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Fractional Derivatives: The Perspective of System Theory

Author

Listed:
  • Manuel Duarte Ortigueira

    (CTS–UNINOVA and DEE of NOVA, School of Science and Technology, Nova University of Lisbon, Campus da FCT da UNL, Quinta da Torre, 2829–516 Caparica, Portugal)

  • José Tenreiro Machado

    (Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, R. Dr. António Bernardino de Almeida, 431, 4249–015 Porto, Portugal)

Abstract

This paper addresses the present day problem of multiple proposals for operators under the umbrella of “fractional derivatives”. Several papers demonstrated that various of those “novel” definitions are incorrect. Here the classical system theory is applied to develop a unified framework to clarify this important topic in Fractional Calculus.

Suggested Citation

  • Manuel Duarte Ortigueira & José Tenreiro Machado, 2019. "Fractional Derivatives: The Perspective of System Theory," Mathematics, MDPI, vol. 7(2), pages 1-14, February.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:2:p:150-:d:203549
    as

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    References listed on IDEAS

    as
    1. 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.
    2. Vasily E. Tarasov, 2015. "Exact Discrete Analogs of Derivatives of Integer Orders: Differences as Infinite Series," Journal of Mathematics, Hindawi, vol. 2015, pages 1-8, November.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Duarte Valério & Manuel D. Ortigueira, 2023. "Variable-Order Fractional Scale Calculus," Mathematics, MDPI, vol. 11(21), pages 1-13, November.
    2. Zaheer Masood & Muhammad Asif Zahoor Raja & Naveed Ishtiaq Chaudhary & Khalid Mehmood Cheema & Ahmad H. Milyani, 2021. "Fractional Dynamics of Stuxnet Virus Propagation in Industrial Control Systems," Mathematics, MDPI, vol. 9(17), pages 1-27, September.
    3. Duarte Valério & Manuel D. Ortigueira & António M. Lopes, 2022. "How Many Fractional Derivatives Are There?," Mathematics, MDPI, vol. 10(5), pages 1-18, February.
    4. María Pilar Velasco & David Usero & Salvador Jiménez & Luis Vázquez & José Luis Vázquez-Poletti & Mina Mortazavi, 2020. "About Some Possible Implementations of the Fractional Calculus," Mathematics, MDPI, vol. 8(6), pages 1-22, June.
    5. Iqbal M. Batiha & Shameseddin Alshorm & Adel Ouannas & Shaher Momani & Osama Y. Ababneh & Meaad Albdareen, 2022. "Modified Three-Point Fractional Formulas with Richardson Extrapolation," Mathematics, MDPI, vol. 10(19), pages 1-16, September.

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