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Similarities in a fifth-order evolution equation with and with no singular kernel

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  • Doungmo Goufo, Emile F.
  • Kumar, Sunil
  • Mugisha, S.B.

Abstract

We perform in this report a comparative analysis between differential fractional operators applied to the non-linear Kaup–Kupershmidt equation. Such operators include the Atangana–Beleanu derivative and the Caputo–Fabrizio derivative which respectively follow the Mittag-Leffler law and the exponential law. We exploit the fixed points of the dynamics and the stability analysis to demonstrate that the exact solution exists and is unique for both types of models. Methods of performing numerical approximations of the solutions are presented and illustrated by graphical representations exhibiting a clear comparison between the dynamics under the influence of Mittag-Leffler law and those under the exponential law. Different cases are presented with respect to values of the derivative order 0 < α ≤ 1. We note a slight difference between both dynamics in terms of individual points, but their global pictures remain similar and close to the traditional and popular traveling wave solution of the standard Kaup–Kupershmidt model (α=1).

Suggested Citation

  • Doungmo Goufo, Emile F. & Kumar, Sunil & Mugisha, S.B., 2020. "Similarities in a fifth-order evolution equation with and with no singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    • Handle: RePEc:eee:chsofr:v:130:y:2020:i:c:s0960077919304138
    DOI: 10.1016/j.chaos.2019.109467
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    References listed on IDEAS

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    1. Amira Rachah & Delfim F. M. Torres, 2015. "Mathematical Modelling, Simulation, and Optimal Control of the 2014 Ebola Outbreak in West Africa," Discrete Dynamics in Nature and Society, Hindawi, vol. 2015, pages 1-9, May.
    2. Doungmo Goufo, Emile Franc, 2017. "Solvability of chaotic fractional systems with 3D four-scroll attractors," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 443-451.
    3. Abdon Atangana & Aydin Secer, 2013. "A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, April.
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    1. Ghanbari, Behzad & Kumar, Sunil & Kumar, Ranbir, 2020. "A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    2. Ghanbari, Behzad & Cattani, Carlo, 2020. "On fractional predator and prey models with mutualistic predation including non-local and nonsingular kernels," Chaos, Solitons & Fractals, Elsevier, vol. 136(C).
    3. Sunil Kumar & Ali Ahmadian & Ranbir Kumar & Devendra Kumar & Jagdev Singh & Dumitru Baleanu & Mehdi Salimi, 2020. "An Efficient Numerical Method for Fractional SIR Epidemic Model of Infectious Disease by Using Bernstein Wavelets," Mathematics, MDPI, vol. 8(4), pages 1-22, April.
    4. De Angelis, Paolo & De Marchis, Roberto & Martire, Antonio Luciano & Oliva, Immacolata, 2020. "A mean-value Approach to solve fractional differential and integral equations," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    5. Fardi, Mojtaba & Khan, Yasir, 2021. "A novel finite difference-spectral method for fractal mobile/immobiletransport model based on Caputo–Fabrizio derivative," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    6. Tabi, C.B. & Ndjawa, P.A.Y. & Motsumi, T.G. & Bansi, C.D.K. & Kofané, T.C., 2020. "Magnetic field effect on a fractionalized blood flow model in the presence of magnetic particles and thermal radiations," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).

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