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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. 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.
    2. 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.
    3. 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.
    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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    Cited by:

    1. Marina Plekhanova & Guzel Baybulatova, 2020. "Multi-Term Fractional Degenerate Evolution Equations and Optimal Control Problems," Mathematics, MDPI, vol. 8(4), pages 1-9, April.
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    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. Faïçal Ndaïrou & Delfim F. M. Torres, 2023. "Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems," Mathematics, MDPI, vol. 11(19), pages 1-12, October.
    5. Isah, Sunday Simon & Fernandez, Arran & Özarslan, Mehmet Ali, 2023. "On bivariate fractional calculus with general univariate analytic kernels," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    6. Shiri, Babak & Baleanu, Dumitru, 2023. "All linear fractional derivatives with power functions’ convolution kernel and interpolation properties," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    7. Kucche, Kishor D. & Mali, Ashwini D. & Fernandez, Arran & Fahad, Hafiz Muhammad, 2022. "On tempered Hilfer fractional derivatives with respect to functions and the associated fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    8. Artion Kashuri & Muhammad Samraiz & Gauhar Rahman & Zareen A. Khan, 2022. "Some New Beesack–Wirtinger-Type Inequalities Pertaining to Different Kinds of Convex Functions," Mathematics, MDPI, vol. 10(5), pages 1-20, February.
    9. Raza, Ali & Ghaffari, Abuzar & Khan, Sami Ullah & Haq, Absar Ul & Khan, M. Ijaz & Khan, M. Riaz, 2022. "Non-singular fractional computations for the radiative heat and mass transfer phenomenon subject to mixed convection and slip boundary effects," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    10. El-Nabulsi, Rami Ahmad & Khalili Golmankhaneh, Alireza & Agarwal, Praveen, 2022. "On a new generalized local fractal derivative operator," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    11. Dumitru Baleanu & Arran Fernandez & Ali Akgül, 2020. "On a Fractional Operator Combining Proportional and Classical Differintegrals," Mathematics, MDPI, vol. 8(3), pages 1-13, March.
    12. Sivashankar, M. & Sabarinathan, S. & Nisar, Kottakkaran Sooppy & Ravichandran, C. & Kumar, B.V. Senthil, 2023. "Some properties and stability of Helmholtz model involved with nonlinear fractional difference equations and its relevance with quadcopter," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    13. Rahaman, Mostafijur & Mondal, Sankar Prasad & Alam, Shariful & Metwally, Ahmed Sayed M. & Salahshour, Soheil & Salimi, Mehdi & Ahmadian, Ali, 2022. "Manifestation of interval uncertainties for fractional differential equations under conformable derivative," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    14. Oscar Martínez-Fuentes & Fidel Meléndez-Vázquez & Guillermo Fernández-Anaya & José Francisco Gómez-Aguilar, 2021. "Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities," Mathematics, MDPI, vol. 9(17), pages 1-29, August.

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