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arXiv · 2406.10524

A simple and fast finite difference method for the integral fractional Laplacian of variable order

Zhaopeng Hao·Siyuan Shi·Zhongqiang Zhang·Rui Du
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Abstract

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.

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Zhaopeng Hao, Siyuan Shi, Zhongqiang Zhang, Rui Du. 2024-06-15. A simple and fast finite difference method for the integral fractional Laplacian of variable order. https://arxiv.org/abs/2406.10524

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