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General Fractional Integrals and Derivatives with the Sonine Kernels

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  • Yuri Luchko

    (Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany)

Abstract

In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the n -fold general fractional integrals and derivatives and study their properties.

Suggested Citation

  • Yuri Luchko, 2021. "General Fractional Integrals and Derivatives with the Sonine Kernels," Mathematics, MDPI, vol. 9(6), pages 1-17, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:6:p:594-:d:514427
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    References listed on IDEAS

    as
    1. Stefan G. Samko & Rogério P. Cardoso, 2003. "Integral equations of the first kind of Sonine type," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-24, January.
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    Cited by:

    1. Maryam Al-Kandari & Latif A-M. Hanna & Yuri Luchko, 2022. "Operational Calculus for the General Fractional Derivatives of Arbitrary Order," Mathematics, MDPI, vol. 10(9), pages 1-17, May.
    2. Vasily E. Tarasov, 2023. "General Fractional Calculus in Multi-Dimensional Space: Riesz Form," Mathematics, MDPI, vol. 11(7), pages 1-20, March.
    3. 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.
    4. 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).
    5. Yuri Luchko, 2023. "Fractional Integrals and Derivatives: “True” versus “False”," Mathematics, MDPI, vol. 11(13), pages 1-2, July.
    6. Vasily E. Tarasov, 2023. "Multi-Kernel General Fractional Calculus of Arbitrary Order," Mathematics, MDPI, vol. 11(7), pages 1-32, April.
    7. Vasily E. Tarasov, 2022. "Fractional Dynamics with Depreciation and Obsolescence: Equations with Prabhakar Fractional Derivatives," Mathematics, MDPI, vol. 10(9), pages 1-34, May.
    8. Muñoz-Vázquez, Aldo Jonathan & Martínez-Fuentes, Oscar & Fernández-Anaya, Guillermo, 2022. "Generalized PI control for robust stabilization of dynamical systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 22-35.
    9. Vasily E. Tarasov, 2023. "General Fractional Noether Theorem and Non-Holonomic Action Principle," Mathematics, MDPI, vol. 11(20), pages 1-35, October.
    10. 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).
    11. 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.
    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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