative (see, Jiang et al., 2017), a fast version of QSC-L1 which is called QSC-FL1 is proposed to improve the computational efficiency. Two numerical examples are provided to support the theoretical results. Furthermore, an inverse problem is considered, in which some parameters of the fractional subdiffusion equations need to be identified. A Levenberg-Marquardt (L-M) method equipped with the QSC-FL1 method is developed for solving the inverse problem. Numerical tests show the effectiveness of the method even for the case that the observation data is contaminated by some levels of random noise.