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On the Generalized Riesz Derivative

Author

Listed:
  • Chenkuan Li

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

  • Joshua Beaudin

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

Abstract

The goal of this paper is to construct an integral representation for the generalized Riesz derivative R Z D x 2 s u ( x ) for k < s < k + 1 with k = 0 , 1 , ⋯ , which is proved to be a one-to-one and linearly continuous mapping from the normed space W k + 1 ( R ) to the Banach space C ( R ) . In addition, we show that R Z D x 2 s u ( x ) is continuous at the end points and well defined for s = 1 2 + k . Furthermore, we extend the generalized Riesz derivative R Z D x 2 s u ( x ) to the space C k ( R n ) , where k is an n -tuple of nonnegative integers, based on the normalization of distribution and surface integrals over the unit sphere. Finally, several examples are presented to demonstrate computations for obtaining the generalized Riesz derivatives.

Suggested Citation

  • Chenkuan Li & Joshua Beaudin, 2020. "On the Generalized Riesz Derivative," Mathematics, MDPI, vol. 8(7), pages 1-22, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1089-:d:379899
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    References listed on IDEAS

    as
    1. 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.
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    Cited by:

    1. Alqhtani, Manal & Owolabi, Kolade M. & Saad, Khaled M. & Pindza, Edson, 2022. "Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).

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