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General Fractional Calculus in Multi-Dimensional Space: Riesz Form

Author

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  • Vasily E. Tarasov

    (Department of Physics, 915, Moscow Aviation Institute, National Research University, 125993 Moscow, Russia)

Abstract

An extension of the general fractional calculus (GFC) is proposed as a generalization of the Riesz fractional calculus, which was suggested by Marsel Riesz in 1949. The proposed Riesz form of GFC can be considered as an extension GFC from the positive real line and the Laplace convolution to the m -dimensional Euclidean space and the Fourier convolution. To formulate the general fractional calculus in the Riesz form, the Luchko approach to construction of the GFC, which was suggested by Yuri Luchko in 2021, is used. The general fractional integrals and derivatives are defined as convolution-type operators. In these definitions the Fourier convolution on m-dimensional Euclidean space is used instead of the Laplace convolution on positive semi-axis. Some properties of these general fractional operators are described. The general fractional analogs of first and second fundamental theorems of fractional calculus are proved. The fractional calculus of the Riesz potential and the fractional Laplacian of the Riesz form are special cases of proposed general fractional calculus of the Riesz form.

Suggested Citation

  • Vasily E. Tarasov, 2023. "General Fractional Calculus in Multi-Dimensional Space: Riesz Form," Mathematics, MDPI, vol. 11(7), pages 1-20, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1651-:d:1110890
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    References listed on IDEAS

    as
    1. Sergey M. Sitnik & Vladimir E. Fedorov & Marina V. Polovinkina & Igor P. Polovinkin, 2023. "On Recovery of the Singular Differential Laplace—Bessel Operator from the Fourier–Bessel Transform," Mathematics, MDPI, vol. 11(5), pages 1-16, February.
    2. 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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    5. 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.
    6. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Economic interpretation of fractional derivatives," Papers 1712.09575, arXiv.org.
    7. Anatoly N. Kochubei & Yuri Kondratiev, 2019. "Growth Equation of the General Fractional Calculus," Mathematics, MDPI, vol. 7(7), pages 1-8, July.
    8. Vasily E. Tarasov, 2021. "General Fractional Vector Calculus," Mathematics, MDPI, vol. 9(21), pages 1-87, November.
    9. 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).
    10. 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.
    11. Vasily E. Tarasov, 2022. "Nonlocal Probability Theory: General Fractional Calculus Approach," Mathematics, MDPI, vol. 10(20), pages 1-82, October.
    12. Tarasov, Vasily E., 2015. "Lattice fractional calculus," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 12-33.
    13. 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.
    14. Vasily E. Tarasov, 2021. "General Fractional Calculus: Multi-Kernel Approach," Mathematics, MDPI, vol. 9(13), pages 1-14, June.
    15. 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.
    16. Vasily E. Tarasov, 2015. "Exact Discrete Analogs of Derivatives of Integer Orders: Differences as Infinite Series," Journal of Mathematics, Hindawi, vol. 2015, pages 1-8, November.
    17. Yuri Luchko, 2021. "General Fractional Integrals and Derivatives with the Sonine Kernels," Mathematics, MDPI, vol. 9(6), pages 1-17, March.
    18. Yuri Luchko & Masahiro Yamamoto, 2020. "The General Fractional Derivative and Related Fractional Differential Equations," Mathematics, MDPI, vol. 8(12), pages 1-20, November.
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    Cited by:

    1. Sergei Sitnik, 2023. "Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”," Mathematics, MDPI, vol. 11(15), pages 1-7, August.

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