Nonlinear Dynamics

, Volume 85, Issue 2, pp 699–715 | Cite as

Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves

Original Paper


In this paper, an analytical method based on the generalized Taylors series formula together with residual error function, namely residual power series method (RPSM), is proposed for finding the numerical solution of the coupled system of time–fractional nonlinear Boussinesq–Burger’s equations. The Boussinesq–Burger’s equations arise in studying the fluid flow in a dynamic system and describe the propagation of the shallow water waves. Subsequently, the approximate solutions of time-fractional nonlinear coupled Boussinesq–Burger’s equations obtained by RPSM are compared with the exact solutions as well as the solutions obtained by modified homotopy analysis transform method. Then, we provide a rigorous convergence analysis and error estimate of RPSM. Numerical simulations of the results are depicted through different graphical representations and tables showing that present scheme is reliable and powerful in finding the numerical solutions of coupled system of fractional nonlinear differential equations like Boussinesq–Burger’s equations.


Fractional Boussinesq–Burger’s equation Residual power series Homotopy analysis transform method Homotopy polynomials  Optimal value 



The authors express their thanks to the referees for carefully reading the paper and helpful comments and suggestions which have improved the paper.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of TechnologyJamshedpurIndia
  2. 2.Department of MathematicsCankya UniversityAnkaraTurkey
  3. 3.Institute of Space ScienceMagurele-BucharestRomania

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