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On bivariate fractional calculus with general univariate analytic kernels

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  • 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
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    References listed on IDEAS

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    1. Dumitru Baleanu & Arran Fernandez, 2019. "On Fractional Operators and Their Classifications," Mathematics, MDPI, vol. 7(9), pages 1-10, September.
    2. Yuri Luchko, 2021. "General Fractional Integrals and Derivatives with the Sonine Kernels," Mathematics, MDPI, vol. 9(6), pages 1-17, March.
    3. 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.
    4. Rudolf Hilfer & Yuri Luchko, 2019. "Desiderata for Fractional Derivatives and Integrals," Mathematics, MDPI, vol. 7(2), pages 1-5, February.
    5. Kürt, Cemaliye & Fernandez, Arran & Özarslan, Mehmet Ali, 2023. "Two unified families of bivariate Mittag-Leffler functions," Applied Mathematics and Computation, Elsevier, vol. 443(C).
    6. 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.
    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.
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