IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v171y2023ics096007792300396x.html
   My bibliography  Save this article

On bivariate fractional calculus with general univariate analytic kernels

Author

Listed:
  • Isah, Sunday Simon
  • Fernandez, Arran
  • Özarslan, Mehmet Ali

Abstract

We introduce a general bivariate fractional calculus, defined using a kernel based on an arbitrary univariate analytic function with an appropriate bivariate substitution. Various properties of the introduced general operators are established, including a series formula, function space mappings, and Fourier and Laplace transforms. A major result of this paper is a fractional Leibniz rule for the new operators, the derivation of which involves correcting a minor error in one of the classic textbooks on fractional calculus. We also solve some fractional differential equations using transform methods, revealing an interesting connection between bivariate type Mittag-Leffler functions.

Suggested Citation

  • Isah, Sunday Simon & Fernandez, Arran & Özarslan, Mehmet Ali, 2023. "On bivariate fractional calculus with general univariate analytic kernels," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    • Handle: RePEc:eee:chsofr:v:171:y:2023:i:c:s096007792300396x
    DOI: 10.1016/j.chaos.2023.113495
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S096007792300396X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2023.113495?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Fernandez, Arran & Özarslan, Mehmet Ali & Baleanu, Dumitru, 2019. "On fractional calculus with general analytic kernels," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 248-265.
    2. Rudolf Hilfer & Yuri Luchko, 2019. "Desiderata for Fractional Derivatives and Integrals," Mathematics, MDPI, vol. 7(2), pages 1-5, February.
    3. Dumitru Baleanu & Arran Fernandez, 2019. "On Fractional Operators and Their Classifications," Mathematics, MDPI, vol. 7(9), pages 1-10, September.
    4. Kürt, Cemaliye & Fernandez, Arran & Özarslan, Mehmet Ali, 2023. "Two unified families of bivariate Mittag-Leffler functions," Applied Mathematics and Computation, Elsevier, vol. 443(C).
    5. Zhao, Dazhi & Luo, Maokang, 2019. "Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 531-544.
    6. Yuri Luchko, 2021. "General Fractional Integrals and Derivatives with the Sonine Kernels," Mathematics, MDPI, vol. 9(6), pages 1-17, March.
    7. Changpin Li & Deliang Qian & YangQuan Chen, 2011. "On Riemann-Liouville and Caputo Derivatives," Discrete Dynamics in Nature and Society, Hindawi, vol. 2011, pages 1-15, March.
    Full references (including those not matched with items on IDEAS)

    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. Shiri, Babak & Baleanu, Dumitru, 2023. "All linear fractional derivatives with power functions’ convolution kernel and interpolation properties," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    2. Oscar Martínez-Fuentes & Fidel Meléndez-Vázquez & Guillermo Fernández-Anaya & José Francisco Gómez-Aguilar, 2021. "Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities," Mathematics, MDPI, vol. 9(17), pages 1-29, August.
    3. 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.
    4. Vasily E. Tarasov, 2023. "General Fractional Calculus in Multi-Dimensional Space: Riesz Form," Mathematics, MDPI, vol. 11(7), pages 1-20, March.
    5. Christopher Nicholas Angstmann & Byron Alexander Jacobs & Bruce Ian Henry & Zhuang Xu, 2020. "Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators," Mathematics, MDPI, vol. 8(11), pages 1-16, November.
    6. Muhammad Samraiz & Ahsan Mehmood & Saima Naheed & Gauhar Rahman & Artion Kashuri & Kamsing Nonlaopon, 2022. "On Novel Fractional Operators Involving the Multivariate Mittag–Leffler Function," Mathematics, MDPI, vol. 10(21), pages 1-19, October.
    7. Dumitru Baleanu & Arran Fernandez & Ali Akgül, 2020. "On a Fractional Operator Combining Proportional and Classical Differintegrals," Mathematics, MDPI, vol. 8(3), pages 1-13, March.
    8. Kucche, Kishor D. & Mali, Ashwini D. & Fernandez, Arran & Fahad, Hafiz Muhammad, 2022. "On tempered Hilfer fractional derivatives with respect to functions and the associated fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    9. 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.
    10. Rahaman, Mostafijur & Mondal, Sankar Prasad & Alam, Shariful & Metwally, Ahmed Sayed M. & Salahshour, Soheil & Salimi, Mehdi & Ahmadian, Ali, 2022. "Manifestation of interval uncertainties for fractional differential equations under conformable derivative," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    11. Samraiz, Muhammad & Mehmood, Ahsan & Iqbal, Sajid & Naheed, Saima & Rahman, Gauhar & Chu, Yu-Ming, 2022. "Generalized fractional operator with applications in mathematical physics," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    12. Vasily E. Tarasov, 2024. "General Fractional Economic Dynamics with Memory," Mathematics, MDPI, vol. 12(15), pages 1-24, August.
    13. Zhao, Dazhi & Yu, Guozhu & Tian, Yan, 2020. "Recursive formulae for the analytic solution of the nonlinear spatial conformable fractional evolution equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
    14. Zhokh, Alexey & Strizhak, Peter, 2018. "Thiele modulus having regard to the anomalous diffusion in a catalyst pellet," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 58-63.
    15. Marina Plekhanova & Guzel Baybulatova, 2020. "Multi-Term Fractional Degenerate Evolution Equations and Optimal Control Problems," Mathematics, MDPI, vol. 8(4), pages 1-9, April.
    16. Chaudhary, Naveed Ishtiaq & Raja, Muhammad Asif Zahoor & Khan, Zeshan Aslam & Mehmood, Ammara & Shah, Syed Muslim, 2022. "Design of fractional hierarchical gradient descent algorithm for parameter estimation of nonlinear control autoregressive systems," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    17. Faïçal Ndaïrou & Delfim F. M. Torres, 2023. "Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems," Mathematics, MDPI, vol. 11(19), pages 1-12, October.
    18. Gómez-Aguilar, J.F., 2020. "Chaos and multiple attractors in a fractal–fractional Shinriki’s oscillator model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    19. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    20. Balasubramaniam, P., 2022. "Solvability of Atangana-Baleanu-Riemann (ABR) fractional stochastic differential equations driven by Rosenblatt process via measure of noncompactness," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).

    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:eee:chsofr:v:171:y:2023:i:c:s096007792300396x. 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: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    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.