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Remarks on the Generalized Fractional Laplacian Operator

Author

Listed:
  • Chenkuan Li

    (Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada)

  • Changpin Li

    (Department of Mathematics, Shanghai University, Shanghai 200444, China)

  • Thomas Humphries

    (Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada)

  • Hunter Plowman

    (Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada)

Abstract

The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighbouring or nearby sites, as well as perform excursions to remote sites by way of Lévy flights. The fractional Laplacian has many applications in the boundary behaviours of solutions to differential equations. The goal of this paper is to investigate the half-order Laplacian operator ( − Δ ) 1 2 in the distributional sense, based on the generalized convolution and Temple’s delta sequence. Several interesting examples related to the fractional Laplacian operator of order 1 / 2 are presented with applications to differential equations, some of which cannot be obtained in the classical sense by the standard definition of the fractional Laplacian via Fourier transform.

Suggested Citation

  • Chenkuan Li & Changpin Li & Thomas Humphries & Hunter Plowman, 2019. "Remarks on the Generalized Fractional Laplacian Operator," Mathematics, MDPI, vol. 7(4), pages 1-17, March.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:4:p:320-:d:218225
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    References listed on IDEAS

    as
    1. Chen, An & Du, Qiang & Li, Changpin & Zhou, Zhi, 2017. "Asymptotically compatible schemes for space-time nonlocal diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 361-371.
    2. Rama Cont & Ekaterina Voltchkova, 2005. "A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models," Post-Print halshs-00445645, HAL.
    3. Hu, Ye & Li, Changpin & Li, Hefeng, 2017. "The finite difference method for Caputo-type parabolic equation with fractional Laplacian: One-dimension case," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 319-326.
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    Cited by:

    1. Chenkuan Li & Joshua Beaudin, 2020. "On the Generalized Riesz Derivative," Mathematics, MDPI, vol. 8(7), pages 1-22, July.
    2. El-Nabulsi, Rami Ahmad & Anukool, Waranont, 2023. "A family of nonlinear Schrodinger equations and their solitons solutions," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    3. Nigus Demelash Melaku & Ali Fares & Ripendra Awal, 2023. "Exploring the Impact of Winter Storm Uri on Power Outage, Air Quality, and Water Systems in Texas, USA," Sustainability, MDPI, vol. 15(5), pages 1-19, February.

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