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**On the solution of two-dimensional coupled Burgers’ equations by variational iteration method.**
*(English)*
Zbl 1197.65203

Summary: By means of variational iteration method the solutions of two-dimensional Burgers’ and inhomogeneous coupled Burgers’ equations are exactly obtained, comparison with the Adomian decomposition method is made, showing that the former is more effective than the later. In this paper, He’s variational iteration method is given approximate solutions that can converge to its exact solutions faster than those of Adomian’s method.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

### MSC:

65N99 | Numerical methods for partial differential equations, boundary value problems |

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

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\textit{A. A. Soliman}, Chaos Solitons Fractals 40, No. 3, 1146--1155 (2009; Zbl 1197.65203)

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### References:

[1] | Ablowitz, M.J.; Clarkson, P.A., Solitons, nonlinear evolution equations and inverse scattering, (1991), Cambridge University Press Cambridge · Zbl 0762.35001 |

[2] | Wadati, M.; Sanuki, H.; Konno, K.; Method, Relationships among Inverse, Backlund transformation and an infinite number of conservation laws, Prog theor phys, 53, 419-436, (1975) |

[3] | Gardner, C.S.; Green, J.M.; Kruskal, M.D.; Miura, R.M., Method for solving the Korteweg-devries equation, Phys rev lett, 19, 1095-1097, (1967) · Zbl 1061.35520 |

[4] | Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys rev lett, 27, 1192-1194, (1971) · Zbl 1168.35423 |

[5] | () |

[6] | E Zayed, E.M.; Zedan, H.A.; Gepreel, K.A., Group analysis and modified extended tanh-function to find the invariant solutions and soliton solutions for nonlinear Euler equations, Int J non-linear sci numer simulat, 5, 3, 221-234, (2004) · Zbl 1401.35014 |

[7] | Abdusalam, H.A., On an improved complex tanh-function method, Int J non-linear sci num simulation, 6, 2, 99-106, (2005) · Zbl 1401.35012 |

[8] | Bender, C.M.; Pinsky, K.S.; Simmons, L.M., A new perturbative approach to nonlinear problems, J math phys, 30, 7, 1447-1455, (1989) · Zbl 0684.34008 |

[9] | Andrianov, I.; Manevitch, L., Asymptotology: ideas, methods, and applications, (2003), Kluwer Academic Publishers |

[10] | He, J.H., Homotopy perturbation technique, Comput methods appl mech eng, 178, 257-262, (1999) · Zbl 0956.70017 |

[11] | He, J.H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int J nonlinear mech, 35, 37-43, (2000) · Zbl 1068.74618 |

[12] | He, J.H., Homotopy perturbation method: a new nonlinear analytical technique, Appl math comput, 135, 1, 73-79, (2003) · Zbl 1030.34013 |

[13] | He, J.H., Homotopy perturbation method for bifurcation of nonlinear problems, Int J non-linear sci num simulat, 6, 207-208, (2005) · Zbl 1401.65085 |

[14] | Fan, E., Soliton solutions for a generalized hirota – satsuma coupled KdV equation and a coupled mkdv equation, Phys lett A, 282, 18-22, (2001) · Zbl 0984.37092 |

[15] | Wu, Y.T.; Geng, X.G.; Hu, X.B.; Zhu, S.M., A generalized hirota – satsuma coupled Korteweg-de Vries equation and miura transformations, Phys lett A, 255, 259-264, (1999) · Zbl 0935.37029 |

[16] | Satsuma, J.; Hirota, R., A coupled KdV equation is one case of the four-reduction of the KP hierarchy, J phys soc jpn, 51, 3390-3397, (1982) |

[17] | Fan, E.G.; Zhang, H.Q., A note on the homogeneous balance method, Phys lett A, 246, 403-406, (1998) · Zbl 1125.35308 |

[18] | Burger, J.M., A mathematical model illustrating the theory of turbulence, Adv appl mech, I, 171-199, (1948) |

[19] | Cole, J.D., On a quasilinear parabolic equations occurring in aerodynamics, Quart appl math, 9, 225-236, (1951) · Zbl 0043.09902 |

[20] | Fletcher, J.D., Generating exact solutions of the two-dimensional burgers’ equations, Int J numer methods fluids, 3, 213-216, (1983) · Zbl 0563.76082 |

[21] | Jain, P.C.; Holla, D.N., Numerical solution of coupled Burgers δ equations, Int J numer methods eng, 12, 213-222, (1978) · Zbl 0388.76049 |

[22] | Wubs, F.W.; de Goede, E.D., An explicit – implicit method for a class of time-dependent partial differential equations, Appl numer math, 9, 157-181, (1992) · Zbl 0749.65068 |

[23] | Bahadir, A.R., A fully implicit finite-difference scheme for two-dimensional burgers’ equations, Appl math comput, 137, 131-137, (2003) · Zbl 1027.65111 |

[24] | Soliman AA. New numerical technique for Burger’s equation based on similarity reductions. In: International conference on computational fluid dynamics; 2000. p. 559-66. |

