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How Many Fractional Derivatives Are There?

Author

Listed:
  • Duarte Valério

    (IDMEC, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal)

  • Manuel D. Ortigueira

    (Centre of Technology and Systems–UNINOVA and Department of Electrical Engineering, NOVA School of Science and Technology of NOVA University of Lisbon, Quinta da Torre, 2829-516 Caparica, Portugal)

  • António M. Lopes

    (LAETA/INEGI, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal)

Abstract

In this paper, we introduce a unified fractional derivative , defined by two parameters (order and asymmetry). From this, all the interesting derivatives can be obtained. We study the one-sided derivatives and show that most known derivatives are particular cases. We consider also some myths of Fractional Calculus and false fractional derivatives. The results are expected to contribute to limit the appearance of derivatives that differ from existing ones just because they are defined on distinct domains, and to prevent the ambiguous use of the concept of fractional derivative.

Suggested Citation

  • Duarte Valério & Manuel D. Ortigueira & António M. Lopes, 2022. "How Many Fractional Derivatives Are There?," Mathematics, MDPI, vol. 10(5), pages 1-18, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:5:p:737-:d:759104
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    References listed on IDEAS

    as
    1. Manuel Duarte Ortigueira & José Tenreiro Machado, 2019. "Fractional Derivatives: The Perspective of System Theory," Mathematics, MDPI, vol. 7(2), pages 1-14, February.
    2. Manuel Duarte Ortigueira & José Tenreiro Machado, 2020. "Revisiting the 1D and 2D Laplace Transforms," Mathematics, MDPI, vol. 8(8), pages 1-24, August.
    3. Chen, W., 2006. "Time–space fabric underlying anomalous diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 923-929.
    4. Edmundo Capelas de Oliveira & José António Tenreiro Machado, 2014. "A Review of Definitions for Fractional Derivatives and Integral," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-6, June.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. J. Alberto Conejero & Jonathan Franceschi & Enric Picó-Marco, 2022. "Fractional vs. Ordinary Control Systems: What Does the Fractional Derivative Provide?," Mathematics, MDPI, vol. 10(15), pages 1-18, August.
    2. Khalili Golmankhaneh, Alireza & Tejado, Inés & Sevli, Hamdullah & Valdés, Juan E. Nápoles, 2023. "On initial value problems of fractal delay equations," Applied Mathematics and Computation, Elsevier, vol. 449(C).
    3. Torres-Hernandez, A. & Brambila-Paz, F. & Montufar-Chaveznava, R., 2022. "Acceleration of the order of convergence of a family of fractional fixed-point methods and its implementation in the solution of a nonlinear algebraic system related to hybrid solar receivers," Applied Mathematics and Computation, Elsevier, vol. 429(C).
    4. Manuel D. Ortigueira, 2022. "A New Look at the Initial Condition Problem," Mathematics, MDPI, vol. 10(10), pages 1-17, May.
    5. Henrique Paz & Duarte Valério, 2022. "Generalization of Reset Controllers to Fractional Orders," Mathematics, MDPI, vol. 10(24), pages 1-18, December.

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