China Ocean Engineering

, Volume 33, Issue 4, pp 477–483 | Cite as

New Wave Solutions of Time-Fractional Coupled Boussinesq–Whitham–Broer–Kaup Equation as A Model of Water Waves

  • Emrah Atilgan
  • Mehmet Senol
  • Ali KurtEmail author
  • Orkun Tasbozan
Technical Notes


The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.

Key words

time fractional coupled Boussinesq–Whitham–Broer–Kaup equation conformable fractional derivative auxiliary equation method 


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Copyright information

© Chinese Ocean Engineering Society and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Emrah Atilgan
    • 1
  • Mehmet Senol
    • 2
  • Ali Kurt
    • 3
    Email author
  • Orkun Tasbozan
    • 4
  1. 1.Department of Management Informatics SystemsMustafa Kemal UniversityHatayTurkey
  2. 2.Department of MathematicsNevsehir Haci Bektas Veli UniversityNevsehirTurkey
  3. 3.Department of MathematicsPamukkale UniversityDenizliTurkey
  4. 4.Department of MathematicsMustafa Kemal UniversityHatayTurkey

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