@Article{NMTMA-15-200,
author = {Zhang , Jia-LiFang , Zhi-Wei and Sun , Hai-Wei},
title = {Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2022},
volume = {15},
number = {1},
pages = {200--226},
abstract = {<p style="text-align: justify;">In this paper, we propose a fast second-order approximation to the variable-order (VO) Caputo fractional derivative, which is developed based on $L2$-$1_σ$ formula and the exponential-sum-approximation technique. The fast evaluation
method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative. This fast
algorithm is applied to construct a relevant fast temporal second-order and spatial
fourth-order scheme ($FL2$-$1_σ$ scheme) for the multi-dimensional VO time-fractional
sub-diffusion equations. Theoretically, $FL2$-$1_σ$ scheme is proved to fulfill the similar
properties of the coefficients as those of the well-studied $L2$-$1_σ$ scheme. Therefore, $FL2$-$1_σ$ scheme is strictly proved to be unconditionally stable and convergent.
A sharp decrease in the computational cost and the acting memory is shown in the
numerical examples to demonstrate the efficiency of the proposed method.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2021-0148},
url = {http://global-sci.org/intro/article_detail/nmtma/20227.html}
}