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Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications

Author

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  • Thabet Abdeljawad
  • Qasem M. Al-Mdallal
  • Mohamed A. Hajji

Abstract

Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo ( ) and Riemann ( ) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order is proved and the ordinary difference Lyapunov inequality then follows as tends to from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given.

Suggested Citation

  • Thabet Abdeljawad & Qasem M. Al-Mdallal & Mohamed A. Hajji, 2017. "Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-8, June.
    • Handle: RePEc:hin:jnddns:4149320
    DOI: 10.1155/2017/4149320
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    Cited by:

    1. Abdeljawad, Thabet, 2018. "Different type kernel h−fractional differences and their fractional h−sums," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 146-156.
    2. Saima Rashid & Thabet Abdeljawad & Fahd Jarad & Muhammad Aslam Noor, 2019. "Some Estimates for Generalized Riemann-Liouville Fractional Integrals of Exponentially Convex Functions and Their Applications," Mathematics, MDPI, vol. 7(9), pages 1-18, September.
    3. 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.
    4. Abdeljawad, Thabet, 2019. "Fractional difference operators with discrete generalized Mittag–Leffler kernels," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 315-324.
    5. Owolabi, Kolade M. & Hammouch, Zakia, 2019. "Spatiotemporal patterns in the Belousov–Zhabotinskii reaction systems with Atangana–Baleanu fractional order derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 1072-1090.
    6. Fall, Aliou Niang & Ndiaye, Seydou Nourou & Sene, Ndolane, 2019. "Black–Scholes option pricing equations described by the Caputo generalized fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 108-118.
    7. Al-Mdallal, Qasem M., 2018. "On fractional-Legendre spectral Galerkin method for fractional Sturm–Liouville problems," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 261-267.

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