A Comparative Analysis of Fractional-Order Kaup–Kupershmidt Equation within Different Operators

Nehad Ali Shah, Yasser S. Hamed, Khadijah M. Abualnaja, Jae Dong Chung, Rasool Shah, Adnan Khan

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


In this paper, we find the solution of the fractional-order Kaup–Kupershmidt (KK) equation by implementing the natural decomposition method with the aid of two different fractional derivatives, namely the Atangana–Baleanu derivative in Caputo manner (ABC) and Caputo–Fabrizio (CF). When investigating capillary gravity waves and nonlinear dispersive waves, the KK equation is extremely important. To demonstrate the accuracy and efficiency of the proposed technique, we study the nonlinear fractional KK equation in three distinct cases. The results are given in the form of a series, which converges quickly. The numerical simulations are presented through tables to illustrate the validity of the suggested technique. Numerical simulations in terms of absolute error are performed to ensure that the proposed methodologies are trustworthy and accurate. The resulting solutions are graphically shown to ensure the applicability and validity of the algorithms under consideration. The results that we obtain confirm that the proposed method is the best tool for handling any nonlinear problems arising in science and technology.

Original languageEnglish
Article number986
Issue number5
StatePublished - May 2022


  • Adomian decomposition method
  • Caputo–Fabrizio and Atangana-Baleanu operators
  • natural transform
  • time-fractional Kaup–Kupershmidt equation


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