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Conservatory of Kaup-Kupershmidt Equation to the Concept of Fractional Derivative with and without Singular Kernel

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Abstract

Making use of the traditional Caputo derivative and the newly introduced Caputo-Fabrizio derivative with fractional order and no singular kernel, we extent the nonlinear Kaup-Kupershmidt to the span of fractional calculus. In the analysis, different methods of fixed-point theorem together with the concept of piccard L-stability are used, allowing us to present the existence and uniqueness of the exact solution to models with both versions of derivatives. Finally, we present techniques to perform some numerical simulations for both non-linear models and graphical simulations are provided for values of the order α = 1.00; 0.90. Solutions are shown to behave similarly to the standard well-known traveling wave solution of Kaup-Kupershmidt equation.

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Authors and Affiliations

  1. Institute for Groundwater Studies, University of the Free State, 9300, Bloemfontein, South Africa

    Abdon Atangana

  2. Department of Mathematical Sciences, University of South Africa, Florida, 0003, South Africa

    Emile Franc Doungmo Goufo

Authors
  1. Abdon Atangana
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  2. Emile Franc Doungmo Goufo
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Correspondence to Abdon Atangana.

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Atangana, A., Goufo, E.F.D. Conservatory of Kaup-Kupershmidt Equation to the Concept of Fractional Derivative with and without Singular Kernel. Acta Math. Appl. Sin. Engl. Ser. 34, 351–361 (2018). https://doi.org/10.1007/s10255-018-0757-7

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  • DOI: https://doi.org/10.1007/s10255-018-0757-7

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