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On a Fractional Operator Combining Proportional and Classical Differintegrals

Author

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  • Dumitru Baleanu

    (Department of Mathematics, Cankaya University, 06530 Balgat, Ankara, Turkey
    Institute of Space Sciences, R76900 Magurele-Bucharest, Romania
    Department of Medical Research, China Medical University, Taichung 40402, Taiwan)

  • Arran Fernandez

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

  • Ali Akgül

    (Department of Mathematics, Faculty of Arts and Sciences, Siirt University, TR-56100 Siirt, Turkey)

Abstract

The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t ) , by a fractional integral operator applied to the derivative f ′ ( t ) . We define a new fractional operator by substituting for this f ′ ( t ) a more general proportional derivative. This new operator can also be written as a Riemann–Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann–Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:360-:d:329555
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    References listed on IDEAS

    as
    1. Dumitru Baleanu & Arran Fernandez, 2019. "On Fractional Operators and Their Classifications," Mathematics, MDPI, vol. 7(9), pages 1-10, September.
    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. Fernandez, Arran & Baleanu, Dumitru & Fokas, Athanassios S., 2018. "Solving PDEs of fractional order using the unified transform method," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 738-749.
    4. Atangana, Abdon, 2017. "Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 396-406.
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    Cited by:

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    2. Shiri, Babak & Baleanu, Dumitru, 2023. "All linear fractional derivatives with power functions’ convolution kernel and interpolation properties," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    3. Xuan, Liu & Ahmad, Shabir & Ullah, Aman & Saifullah, Sayed & Akgül, Ali & Qu, Haidong, 2022. "Bifurcations, stability analysis and complex dynamics of Caputo fractal-fractional cancer model," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    4. Acay, Bahar & Inc, Mustafa & Mustapha, Umar Tasiu & Yusuf, Abdullahi, 2021. "Fractional dynamics and analysis for a lana fever infectious ailment with Caputo operator," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    5. Saifullah, Sayed & Ali, Amir & Franc Doungmo Goufo, Emile, 2021. "Investigation of complex behaviour of fractal fractional chaotic attractor with mittag-leffler Kernel," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
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    7. Nisar, Kottakkaran Sooppy & Farman, Muhammad & Hincal, Evren & Shehzad, Aamir, 2023. "Modelling and analysis of bad impact of smoking in society with Constant Proportional-Caputo Fabrizio operator," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).

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