TY - JOUR
T1 - Analysis and Efficient Implementation of Quadratic Spline Collocation ADI Methods for Variable-Order Time-Fractional Mobile-Immobile Diffusion Equations
AU - Liu , Jun
AU - Fu , Hongfei
AU - Zhang , Bingyin
AU - Zhang , Jiansong
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 226
EP - 258
PY - 2025
DA - 2025/04
SN - 18
DO - http://doi.org/10.4208/nmtma.OA-2024-0086
UR - https://global-sci.org/intro/article_detail/nmtma/23948.html
KW - Time-fractional mobile-immobile diffusion equations, variable-order, QSC method, ADI, stability and convergence, fast implementation.
AB - <p style="text-align: justify;">In this paper, a quadratic spline collocation (QSC) method combined with $L1$ time discretization in the framework of alternating direction implicit (ADI) approach, namely ADI-QSC-$L$1 method, is developed to solve the variable-order time-fractional mobile-immobile diffusion equations in multi-dimensional spaces. Discrete $L_2$ norm-based stability and error estimate are carefully discussed, which
show that the proposed method is unconditionally stable and convergent with first-order accuracy in time and second-order accuracy in space. Then, based on the
exponential-sum-approximation technique for the fast evaluation of the variable-order Caputo fractional derivative, an efficient implementation strategy of the ADI-QSC-$L1$ method, named ADI-QSC-${\rm F}L1$ is presented, which further improves the
computational efficiency by reduced memory requirement and computational cost.
Finally, numerical examples are provided to support both the theoretical results and
efficiency of the developed method.</p>