IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i13p1501-d582777.html
   My bibliography  Save this article

General Fractional Calculus: Multi-Kernel Approach

Author

Listed:
  • Vasily E. Tarasov

    (Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Faculty “Information Technologies and Applied Mathematics”, Moscow Aviation Institute, National Research University, 125993 Moscow, Russia)

Abstract

For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in 2021. In Luchko works, the proposed approaches to formulate this calculus are based either on the power of one Sonin kernel or the convolution of one Sonin kernel with the kernels of the integer-order integrals. To apply general fractional calculus, it is useful to have a wider range of operators, for example, by using the Laplace convolution of different types of kernels. In this paper, an extended formulation of the general fractional calculus of arbitrary order is proposed. Extension is achieved by using different types (subsets) of pairs of operator kernels in definitions general fractional integrals and derivatives. For this, the definition of the Luchko pair of kernels is somewhat broadened, which leads to the symmetry of the definition of the Luchko pair. The proposed set of kernel pairs are subsets of the Luchko set of kernel pairs. The fundamental theorems for the proposed general fractional derivatives and integrals are proved.

Suggested Citation

  • Vasily E. Tarasov, 2021. "General Fractional Calculus: Multi-Kernel Approach," Mathematics, MDPI, vol. 9(13), pages 1-14, June.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:13:p:1501-:d:582777
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/13/1501/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/13/1501/
    Download Restriction: no
    ---><---

    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. 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.
    3. Vasily E. Tarasov, 2020. "Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 8(5), pages 1-3, April.
    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. Vasily E. Tarasov, 2024. "General Fractional Economic Dynamics with Memory," Mathematics, MDPI, vol. 12(15), pages 1-24, August.
    2. Vasily E. Tarasov, 2021. "General Fractional Vector Calculus," Mathematics, MDPI, vol. 9(21), pages 1-87, November.
    3. Vasily E. Tarasov, 2023. "General Fractional Calculus in Multi-Dimensional Space: Riesz Form," Mathematics, MDPI, vol. 11(7), pages 1-20, March.
    4. Vasily E. Tarasov, 2023. "General Fractional Noether Theorem and Non-Holonomic Action Principle," Mathematics, MDPI, vol. 11(20), pages 1-35, October.
    5. 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).
    6. Vasily E. Tarasov, 2023. "Multi-Kernel General Fractional Calculus of Arbitrary Order," Mathematics, MDPI, vol. 11(7), pages 1-32, April.
    7. Yuri Luchko, 2024. "On Symmetrical Sonin Kernels in Terms of Hypergeometric-Type Functions," Mathematics, MDPI, vol. 12(24), pages 1-18, December.

    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. Vasily E. Tarasov, 2023. "General Fractional Calculus in Multi-Dimensional Space: Riesz Form," Mathematics, MDPI, vol. 11(7), pages 1-20, March.
    4. 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).
    5. 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.
    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. Vasily E. Tarasov, 2021. "General Fractional Vector Calculus," Mathematics, MDPI, vol. 9(21), pages 1-87, November.
    8. Vasily E. Tarasov, 2021. "General Fractional Dynamics," Mathematics, MDPI, vol. 9(13), pages 1-26, June.
    9. Vasily E. Tarasov, 2024. "General Fractional Economic Dynamics with Memory," Mathematics, MDPI, vol. 12(15), pages 1-24, August.
    10. Beghin, Luisa & Cristofaro, Lorenzo & Mishura, Yuliya, 2024. "A class of processes defined in the white noise space through generalized fractional operators," Stochastic Processes and their Applications, Elsevier, vol. 178(C).
    11. Vasily E. Tarasov, 2021. "Integral Equations of Non-Integer Orders and Discrete Maps with Memory," Mathematics, MDPI, vol. 9(11), pages 1-12, May.
    12. Giacomo Ascione & Nikolai Leonenko & Enrica Pirozzi, 2022. "Non-local Solvable Birth–Death Processes," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1284-1323, June.
    13. Vasily E. Tarasov, 2023. "General Fractional Noether Theorem and Non-Holonomic Action Principle," Mathematics, MDPI, vol. 11(20), pages 1-35, October.
    14. Zain-Aldeen S. A. Rahman & Basil H. Jasim & Yasir I. A. Al-Yasir & Yim-Fun Hu & Raed A. Abd-Alhameed & Bilal Naji Alhasnawi, 2021. "A New Fractional-Order Chaotic System with Its Analysis, Synchronization, and Circuit Realization for Secure Communication Applications," Mathematics, MDPI, vol. 9(20), pages 1-25, October.
    15. Borin, Daniel, 2024. "Caputo fractional standard map: Scaling invariance analyses," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    16. Yuri Luchko, 2021. "General Fractional Integrals and Derivatives with the Sonine Kernels," Mathematics, MDPI, vol. 9(6), pages 1-17, March.
    17. Vasily E. Tarasov, 2020. "Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients," Mathematics, MDPI, vol. 8(12), pages 1-11, December.
    18. Dmitrii Tverdyi & Roman Parovik, 2023. "Hybrid GPU–CPU Efficient Implementation of a Parallel Numerical Algorithm for Solving the Cauchy Problem for a Nonlinear Differential Riccati Equation of Fractional Variable Order," Mathematics, MDPI, vol. 11(15), pages 1-21, July.
    19. Mofareh Alhazmi & Yasser Alrashedi & Hamed Ould Sidi & Maawiya Ould Sidi, 2025. "Detection of a Spatial Source Term Within a Multi-Dimensional, Multi-Term Time-Space Fractional Diffusion Equation," Mathematics, MDPI, vol. 13(5), pages 1-16, February.
    20. Jimin Yu & Zeming Zhao & Yabin Shao, 2023. "On Cauchy Problems of Caputo Fractional Differential Inclusion with an Application to Fractional Non-Smooth Systems," Mathematics, MDPI, vol. 11(3), pages 1-18, January.

    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:9:y:2021:i:13:p:1501-:d:582777. 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.