Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves
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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.
KeywordsFractional 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.
- 5.Yang, X.J., Baleanu, D., Srivastava, H.M.: Local Fractional Integral Transform and their Applications. Academic Press, New York (2015)Google Scholar
- 14.Kumar, S., Kocak, H., Yildirim, A.: A fractional model of gas dynamics equation and its approximate solution by using Laplace transform. Z. Naturforsch. 67a, 389–396 (2012)Google Scholar
- 18.Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S., Gorder, R.A.V.: A new Jacobi spectral collocation method for solving 1+1 fractional Schrodinger equations and fractional coupled Schrodinger systems. Eur. Phys. J. Plus. 129(12), 1–21 (2014)Google Scholar
- 20.Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: An efficient Legendre spectral tau matrix formulation for solving fractional subdiffusion and reaction subdiffusion equations. J. Comput. Nonlinear Dyn. 10, 021019 (1–8) (2015)Google Scholar
- 21.Bhrawy, A.H., Zaky, M.A., Machado, J.A.T.: Efficient Legendre spectral tau algorithm for solving two-sided space-time Caputo fractional advection-dispersion equation. J. Vib. Control. (2015). doi:10.1177/1077546314566835
- 22.Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S., Abdelkawy, M.A.: A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equation. Calcolo. (2015). doi:10.1007/s10092-014-0132-x
- 23.Bhrawy, A.H., Ezz-Eldien, S.S.: A new Legendre operational technique for delay fractional optimal control problems. Calcolo. (2015). doi:10.1007/s10092-015-0160-1
- 24.Bhrawy, A.H., Doha, E.H., Machado, J.A.T., Ezz-Eldien, S.S.: An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index. Asian J. Control. (2016). doi:10.1002/asjc.1109
- 29.Wang, P., Tian, B., Liu, W., Lü, X., Jiang, Y.: Lax pair Bcklund transformation and multi-soliton solutions for the Boussinesq-Burgers equations from shallow water waves. Appl. Math. Comput. 218, 1726–1734 (2011)Google Scholar
- 36.El-Ajou, A., Abu Arqub, O., Momani, S., Baleanu, D., Alsaedi, A.: A novel expansion iterative method for solving linear partial differential equation of fractional order. Appl. Math. Comput. (2015). doi:10.1016/j.amc.2014.12.121