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Fuzzy fractional differential equations under the Mittag-Leffler kernel differential operator of the ABC approach: Theorems and applications

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  • Al-Smadi, Mohammed
  • Arqub, Omar Abu
  • Zeidan, Dia

Abstract

In this manuscript, we introduced, analyzed, and studied fuzzy fractional differential equations in terms of Atangana-Baleanu-Caputo differential operator equipped with uncertain constraints coefficients and initial conditions. To this end, we discussed both the fuzzy Atangana-Baleanu-Caputo fractional derivative and integral. Also, Newton-Leibniz fuzzy inversion formulas for both derivative and integral are proved. Using Banach fixed point theorem, existence and uniqueness results of solution are established by means of fuzzy strongly generalized differentiability of fuzzy fractional differential equation with Atangana-Baleanu fractional derivative under the Lipschitz condition. To achieve the above results, some prerequisite provisions for characterizing the solution in synonymous systems of crisp Atangana-Baleanu-Caputo fractional differential equations are argued. In this tendency, a new computational algorithm is proposed to obtain analytic solutions of the studied equations. To grasp the debated approach, some illustrative examples are provided and analyzed by the figures to visualize and support the theoretical results.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:chsofr:v:146:y:2021:i:c:s0960077921002447
    DOI: 10.1016/j.chaos.2021.110891
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    1. Abu Arqub, Omar & Maayah, Banan, 2019. "Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 163-170.
    2. Abu Arqub, Omar & Al-Smadi, Mohammed, 2018. "Atangana–Baleanu fractional approach to the solutions of Bagley–Torvik and Painlevé equations in Hilbert space," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 161-167.
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    6. Al-Smadi, Mohammed & Arqub, Omar Abu, 2019. "Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 280-294.
    7. Abdeljawad, Thabet, 2019. "Fractional difference operators with discrete generalized Mittag–Leffler kernels," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 315-324.
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    9. Arqub, Omar Abu & Maayah, Banan, 2018. "Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana–Baleanu fractional operator," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 117-124.
    10. Abu Arqub, Omar & Al-Smadi, Mohammed, 2020. "An adaptive numerical approach for the solutions of fractional advection–diffusion and dispersion equations in singular case under Riesz’s derivative operator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    11. Arqub, Omar Abu & Maayah, Banan, 2019. "Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC – Fractional Volterra integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 394-402.
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    Cited by:

    1. Hussam Aljarrah & Mohammad Alaroud & Anuar Ishak & Maslina Darus, 2022. "Approximate Solution of Nonlinear Time-Fractional PDEs by Laplace Residual Power Series Method," Mathematics, MDPI, vol. 10(12), pages 1-16, June.
    2. Gao, Wei & Baskonus, Haci Mehmet, 2022. "Deeper investigation of modified epidemiological computer virus model containing the Caputo operator," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).
    3. Eghlimi, Hadi & Asgari, Mohammad Sadegh, 2023. "A study of the time-fractional heat equation under the generalized Hukuhara conformable fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    4. Kumar, Sachin & Nieto, Juan J. & Ahmad, Bashir, 2022. "Chebyshev spectral method for solving fuzzy fractional Fredholm–Volterra integro-differential equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 501-513.
    5. Ahmad, Zubair & Ali, Farhad & Khan, Naveed & Khan, Ilyas, 2021. "Dynamics of fractal-fractional model of a new chaotic system of integrated circuit with Mittag-Leffler kernel," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    6. Duc, Tran Minh & Van Hoa, Ngo, 2021. "Stabilization of impulsive fractional-order dynamic systems involving the Caputo fractional derivative of variable-order via a linear feedback controller," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    7. Hussam Aljarrah & Mohammad Alaroud & Anuar Ishak & Maslina Darus, 2021. "Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations," Mathematics, MDPI, vol. 9(22), pages 1-17, November.

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