[25] | Fletcher, C.J., A comparison of finite element and finite difference solutions of the one-and two-dimensional burgers’ equations, J comput phys, 51, 159, (1983) · Zbl 0525.65077 |

[26] | Esipov, S.E., Coupled burgers’ equations: a model of polydispersive sedimentation, Phys rev E, 52, 3711-3718, (1995) |

[27] | Nee, J.; Duan, J., Limit set of trajectories of the coupled viscous burgers’ equations, Appl math lett, 11, 1, 57-61, (1998) · Zbl 1076.35537 |

[28] | He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput methods appl mech eng, 167, 57-68, (1998) · Zbl 0942.76077 |

[29] | He, J.H., Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput methods appl mech eng, 167, 69-73, (1998) · Zbl 0932.65143 |

[30] | He, J.H., Variational iteration method – a kind of non-linear analytical technique: some examples, In J non-linear mech, 34, 699-708, (1999) · Zbl 1342.34005 |

[31] | He, J.H., A new approach to non-linear partial differential equations, Commun non-linear sci numer simulat, 2, 4, 230-235, (1997) |

[32] | He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl math comput, 114, 2,3, 115-123, (2000) · Zbl 1027.34009 |

[33] | Marinca, V., An approximate solution for one-dimensional weakly nonlinear oscillations, Int J non-linear sci num simulat, 3, 107-110, (2002) · Zbl 1079.34028 |

[34] | Abdou, M.A.; Soliman, A.A., Variational iteration method for solving burger’s and coupled burger’s equations, J comput appl math, 181, 2, 245-251, (2004) · Zbl 1072.65127 |

[35] | Soliman, A.A., Numerical simulation of the generalized regularized long wave equation by he’s variational iteration method, Math comput simulat, 70, 2, 119-124, (2005) · Zbl 1152.65467 |

[36] | Soliman, A.A., A numerical simulation and explicit solutions of kdv – burgers’ and lax’s seventh-order KdV equations, Chaos solitons and fractals, 29, 294-302, (2006) · Zbl 1099.35521 |

[37] | Abdou, M.A.; Soliman, A.A., New applications of variational iteration method, Physica D, 211, 1-8, (2005) · Zbl 1084.35539 |

[38] | Yusufoglu, E., Variational iteration method for construction of some compact and noncompact structures of klein – gordon equations, Int J non-linear sci num simulat, 8, 2, 152-158, (2007) |

[39] | Sweilam, N.H.; Khader, M.M., Variational iteration method for one dimensional nonlinear thermoelasticity, Chaos soliton & fractals, 32, 145-149, (2007) · Zbl 1131.74018 |

[40] | Tari, H.; Ganji, D.D.; Rostamian, M., Approximate solutions of K(2,2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method, Int J non-linear sci num simulat, 8, 203-210, (2007) |

[41] | Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos soliton & fractals, 31, 1248-1255, (2007) · Zbl 1137.65450 |

[42] | Momani, S.; Abuasad, S., Application of he’s variational iteration method to Helmholtz equation, Chaos soliton & fractals, 27, 1119-1123, (2006) · Zbl 1086.65113 |

[43] | Odibat, Z.M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Int J non-linear sci num simulation, 7, 27-34, (2006) · Zbl 1401.65087 |

[44] | Bildik, N.; Konuralp, A., The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, Int J non-linear sci num simulation, 7, 65-70, (2006) · Zbl 1401.35010 |

[45] | He, J.H., Some asymptotic methods for strongly nonlinear equations, Int J mod phys B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039 |

[46] | He, J.H.; Wu, X.H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos soliton & fractals, 29, 108-113, (2006) · Zbl 1147.35338 |

[47] | Draganescu, Gh.E.; Capalnasan, V., Nonlinear relaxation phenomena in polycrystalline solids, Int J non-linear sci num simulat, 4, 219-226, (2004) |

[48] | Wazwaz, A.M., A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems, Comput math appl, 4, 1237-1244, (2001) · Zbl 0983.65090 |

[49] | He, J.H., Approximate analytical methods in science and engineering, (2002), Henan Sci. & Tech. Press Berlin, (in Chinese) |

[50] | He, J.H., Generalized variational principles in fluids, (2003), Science & Culture Publishing House of China Hongkong, (in Chinese) · Zbl 1054.76001 |

[51] | EL-Sayed, S.M.; Kaya, D., On the numerical solution of the system of two-dimensional burgers’ equations by the decomposition method, Appl math comput, 158, 101-109, (2004) · Zbl 1061.65099 |

[52] | Kaya, D., An explicit solution of coupled viscous burgers’ equation by the decomposition method, Intj math sci, 27, 675-680, (2001) · Zbl 0997.35077 |

[53] | Wazwaz, A.M., Necessary conditions for appearance of noise terms in decomposition solution series, J math anal appl, 5, 265-274, (1997) · Zbl 0882.65132 |

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.