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On Delta and Nabla Caputo Fractional Differences and Dual Identities

Author

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  • Thabet Abdeljawad

Abstract

We investigate two types of dual identities for Caputo fractional differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by the Q -operator relates left and right fractional sums and differences. Two types of Caputo fractional differences are introduced; one of them (dual one) is defined so that it obeys the investigated dual identities. The relation between Riemann and Caputo fractional differences is investigated, and the delta and nabla discrete Mittag-Leffler functions are confirmed by solving Caputo type linear fractional difference equations. A nabla integration by parts formula is obtained for Caputo fractional differences as well.

Suggested Citation

  • Thabet Abdeljawad, 2013. "On Delta and Nabla Caputo Fractional Differences and Dual Identities," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-12, July.
    • Handle: RePEc:hin:jnddns:406910
    DOI: 10.1155/2013/406910
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    Cited by:

    1. Thabet Abdeljawad & Arran Fernandez, 2019. "On a New Class of Fractional Difference-Sum Operators with Discrete Mittag-Leffler Kernels," Mathematics, MDPI, vol. 7(9), pages 1-13, August.
    2. Rashid, Saima & Sultana, Sobia & Jarad, Fahd & Jafari, Hossein & Hamed, Y.S., 2021. "More efficient estimates via ℏ-discrete fractional calculus theory and applications," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    3. Almusawa, Musawa Yahya & Mohammed, Pshtiwan Othman, 2023. "Approximation of sequential fractional systems of Liouville–Caputo type by discrete delta difference operators," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    4. Rashid, Saima & Sultana, Sobia & Hammouch, Zakia & Jarad, Fahd & Hamed, Y.S., 2021. "Novel aspects of discrete dynamical type inequalities within fractional operators having generalized ℏ-discrete Mittag-Leffler kernels and application," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    5. Pshtiwan Othman Mohammed & Thabet Abdeljawad & Faraidun Kadir Hamasalh, 2021. "On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis," Mathematics, MDPI, vol. 9(11), pages 1-17, June.
    6. Abdeljawad, Thabet & Baleanu, Dumitru, 2017. "Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 106-110.
    7. Jiraporn Reunsumrit & Thanin Sitthiwirattham, 2020. "On the Nonlocal Fractional Delta-Nabla Sum Boundary Value Problem for Sequential Fractional Delta-Nabla Sum-Difference Equations," Mathematics, MDPI, vol. 8(4), pages 1-13, March.
    8. Suwan, Iyad & Abdeljawad, Thabet & Jarad, Fahd, 2018. "Monotonicity analysis for nabla h-discrete fractional Atangana–Baleanu differences," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 50-59.
    9. Abdeljawad, Thabet, 2018. "Different type kernel h−fractional differences and their fractional h−sums," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 146-156.
    10. Abdeljawad, Thabet, 2019. "Fractional difference operators with discrete generalized Mittag–Leffler kernels," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 315-324.
    11. Qiushuang Wang & Run Xu, 2022. "On Hilfer Generalized Proportional Nabla Fractional Difference Operators," Mathematics, MDPI, vol. 10(15), pages 1-16, July.
    12. Mohammed, Pshtiwan Othman & Abdeljawad, Thabet & Hamasalh, Faraidun Kadir, 2021. "Discrete Prabhakar fractional difference and sum operators," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    13. Kamsing Nonlaopon & Pshtiwan Othman Mohammed & Y. S. Hamed & Rebwar Salih Muhammad & Aram Bahroz Brzo & Hassen Aydi, 2022. "Analytical and Numerical Monotonicity Analyses for Discrete Delta Fractional Operators," Mathematics, MDPI, vol. 10(10), pages 1-9, May.
    14. Jarunee Soontharanon & Saowaluck Chasreechai & Thanin Sitthiwirattham, 2019. "A Coupled System of Fractional Difference Equations with Nonlocal Fractional Sum Boundary Conditions on the Discrete Half-Line," Mathematics, MDPI, vol. 7(3), pages 1-22, March.
    15. Panda, Sumati Kumari & Vijayakumar, Velusamy & Nagy, A.M., 2023. "Complex-valued neural networks with time delays in the Lp sense: Numerical simulations and finite time stability," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    16. Cui, Xueke & Li, Hong-Li & Zhang, Long & Hu, Cheng & Bao, Haibo, 2023. "Complete synchronization for discrete-time fractional-order coupled neural networks with time delays," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).

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