IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i2p150-d203549.html
   My bibliography  Save this article

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

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/2/150/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/7/2/150/
    Download Restriction: no
    ---><---

    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)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. 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.
    4. Duarte Valério & Manuel D. Ortigueira & António M. Lopes, 2022. "How Many Fractional Derivatives Are There?," Mathematics, MDPI, vol. 10(5), pages 1-18, February.
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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).
    3. Michelitsch, T.M. & Collet, B.A. & Riascos, A.P. & Nowakowski, A.F. & Nicolleau, F.C.G.A., 2016. "A fractional generalization of the classical lattice dynamics approach," Chaos, Solitons & Fractals, Elsevier, vol. 92(C), pages 43-50.
    4. Vasily E. Tarasov, 2016. "Exact Discrete Analogs of Canonical Commutation and Uncertainty Relations," Mathematics, MDPI, vol. 4(3), pages 1-13, June.
    5. 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.
    6. Vasily E. Tarasov, 2023. "General Fractional Calculus in Multi-Dimensional Space: Riesz Form," Mathematics, MDPI, vol. 11(7), pages 1-20, March.
    7. 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.
    8. Tarasova, Valentina V. & Tarasov, Vasily E., 2017. "Logistic map with memory from economic model," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 84-91.
    9. 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.
    10. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Accelerators in Macroeconomics: Comparison of Discrete and Continuous Approaches," American Journal of Economics and Business Administration, Science Publications, vol. 9(3), pages 47-55, November.
    11. 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.
    12. Vasily E. Tarasov, 2024. "Exact Finite-Difference Calculus: Beyond Set of Entire Functions," Mathematics, MDPI, vol. 12(7), pages 1-37, March.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. 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).
    18. 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).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:2:p:150-:d:203549. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.