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arXiv · 2102.02960

Fast second-order evaluation for variable-order Caputo fractional derivative with applications to fractional sub-diffusion equations

Jia-li Zhang·Zhi-wei Fang·Hai-wei Sun
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Abstract

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.

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Jia-li Zhang, Zhi-wei Fang, Hai-wei Sun. 2021-02-05. Fast second-order evaluation for variable-order Caputo fractional derivative with applications to fractional sub-diffusion equations. https://doi.org/10.4208/nmtma.oa-2021-0148

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