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Fractional difference operators with discrete generalized Mittag–Leffler kernels

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  • Abdeljawad, Thabet

Abstract

In this article we define fractional difference operators with discrete generalized Mittag–Leffler kernels of the form (Eθ,μ¯γ(λ,t−ρ(s)) for both Riemann type (ABR) and Caputo type (ABC) cases, where AB stands for Atangana–Baleanu. Then, we employ the discrete Laplace transforms to formulate their corresponding AB−fractional sums, and prove useful and applicable versions of their semi-group properties. The action of fractional sums on the ABC type fractional differences is proved and used to solve the ABC−fractional difference initial value problems. The nonhomogeneous linear ABC fractional difference equation with constant coefficient is solved by both the discrete Laplace transforms and the successive approximation, and the Laplace transform method is remarked for the continuous counterpart. In fact, for the case μ ≠ 1, we obtain a nontrivial solution for the homogeneous linear ABC− type initial value problem with constant coefficient. The relation between the ABC and ABR fractional differences are formulated by using the discrete Laplace transform. We iterate the fractional sums of order −(θ,μ,1) to generate fractional sum-differences for which a semigroup property is proved. The nabla discrete transforms for the AB−fractional sums and the AB−iterated fractional sum-differences are calculated. Examples and remarks are given to clarify and confirm the obtained results and some of their particular cases are highlighted. Finally, the discrete extension to the higher order case is discussed.

Suggested Citation

  • Abdeljawad, Thabet, 2019. "Fractional difference operators with discrete generalized Mittag–Leffler kernels," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 315-324.
  • Handle: RePEc:eee:chsofr:v:126:y:2019:i:c:p:315-324
    DOI: 10.1016/j.chaos.2019.06.012
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    References listed on IDEAS

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    1. 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.
    2. Atangana, Abdon, 2016. "On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 948-956.
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    4. 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.
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    7. 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.
    8. Abdeljawad, Thabet, 2018. "Different type kernel h−fractional differences and their fractional h−sums," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 146-156.
    9. Jarad, Fahd & Abdeljawad, Thabet & Hammouch, Zakia, 2018. "On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 16-20.
    10. Al-Mdallal, Qasem M., 2009. "An efficient method for solving fractional Sturm–Liouville problems," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 183-189.
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    1. Abdalla, Bahaaeldin & Abdeljawad, Thabet, 2019. "On the oscillation of Caputo fractional differential equations with Mittag–Leffler nonsingular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 173-177.
    2. Al-Smadi, Mohammed & Arqub, Omar Abu & Zeidan, Dia, 2021. "Fuzzy fractional differential equations under the Mittag-Leffler kernel differential operator of the ABC approach: Theorems and applications," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    3. Du, Feifei & Jia, Baoguo, 2020. "Finite time stability of fractional delay difference systems: A discrete delayed Mittag-Leffler matrix function approach," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    4. Mohammed, Pshtiwan Othman & Abdeljawad, Thabet & Hamasalh, Faraidun Kadir, 2021. "Discrete Prabhakar fractional difference and sum operators," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    5. 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).
    6. 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).
    7. Acay, Bahar & Bas, Erdal & Abdeljawad, Thabet, 2020. "Fractional economic models based on market equilibrium in the frame of different type kernels," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    8. Khan, Hasib & Jarad, Fahd & Abdeljawad, Thabet & Khan, Aziz, 2019. "A singular ABC-fractional differential equation with p-Laplacian operator," Chaos, Solitons & Fractals, Elsevier, vol. 129(C), pages 56-61.
    9. Ullah, Malik Zaka & Mallawi, Fouad & Baleanu, Dumitru & Alshomrani, Ali Saleh, 2020. "A new fractional study on the chaotic vibration and state-feedback control of a nonlinear suspension system," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    10. Panda, Sumati Kumari & Abdeljawad, Thabet & Ravichandran, C., 2020. "A complex valued approach to the solutions of Riemann-Liouville integral, Atangana-Baleanu integral operator and non-linear Telegraph equation via fixed point method," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    11. Logeswari, K. & Ravichandran, C., 2020. "A new exploration on existence of fractional neutral integro- differential equations in the concept of Atangana–Baleanu derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 544(C).

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