Nonlinear Dynamics

, Volume 85, Issue 2, pp 699–715

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

Original Paper

## Abstract

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.

### Keywords

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

## Notes

### Acknowledgments

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

### References

1. 1.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier (Nort h-Holland), Sci. Publishers, Amsterdam (2006)
2. 2.
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
3. 3.
Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007)
4. 4.
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
5. 5.
Yang, X.J., Baleanu, D., Srivastava, H.M.: Local Fractional Integral Transform and their Applications. Academic Press, New York (2015)Google Scholar
6. 6.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
7. 7.
Saha Ray, S., Bera, R.K.: Analytical solution of a fractional diffusion equation by Adomian decomposition method. Appl. Math. Comput. 174, 329–336 (2006)
8. 8.
Odibat, Z., Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simulat. 7, 27–34 (2006)
9. 9.
He, J.H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135, 73–79 (2003)
10. 10.
Liao, S.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004)
11. 11.
Vishal, K., Kumar, S., Das, S.: Application of homotopy analysis method for fractional swift Hohenberg equation-revisited. Appl. Math. Model. 36, 3630–3637 (2012)
12. 12.
Pandey, R.K., Singh, O.P., Baranwal, V.K.: An analytic algorithm for the space-time fractional advection-dispersion equation. Comput. Phys. Commun. 182, 134–144 (2011)
13. 13.
Srivastava, V.K., Awasthi, M.K., Kumar, S.: Analytical approximations of two and three dimensional time-fractional telegraphic equation by reduced differential transform method. Egypt. J. Basic Appl. Sci. 1, 60–66 (2014)
14. 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
15. 15.
Kumar, S.: A numerical study for solution of time fractional nonlinear shallow water equation in oceans. Z. Naturforsch. 68a, 1–7 (2013)
16. 16.
Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 62, 2364–2373 (2011)
17. 17.
Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 36, 4931–4943 (2012)
18. 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
19. 19.
Bhrawy, A.H., Taha, T.M., Machado, J.A.T.: A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dyn. 81, 1023–1052 (2015)
20. 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. 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. 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. 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. 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
25. 25.
Zhang, J., Wu, Y., Li, X.: Quasi-periodic solution of the (2+1)-dimensional Boussinesq–Burgers soliton equation. Phys. A Stat. Mech. Appl. 319, 213–232 (2003)
26. 26.
Zhang, L., Zhang, L.F., Li, C.: Some new exact solutions of Jacobian elliptic function about the generalized Boussinesq equation and Boussinesq-Burgers equation. Chin. Phys. B 17, 403–410 (2008)
27. 27.
Rady, A.S.A., Khalfallah, M.: On soliton solutions for Boussinesq-Burgers equations. Commun. Nonlinear Sci. Numer. Simul. 15, 886–894 (2010)
28. 28.
Chen, A., Li, X.: Darboux transformation and soliton solutions of Boussinesq–Burgers equation. Chaos. Soliton Fract. 27, 43–52 (2006)
29. 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
30. 30.
Gupta, A.K., Saha Ray, S.: Comparison between homotopy perturbation method and optimal homotopy asymptotic method for the soliton solutions of Boussinesq–Burger equations. Comput. Fluids 103, 34–41 (2014)
31. 31.
Kumar, S., Rashidi, M.M.: New analytical method for gas dynamic equation arising in shock fronts. Comput. Phys. Commun. 185, 1947–1954 (2014)
32. 32.
Kumar, S.: A new analytical modeling for telegraph equation via Laplace transform. Appl. Math. Model. 38, 3154–3163 (2014)
33. 33.
Odibat, Z., Bataineh, A.S.: An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials. Math. Methods Appl. Sci. 38(5), 991–1000 (2015)
34. 34.
Abu Arqub, O.: Series solution of fuzzy differential equations under strongly generalized differentiability. J. Adv. Res. Appl. Math. 5, 31–52 (2013)
35. 35.
El-Ajou, A., Abu Arqub, O., Momani, S.: Approximate analytical solution of the nonlinear fractional Kdv-Burger equation: a new iterative algorithm. J. Comput. Phys. 293, 81–95 (2015)
36. 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
37. 37.
Liao, S.: An optimal homotopy—analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 2003–2016 (2010)
38. 38.
Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylors formula. Appl. Math. Comput. 186, 286–293 (2007)