A simple and fast finite difference method for the integral fractional Laplacian of variable order
<|reference_start|>A simple and fast finite difference method for the integral fractional Laplacian of variable order: For the fractional Laplacian of variable order, an efficient and accurate numerical evaluation in multi-dimension is a challenge for the nature of a singular integral. We propose a simple and easy-to-implement finite difference scheme for the multi-dimensional variable-order fractional Laplacian defined by a hypersingular integral. We prove that the scheme is of second-order convergence and apply the developed finite difference scheme to solve various equations with the variable-order fractional Laplacian. We present a fast solver with quasi-linear complexity of the scheme for computing variable-order fractional Laplacian and corresponding PDEs. Several numerical examples demonstrate the accuracy and efficiency of our algorithm and verify our theory.<|reference_end|>
@article{hao2024a,
    title={A simple and fast finite difference method for the integral fractional
  Laplacian of variable order},
    author={Zhaopeng Hao and Siyuan Shi and Zhongqiang Zhang and Rui Du},
    journal={arXiv preprint arXiv:2406.10524},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.10524},
    primaryClass={math.NA cs.NA}
}