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Operational Calculus for the General Fractional Derivatives of Arbitrary Order

Author

Listed:
  • Maryam Al-Kandari

    (Department of Mathematics, Kuwait University, Kuwait City 12037, Kuwait)

  • Latif A-M. Hanna

    (Department of Mathematics, Kuwait University, Kuwait City 12037, Kuwait)

  • Yuri Luchko

    (Department of Mathematics, Physics, and Chemistry, Berlin University of Applied Sciences and Technology, 10587 Berlin, Germany)

Abstract

In this paper, we deal with the general fractional integrals and the general fractional derivatives of arbitrary order with the kernels from a class of functions that have an integrable singularity of power function type at the origin. In particular, we introduce the sequential fractional derivatives of this type and derive an explicit formula for their projector operator. The main contribution of this paper is a construction of an operational calculus of Mikusiński type for the general fractional derivatives of arbitrary order. In particular, we present a representation of the m -fold sequential general fractional derivatives of arbitrary order as algebraic operations in the field of convolution quotients and derive some important operational relations.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1590-:d:810705
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    References listed on IDEAS

    as
    1. 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.
    2. Yuri Luchko, 2021. "General Fractional Integrals and Derivatives with the Sonine Kernels," Mathematics, MDPI, vol. 9(6), pages 1-17, March.
    3. Yuri Luchko & Masahiro Yamamoto, 2020. "The General Fractional Derivative and Related Fractional Differential Equations," Mathematics, MDPI, vol. 8(12), pages 1-20, November.
    4. Yuri Luchko, 2021. "Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications," Mathematics, MDPI, vol. 9(17), pages 1-15, September.
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    Citations

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    Cited by:

    1. Vasily E. Tarasov, 2024. "General Fractional Economic Dynamics with Memory," Mathematics, MDPI, vol. 12(15), pages 1-24, August.
    2. Vasily E. Tarasov, 2023. "General Fractional Calculus in Multi-Dimensional Space: Riesz Form," Mathematics, MDPI, vol. 11(7), pages 1-20, March.
    3. 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).
    4. Vasily E. Tarasov, 2023. "Multi-Kernel General Fractional Calculus of Arbitrary Order," Mathematics, MDPI, vol. 11(7), pages 1-32, April.

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