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Variable-Order Fractional Laplacian and its Accurate and Efficient Computations with Meshfree Methods

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Abstract

The variable-order fractional Laplacian plays an important role in the study of heterogeneous systems. In this paper, we propose the first numerical methods for the variable-order Laplacian \((-\varDelta )^{\alpha (\textbf{x} )/2}\) with \(0 < \alpha (\textbf{x} ) \le 2\), which will also be referred as the variable-order fractional Laplacian if \(\alpha (\textbf{x} )\) is strictly less than 2. We present a class of hypergeometric functions whose variable-order Laplacian can be analytically expressed. Building on these analytical results, we design the meshfree methods based on globally supported radial basis functions (RBFs), including Gaussian, generalized inverse multiquadric, and Bessel-type RBFs, to approximate the variable-order Laplacian \((-\varDelta )^{\alpha (\textbf{x} )/2}\). Our meshfree methods integrate the advantages of both pseudo-differential and hypersingular integral forms of the variable-order fractional Laplacian, and thus avoid numerically approximating the hypersingular integral. Moreover, our methods are simple and flexible of domain geometry, and their computer implementation remains the same for any dimension \(d \ge 1\). Compared to finite difference methods, our methods can achieve a desired accuracy with much fewer points. This fact makes our method much attractive for problems involving variable-order fractional Laplacian where the number of points required is a critical cost. We then apply our method to study solution behaviors of variable-order fractional PDEs arising in different fields, including transition of waves between classical and fractional media, and coexistence of anomalous and normal diffusion in both diffusion equation and the Allen–Cahn equation. These results would provide insights for further understanding and applications of variable-order fractional derivatives.

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Funding

This work was partially supported by the US National Science Foundation under Grant Numbers DMS-1913293 and DMS-1953177.

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  1. Department of Pharmacology, University of California Davis, Davis, CA, 95616-8636, USA

    Yixuan Wu

  2. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, 65409, USA

    Yanzhi Zhang

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  1. Yixuan Wu
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Both authors contributed to the study conception and design. The first draft of the manuscript was written by Yixuan Wu and both authors commented on previous versions of the manuscript. Both authors read and approved the final manuscript.

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Correspondence to Yanzhi Zhang.

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Wu, Y., Zhang, Y. Variable-Order Fractional Laplacian and its Accurate and Efficient Computations with Meshfree Methods. J Sci Comput 99, 18 (2024). https://doi.org/10.1007/s10915-024-02472-x

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