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Growth Equation of the General Fractional Calculus

Author

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  • Anatoly N. Kochubei

    (Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska 3, 01024 Kyiv, Ukraine)

  • Yuri Kondratiev

    (Department of Mathematics, University of Bielefeld, D-33615 Bielefeld, Germany)

Abstract

We consider the Cauchy problem ( D ( k ) u ) ( t ) = λ u ( t ) , u ( 0 ) = 1 , where D ( k ) is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory 71 (2011), 583–600), λ > 0 . The solution is a generalization of the function t ↦ E α ( λ t α ) , where 0 < α < 1 , E α is the Mittag–Leffler function. The asymptotics of this solution, as t → ∞ , are studied.

Suggested Citation

  • Anatoly N. Kochubei & Yuri Kondratiev, 2019. "Growth Equation of the General Fractional Calculus," Mathematics, MDPI, vol. 7(7), pages 1-8, July.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:7:p:615-:d:247490
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    References listed on IDEAS

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    1. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
    2. Tarasov, Vasily E. & Tarasova, Valentina V., 2018. "Macroeconomic models with long dynamic memory: Fractional calculus approach," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 466-486.
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    Cited by:

    1. Vasily E. Tarasov, 2023. "General Fractional Calculus in Multi-Dimensional Space: Riesz Form," Mathematics, MDPI, vol. 11(7), pages 1-20, March.
    2. Yuri Luchko, 2022. "Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense," Mathematics, MDPI, vol. 10(6), pages 1-24, March.
    3. Giacomo Ascione & Nikolai Leonenko & Enrica Pirozzi, 2022. "Non-local Solvable Birth–Death Processes," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1284-1323, June.
    4. Vasily E. Tarasov, 2021. "General Fractional Dynamics," Mathematics, MDPI, vol. 9(13), pages 1-26, June.
    5. Vasily E. Tarasov, 2023. "Multi-Kernel General Fractional Calculus of Arbitrary Order," Mathematics, MDPI, vol. 11(7), pages 1-32, April.
    6. Giacomo Ascione, 2020. "On the Construction of Some Deterministic and Stochastic Non-Local SIR Models," Mathematics, MDPI, vol. 8(12), pages 1-28, November.
    7. Giacomo Ascione & Enrica Pirozzi, 2021. "Generalized Fractional Calculus for Gompertz-Type Models," Mathematics, MDPI, vol. 9(17), pages 1-32, September.
    8. Tarasov, Vasily E., 2023. "Nonlocal statistical mechanics: General fractional Liouville equations and their solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    9. Mohammed Al-Refai & Yuri Luchko, 2023. "The General Fractional Integrals and Derivatives on a Finite Interval," Mathematics, MDPI, vol. 11(4), pages 1-13, February.
    10. Vasily E. Tarasov, 2021. "General Fractional Calculus: Multi-Kernel Approach," Mathematics, MDPI, vol. 9(13), pages 1-14, June.
    11. Vasily E. Tarasov & Svetlana S. Tarasova, 2020. "Fractional Derivatives and Integrals: What Are They Needed For?," Mathematics, MDPI, vol. 8(2), pages 1-22, January.
    12. Vasily E. Tarasov, 2022. "General Non-Local Continuum Mechanics: Derivation of Balance Equations," Mathematics, MDPI, vol. 10(9), pages 1-43, April.

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