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Numerical approximations of Atangana–Baleanu Caputo derivative and its application

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  • Yadav, Swati
  • Pandey, Rajesh K.
  • Shukla, Anil K.

Abstract

To solve the problems of non-local dynamical systems, Caputo and Fabrizio proposed a new definition for the fractional derivative. Atangana and Baleanu generalized the Caputo-Fabrizio derivative using the Mittag–Leffler function as the kernel which is both non-singular and non-local. In this paper, we investigate numerical schemes for the Atangana–Baleanu Caputo derivative in two ways and use the same for solving Advection-Diffusion equation whose time derivative is Atangana–Baleanu Caputo derivative. The stability of the schemes is established numerically. Numerical examples are provided to support the theory presented in the paper.

Suggested Citation

  • Yadav, Swati & Pandey, Rajesh K. & Shukla, Anil K., 2019. "Numerical approximations of Atangana–Baleanu Caputo derivative and its application," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 58-64.
    • Handle: RePEc:eee:chsofr:v:118:y:2019:i:c:p:58-64
    DOI: 10.1016/j.chaos.2018.11.009
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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. Dumitru Baleanu & Amin Jajarmi & Mojtaba Hajipour, 2017. "A New Formulation of the Fractional Optimal Control Problems Involving Mittag–Leffler Nonsingular Kernel," Journal of Optimization Theory and Applications, Springer, vol. 175(3), pages 718-737, December.
    3. Atangana, Abdon & Koca, Ilknur, 2016. "Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 447-454.
    4. Solís-Pérez, J.E. & Gómez-Aguilar, J.F. & Atangana, A., 2018. "Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 175-185.
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    Cited by:

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    9. 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).
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