Approximate Analytical Solution of the Damped Burgers and Boussinesq-Burgers Equations

Abstract

In this paper, the Homotopy Analysis Method (HAM) is applied to the damped Burgers and Boussinesq-Burgers equations to obtain their approximate analytical solutions. The HAM solution includes an auxiliary parameter h¯ which provides a convenient way to adjust and control the convergence region of the solution series. An appropriate choice of the auxiliary parameter in the model problems for increasing time is investigated.

Keywords

• Homotopy analysis method,approximate analytical solution,damped Burgers equation,Boussinesq-Burgers equation

References

1. [1] S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D thesis, Shanghai Jiao Tong University, (1992).
2. [2] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, (2003).
3. [3] S. J. Liao, Homotopy analysis method: A new analytical technique for nonlinear problems, Commun. Nonlinear Sci. Numer. Simulat. 2(2), (1997), 95-100.
4. [4] S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147, (2004), 499-513.
5. [5] S. J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Commun. Nonlinear Sci. Numer. Simulat., 14, (2009), 983-997.
6. [6] S. Abbasbandy, The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation, Phys. Lett. A, 361, (2007), 478-483.
7. [7] E. Babolian, J. Saeidian, Analytic approximate solutions to Burgers, Fisher, Huxley equations and two combined forms of these eqautions, Commun. Nonlinear. Sci. Numer. Simulat., 14, (2009), 1984-1992.
8. [8] A. Fakhari, G. Domairry, Ebrahimpour, Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution, Phys. Lett. A, 368, (2007), 64-68.

9. [9] M. M. Rashidi, G. Domairry, A. DoostHosseini, S. Dinarvand, Explicit Approximate Solution of the Coupled KdV Equations by using the Homotopy Analysis Method, Int. Journal of Math. Analysis, 2(12), (2008), 581-589.
10. [10] M. Inc, On numerical solution of Burgers equation by homotopy analysis method, Phys Lett A, 372, (2008), 356-360.
11. [11] A. S. Bataineh, M. S. M. Noorani, I. Hashim, Approximate analytical solutions of systems of PDEs by homotopy analysis method, Comp. Math. Appl., 55, (2008), 2913-2923.
12. [12] S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A, 360, (2006), 109-113.
13. [13] T. Hayat, M. Sajid, On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder, Phys. Lett. A, 361, (2007), 316-322.
14. [14] S. J. Liao, A. Y. Tan, A general approach to obtain series solutions of nonlinear differential equations, Commun. Nonlinear. Sci. Numer. Simulat., 14, (2009), 983-997.
15. [15] A. Esen, O. Tas¸bozan and N.M. Ya˘gmurlu, Approximate Analytical Solutions of the Fractional Sharmo-TassoOlver Equation Using Homotopy Analysis Method and a Comparison with Other Methods, C¸ ankaya University Journal of Science and Engineering, 9(2), (2012), 139-147.
16. [16] A. Esen, N.M. Ya˘gmurlu and O. Tas¸bozan, Approximate Analytical Solution to Time-Fractional Damped Burger and Cahn-Allen Equations, Applied Mathematics Information Sciences, 7(5), (2013), 1951-1956.
17. [17] O. Tas¸bozan, A. Esen and N.M. Ya˘gmurlu, Approximate analytical solutions of fractional coupled mKdV equation by homotopy analysis method, Open Journal of Applied Science, 2(3), (2012), 193-197.
18. [18] P. L. Sachdev, Nonlinear Diffusive Waves, Cambridge University Press, (1987).
19. [19] W. Malfliet, Approximate solution of the damped Burgers equation, J. Phys. A: Math. Gen., 26, (1993), L723- L728.
20. [20] Y. Peng, W. Chen, A new similarity solution of the Burgers equation with linear damping Czech. J, Phys., 56, (2008), 317-428.
21. [21] B. M. Vaganan, M. S. Kumaran, Similarity Solutions of the Burgers Equation with linear damping, Appl. Math. Lett. 17, (2004), 1191-1196.
22. [22] B. M. Vaganan, M. S. Kumaran, Kummer function solutions of damped Burgers equations with time-dependent viscosity by exact linearization, Nonlinear Anal. Real World Appl., 4, (2003), 723-741.
23. [23] X. M. Li, A. H. Chen, Darboux transformation and multi-soliton solutions of Boussinesq-Burgers equation, Phys. Lett. A, 342, (2005), 413-420.
24. [24] L. Gao, W. Xu, Y. Tang, G. Meng, New families of travelling wave solutions for Boussinesq-Burgers equation and (3 +1)-dimensional Kadomtsev-Petviashvili equation, Phys. Lett. A, 366, (2007), 411-421.
25. [25] A. S. A. Rady, M. Khalfallah, On soliton solutions for Boussinesq-Burgers equations, Commun. Nonlinear Sci. Numer. Simulat., 15, (2010), 886-894.