##
**A residual power series technique for solving Boussinesq-Burgers equations.**
*(English)*
Zbl 1438.35363

Summary: In this paper, a residual power series method (RPSM) is combining Taylor’s formula series with residual error function, and is investigated to find a novel analytical solution of the coupled strong system nonlinear Boussinesq-Burgers equations according to the time. Analytical solution was purposed to find approximate solutions by RPSM and compared with the exact solutions and approximate solutions obtained by the homotopy perturbation method and optimal homotopy asymptotic method at different time and concluded that the present results are more accurate and efficient than analytical methods studied. Then, analytical simulations of the results are studied graphically through representations for action of time and accuracy of method.

### MSC:

35Q53 | KdV equations (Korteweg-de Vries equations) |

35C10 | Series solutions to PDEs |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

94A20 | Sampling theory in information and communication theory |

PDF
BibTeX
XML
Cite

\textit{B. A. Mahmood} and \textit{M. A. Yousif}, Cogent Math. 4, Article ID 1279398, 11 p. (2017; Zbl 1438.35363)

Full Text:
DOI

### References:

[1] | Abdel Rady, A. S. A.; Khalfallah, M., On soliton solutions for Boussinesq-Burgers equations, Communications in Nonlinear Science and Numerical Simulation, 15, 886-894 (2010) · Zbl 1221.35357 |

[2] | Abdel Rady, A. S. A.; Osman, E. S.; Khalfallah, M., Multi-soliton solution, rational solution of the Boussinesq-Burgers equations, Communications in Nonlinear Science and Numerical Simulation, 15, 1172-1176 (2010) · Zbl 1221.35359 |

[3] | Arqub, O. A., Series solution of fuzzy differential equations under strongly generalized differentiability, Journal of Advanced Research in Applied Mathematics, 5, 31-52 (2013) |

[4] | Arqub, O. A.; Abo-Hammour, Z.; Al-Badarneh, R.; Momani, S., A reliable analytical method for solving higher-order initial value problems, Discrete Dynamics in Nature and Society, 12 pages (2013) · Zbl 1417.34044 |

[5] | Arqub, O. A.; El-Ajou, A.; Bataineh, A.; Hashim, I., A representation of the exact solution of generalized Lane-Emden equations using a new analytical method, Abstract and Applied Analysis, 10 pages (2013) · Zbl 1291.34024 |

[6] | Arqub, O. A.; El-Ajou, A.; Momani, S., Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations, Journal of Computational Physics, 293, 385-399 (2015) · Zbl 1349.35394 |

[7] | Arqub, O. A.; Maayah, B., Solutions of Bagley-Torvik and Painlevé equations of fractional order using iterative reproducing kernel algorithm, Neural Computing & Applications, 27, 1-15 (2016) |

[8] | Changjiang, Z.; Renjun, D., Existence and uniqueness of entropy solution to initial boundary value problem for the inviscid Burgers equation, Journal of Physics A: Mathematical and General, 36, 2099-2107 (2003) · Zbl 1168.35393 |

[9] | Chen, A. H.; Li, X. M., Darboux transformation and soliton solutions for Boussinesq-Burgers equation, Chaos, Solitons & Fractals, 27, 43-49 (2006) · Zbl 1088.35527 |

[10] | Ding, W.; Wang, Z., Global existence and asymptotic behavior of the Boussinesq-Burgers system, Journal of Mathematical Analysis and Applications, 424, 584-597 (2015) · Zbl 1307.35044 |

[11] | El-Ajou, A.; Arqub, O. A.; Al-Smadi, M., A general form of the generalized Taylor’s formula with some applications, Applied Mathematics and Computation, 256, 851-859 (2015) · Zbl 1338.40007 |

[12] | El-Ajou, A.; Arqub, O. A.; Momani, S., Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: A new iterative algorithm, Journal of Computational Physics, 293, 81-95 (2015) · Zbl 1349.65546 |

[13] | El-Ajou, A.; Arqub, O. A.; Momani, S.; Baleanu, D.; Alsaedi, A., A novel expansion iterative method for solving linear partial differential equations of fractional order, Applied Mathematics and Computation, 257, 119-133 (2015) · Zbl 1339.65201 |

[14] | El-Ajou, A.; Arqub, O. A.; Zhour, Z. A.; Momani, S., New results on fractional power series: Theory and applications, Entropy, 15, 5305-5323 (2013) · Zbl 1337.26010 |

[15] | Gao, L.; Xu, W.; Tang, Y.; Meng, G., New families of travelling wave solutions for Boussinesq-Burger’s equation and (3 + 1)-dimensional Kadomtsev-Petviashvili equation, Physics Letters A, 366, 411-421 (2007) · Zbl 1203.35217 |

[16] | Hardik, S. P.; Meher, R., Application of laplace adomian decompositionmethod for the soliton solutions of Boussinesq-Burger equations, International Journal of Advances in Applied Mathematics and Mechanics, 3, 50-58 (2015) · Zbl 1359.65190 |

[17] | Khalfallah, M., Exact traveling wave solutions of the Boussinesq-Burgers equation, Mathematical and Computer Modelling, 49, 666-671 (2009) · Zbl 1165.35445 |

[18] | Komashynska, I.; Al-Smadi, M.; Al-Habahbeh, A.; Ateiwi, A., Analytical approximate solutions of systems of multi-pantograph delay differential equations using residual power-series method, Australian Journal of Basic and Applied Sciences, 8, 664-675 (2014) |

[19] | Komashynska, I.; Al-Smadi, M.; Ateiwi, A.; Al-Obaidy, S., Approximate analytical solution by residual power series method for system of Fredholm integral equations, Applied Mathematics & Information Sciences, 10, 975-985 (2016) |

[20] | Li, X. M.; Chen, A. H., Darboux transformation and multi-soliton solutions of Boussinesq-Burger’s equation, Physics Letters A, 342, 413-420 (2005) · Zbl 1222.35175 |

[21] | Liu, X., Global existence and uniqueness of solutions to the three-dimensional Boussinesq equations. Boundary Value Problems, 2016, 85 (2016) |

[22] | Wang, L.; Chen, X., Approximate analytical solutions of time fractional Whitham-Broer-Kaup equations by a residual power series method, Entropy, 17, 6519-6533 (2015) · Zbl 1338.35091 |

[23] | Wang, P.; Tian, B.; Liu, W.; Lü, X.; Jiang, Y., Lax pair, Bäcklund transformation and multi-soliton solutions for the Boussinesq-Burgers equations from shallow water waves, Applied Mathematics and Computation, 218, 1726-1734 (2011) · Zbl 1433.35302 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.