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Operational Calculus for the 1st-Level General Fractional Derivatives and Its Applications

Author

Listed:
  • Maryam Alkandari

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

  • Yuri Luchko

    (Department of Mathematics, Physics and Chemistry, Berlin University of Applied Sciences and Technology, 13353 Berlin, Germany)

Abstract

The 1st-level General Fractional Derivatives (GFDs) combine in one definition the GFDs of the Riemann–Liouville type and the regularized GFDs (or the GFDs of the Caputo type) that have been recently introduced and actively studied in the fractional calculus literature. In this paper, we first construct an operational calculus of the Mikusiński type for the 1st-level GFDs. In particular, it includes the operational calculi for the GFDs of the Riemann–Liouville type and for the regularized GFDs as its particular cases. In the second part of the paper, this calculus is applied for the derivation of the closed-form solution formulas to the initial-value problems for the linear fractional differential equations with the 1st-level GFDs.

Suggested Citation

  • Maryam Alkandari & Yuri Luchko, 2024. "Operational Calculus for the 1st-Level General Fractional Derivatives and Its Applications," Mathematics, MDPI, vol. 12(17), pages 1-23, August.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:17:p:2626-:d:1463213
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    References listed on IDEAS

    as
    1. Anatoly N. Kochubei & Yuri Kondratiev, 2019. "Growth Equation of the General Fractional Calculus," Mathematics, MDPI, vol. 7(7), pages 1-8, July.
    2. Vasily E. Tarasov, 2022. "Nonlocal Probability Theory: General Fractional Calculus Approach," Mathematics, MDPI, vol. 10(20), pages 1-82, October.
    3. Fahad, Hafiz Muhammad & Fernandez, Arran, 2021. "Operational calculus for Caputo fractional calculus with respect to functions and the associated fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    4. Tarasov, Vasily E., 2020. "Fractional econophysics: Market price dynamics with memory effects," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
    5. Vasily E. Tarasov, 2022. "General Non-Local Continuum Mechanics: Derivation of Balance Equations," Mathematics, MDPI, vol. 10(9), pages 1-43, April.
    6. 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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