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On Fractional Operators and Their Classifications

Author

Listed:
  • Dumitru Baleanu

    (Department of Mathematics, Cankaya University, Balgat 06530, Ankara, Turkey
    Institute of Space Sciences, R76900 Magurele-Bucharest, Romania)

  • Arran Fernandez

    (Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, 99628 Famagusta, Northern Cyprus, via Mersin-10, Turkey)

Abstract

Fractional calculus dates its inception to a correspondence between Leibniz and L’Hopital in 1695, when Leibniz described “paradoxes” and predicted that “one day useful consequences will be drawn” from them. In today’s world, the study of non-integer orders of differentiation has become a thriving field of research, not only in mathematics but also in other parts of science such as physics, biology, and engineering: many of the “useful consequences” predicted by Leibniz have been discovered. However, the field has grown so far that researchers cannot yet agree on what a “fractional derivative” can be. In this manuscript, we suggest and justify the idea of classification of fractional calculus into distinct classes of operators.

Suggested Citation

  • Dumitru Baleanu & Arran Fernandez, 2019. "On Fractional Operators and Their Classifications," Mathematics, MDPI, vol. 7(9), pages 1-10, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:830-:d:265366
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    References listed on IDEAS

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    1. Fernandez, Arran & Özarslan, Mehmet Ali & Baleanu, Dumitru, 2019. "On fractional calculus with general analytic kernels," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 248-265.
    2. Atangana, Abdon & Gómez-Aguilar, J.F., 2018. "Fractional derivatives with no-index law property: Application to chaos and statistics," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 516-535.
    3. Uçar, Sümeyra & Uçar, Esmehan & Özdemir, Necati & Hammouch, Zakia, 2019. "Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 300-306.
    4. Hari M. Srivastava & Arran Fernandez & Dumitru Baleanu, 2019. "Some New Fractional-Calculus Connections between Mittag–Leffler Functions," Mathematics, MDPI, vol. 7(6), pages 1-10, May.
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