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New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger’s equations

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Abstract

In this paper, we extend the model of the Korteweg–de Vries (KDV) and Korteweg–de Vries–Burger’s (KDVB) to new model time fractional Korteweg–de Vries (TFKDV) and time fractional Korteweg–de Vries-Burger’s (TFKDVB) with Liouville–Caputo (LC), Caputo–Fabrizio (CF), and Atangana-Baleanu (AB) fractional time derivative equations, respectively. We utilize the q-homotopy analysis transform method (q-HATM) to compute the approximate solutions of TFKDV and TFKDVB using LC, CF and AB in Liouville–Caputo sense. We study the convergence analysis of q-HATM by computing the Residual Error Function (REF) and finding the interval of the convergence through the h-curves. Also, we find the optimal values of h so that, we assure the convergence of the approximate solutions. The results are very effective and accurate in solving the TFKDV and TFKDVB.

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences and Arts, Najran University, Najran, 55461, Saudi Arabia

    Khaled M. Saad

  2. Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen

    Khaled M. Saad

  3. Department of Mathematics, Faculty of Sciences, Cankaya University, Balgat, 06530, Ankara, Turkey

    Dumitru Baleanu

  4. Institute of Space Sciences, Magurele-Bucharest, Romania

    Dumitru Baleanu

  5. Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, 9300, South Africa

    Abdon Atangana

Authors
  1. Khaled M. Saad
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  2. Dumitru Baleanu
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  3. Abdon Atangana
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Correspondence to Khaled M. Saad.

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Communicated by José Tenreiro Machado.

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Saad, K.M., Baleanu, D. & Atangana, A. New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger’s equations. Comp. Appl. Math. 37, 5203–5216 (2018). https://doi.org/10.1007/s40314-018-0627-1

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