TY - JOUR
T1 - Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations
AU - Zhang , Jia-Li
AU - Fang , Zhi-Wei
AU - Sun , Hai-Wei
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 200
EP - 226
PY - 2022
DA - 2022/02
SN - 15
DO - http://doi.org/10.4208/nmtma.OA-2021-0148
UR - https://global-sci.org/intro/article_detail/nmtma/20227.html
KW - Variable-order Caputo fractional derivative, exponential-sum-approximation method, fast algorithm, time-fractional sub-diffusion equation, stability and convergence.
AB - <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>