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Riesz potential operators and inverses via fractional centred derivatives

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  • Manuel Duarte Ortigueira

Abstract

Fractional centred differences and derivatives definitions are proposed, generalizing to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalizations of the Cauchy integral derivative. To compute this integral, a special two-straight-line path was used. With this the referred integrals lead to the well-known Riesz potential operators and their inverses that emerge as true fractional centred derivatives, but that can be computed through summations similar to the well-known Grünwald-Letnikov derivatives.

Suggested Citation

  • Manuel Duarte Ortigueira, 2006. "Riesz potential operators and inverses via fractional centred derivatives," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2006, pages 1-12, August.
    • Handle: RePEc:hin:jijmms:048391
    DOI: 10.1155/IJMMS/2006/48391
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    Cited by:

    1. Adán J. Serna-Reyes & Jorge E. Macías-Díaz, 2021. "A Mass- and Energy-Conserving Numerical Model for a Fractional Gross–Pitaevskii System in Multiple Dimensions," Mathematics, MDPI, vol. 9(15), pages 1-31, July.
    2. Jorge E. Macías-Díaz & Nuria Reguera & Adán J. Serna-Reyes, 2021. "An Efficient Discrete Model to Approximate the Solutions of a Nonlinear Double-Fractional Two-Component Gross–Pitaevskii-Type System," Mathematics, MDPI, vol. 9(21), pages 1-14, October.
    3. Zhang, Xue & Gu, Xian-Ming & Zhao, Yong-Liang & Li, Hu & Gu, Chuan-Yun, 2024. "Two fast and unconditionally stable finite difference methods for Riesz fractional diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 462(C).
    4. Fu, Yayun & Song, Yongzhong & Wang, Yushun, 2019. "Maximum-norm error analysis of a conservative scheme for the damped nonlinear fractional Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 206-223.
    5. Macías-Díaz, J.E., 2018. "A numerically efficient Hamiltonian method for fractional wave equations," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 231-248.
    6. Owolabi, Kolade M. & Gómez-Aguilar, J.F., 2018. "Numerical simulations of multilingual competition dynamics with nonlocal derivative," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 175-182.
    7. Manuel Duarte Ortigueira & José Tenreiro Machado, 2019. "Fractional Derivatives: The Perspective of System Theory," Mathematics, MDPI, vol. 7(2), pages 1-14, February.
    8. Joel Alba-Pérez & Jorge E. Macías-Díaz, 2019. "Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion," Mathematics, MDPI, vol. 7(12), pages 1-31, December.
    9. Adán J. Serna-Reyes & Jorge E. Macías-Díaz & Nuria Reguera, 2021. "A Convergent Three-Step Numerical Method to Solve a Double-Fractional Two-Component Bose–Einstein Condensate," Mathematics, MDPI, vol. 9(12), pages 1-22, June.
    10. Qiang Yu & Viktor Vegh & Fawang Liu & Ian Turner, 2015. "A Variable Order Fractional Differential-Based Texture Enhancement Algorithm with Application in Medical Imaging," PLOS ONE, Public Library of Science, vol. 10(7), pages 1-35, July.
    11. Martínez, Romeo & Macías-Díaz, Jorge E. & Sheng, Qin, 2022. "A nonlinear discrete model for approximating a conservative multi-fractional Zakharov system: Analysis and computational simulations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 1-21.
    12. Owolabi, Kolade M. & Jain, Sonal, 2023. "Spatial patterns through diffusion-driven instability in modified predator–prey models with chaotic behaviors," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    13. 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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