IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i7p1651-d1110890.html
   My bibliography  Save this article

General Fractional Calculus in Multi-Dimensional Space: Riesz Form

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/7/1651/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/7/1651/
    Download Restriction: no
    ---><---

    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.
    3. Tarasov, Vasily E., 2015. "Lattice fractional calculus," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 12-33.
    4. Rudolf Hilfer & Yuri Luchko, 2019. "Desiderata for Fractional Derivatives and Integrals," Mathematics, MDPI, vol. 7(2), pages 1-5, February.
    5. Hameed Ur Rehman & Maslina Darus & Jamal Salah, 2018. "A Note on Caputo’s Derivative Operator Interpretation in Economy," Journal of Applied Mathematics, Hindawi, vol. 2018, pages 1-7, October.
    6. 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.
    7. Vasily E. Tarasov, 2021. "General Fractional Calculus: Multi-Kernel Approach," Mathematics, MDPI, vol. 9(13), pages 1-14, June.
    8. 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.
    9. 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.
    10. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Economic interpretation of fractional derivatives," Papers 1712.09575, arXiv.org.
    11. Anatoly N. Kochubei & Yuri Kondratiev, 2019. "Growth Equation of the General Fractional Calculus," Mathematics, MDPI, vol. 7(7), pages 1-8, July.
    12. Vasily E. Tarasov, 2021. "General Fractional Vector Calculus," Mathematics, MDPI, vol. 9(21), pages 1-87, November.
    13. 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.
    14. Yuri Luchko, 2021. "General Fractional Integrals and Derivatives with the Sonine Kernels," Mathematics, MDPI, vol. 9(6), pages 1-17, March.
    15. 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).
    16. Yuri Luchko & Masahiro Yamamoto, 2020. "The General Fractional Derivative and Related Fractional Differential Equations," Mathematics, MDPI, vol. 8(12), pages 1-20, November.
    17. 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.
    18. Vasily E. Tarasov, 2022. "Nonlocal Probability Theory: General Fractional Calculus Approach," Mathematics, MDPI, vol. 10(20), pages 1-82, October.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Vasily E. Tarasov, 2023. "Multi-Kernel General Fractional Calculus of Arbitrary Order," Mathematics, MDPI, vol. 11(7), pages 1-32, April.
    2. 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.
    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, 2024. "General Fractional Economic Dynamics with Memory," Mathematics, MDPI, vol. 12(15), pages 1-24, August.
    5. Vasily E. Tarasov, 2023. "General Fractional Noether Theorem and Non-Holonomic Action Principle," Mathematics, MDPI, vol. 11(20), pages 1-35, October.
    6. 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.
    7. 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.
    8. Vasily E. Tarasov, 2022. "General Non-Local Continuum Mechanics: Derivation of Balance Equations," Mathematics, MDPI, vol. 10(9), pages 1-43, April.
    9. Tarasova, Valentina V. & Tarasov, Vasily E., 2017. "Logistic map with memory from economic model," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 84-91.
    10. 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.
    11. Yuri Luchko, 2023. "Fractional Integrals and Derivatives: “True” versus “False”," Mathematics, MDPI, vol. 11(13), pages 1-2, July.
    12. Michelitsch, T.M. & Collet, B.A. & Riascos, A.P. & Nowakowski, A.F. & Nicolleau, F.C.G.A., 2016. "A fractional generalization of the classical lattice dynamics approach," Chaos, Solitons & Fractals, Elsevier, vol. 92(C), pages 43-50.
    13. Vasily E. Tarasov, 2016. "Exact Discrete Analogs of Canonical Commutation and Uncertainty Relations," Mathematics, MDPI, vol. 4(3), pages 1-13, June.
    14. Vasily E. Tarasov, 2019. "Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models," Mathematics, MDPI, vol. 7(6), pages 1-50, June.
    15. Shi, Jianping & He, Ke & Fang, Hui, 2022. "Chaos, Hopf bifurcation and control of a fractional-order delay financial system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 348-364.
    16. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Logistic map with memory from economic model," Papers 1712.09092, arXiv.org.
    17. 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.
    18. Vasily E. Tarasov, 2021. "General Fractional Calculus: Multi-Kernel Approach," Mathematics, MDPI, vol. 9(13), pages 1-14, June.
    19. 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.
    20. Vasily E. Tarasov & Valentina V. Tarasova, 2017. "Accelerators in Macroeconomics: Comparison of Discrete and Continuous Approaches," American Journal of Economics and Business Administration, Science Publications, vol. 9(3), pages 47-55, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1651-:d:1110890. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.