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Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order

Author

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  • Alkahtani, B.S.T.
  • Atangana, A.

Abstract

In order to control the movement of waves on the area of shallow water, the newly derivative with fractional order proposed by Caputo and Fabrizio was used. To achieve this, we first proposed a transition from ordinary to fractional differential equation. We proved the existence and uniqueness of the coupled solutions of the modified system using the fixed-point theorem. We derive the special solution of the modified system using an iterative method. We proved the stability of the used method and also the uniqueness of the special solution. We presented the numerical simulations for different values of alpha.

Suggested Citation

  • Alkahtani, B.S.T. & Atangana, A., 2016. "Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 539-546.
    • Handle: RePEc:eee:chsofr:v:89:y:2016:i:c:p:539-546
    DOI: 10.1016/j.chaos.2016.03.012
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    References listed on IDEAS

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    1. Hasan Bulut & Haci Mehmet Baskonus & Fethi Bin Muhammad Belgacem, 2013. "The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu Transform Method," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-6, September.
    2. Abdon Atangana & Necdet Bildik, 2013. "The Use of Fractional Order Derivative to Predict the Groundwater Flow," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-9, October.
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    Cited by:

    1. Owolabi, Kolade M., 2021. "Computational analysis of different Pseudoplatystoma species patterns the Caputo-Fabrizio derivative," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    2. Owolabi, Kolade M. & Atangana, Abdon, 2017. "Numerical approximation of nonlinear fractional parabolic differential equations with Caputo–Fabrizio derivative in Riemann–Liouville sense," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 171-179.
    3. Tian, Xue & Zhang, Yi, 2019. "Noether’s theorem for fractional Herglotz variational principle in phase space," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 50-54.
    4. Avcı, Derya & Yetim, Aylin, 2019. "Cauchy and source problems for an advection-diffusion equation with Atangana–Baleanu derivative on the real line," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 361-365.
    5. Abro, Kashif Ali & Abro, Irfan Ali & Yıldırım, Ahmet, 2020. "A comparative analysis of sulfate SO4−2 ion concentration via modern fractional derivatives: An industrial application to cooling system of power plant," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    6. Atangana, Abdon & Gómez-Aguilar, J.F., 2017. "Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 285-294.
    7. Costa, F.S. & Oliveira, D.S. & Rodrigues, F.G. & de Oliveira, E.C., 2019. "The fractional space–time radial diffusion equation in terms of the Fox’s H-function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 403-418.
    8. Ding, Juan-Juan & Zhang, Yi, 2020. "Noether’s theorem for fractional Birkhoffian system of Herglotz type with time delay," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    9. Panwar, Virender Singh & Sheik Uduman, P.S. & Gómez-Aguilar, J.F., 2021. "Mathematical modeling of coronavirus disease COVID-19 dynamics using CF and ABC non-singular fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    10. Arshad, Sadia & Defterli, Ozlem & Baleanu, Dumitru, 2020. "A second order accurate approximation for fractional derivatives with singular and non-singular kernel applied to a HIV model," Applied Mathematics and Computation, Elsevier, vol. 374(C).
    11. Cai, Rui-Yang & Zhou, Hua-Cheng & Kou, Chun-Hai, 2021. "Boundary control strategy for three kinds of fractional heat equations with control-matched disturbances," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    12. Al-Refai, Mohammed & Jarrah, Abdulla M., 2019. "Fundamental results on weighted Caputo–Fabrizio fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 7-11.

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