zbMATH — the first resource for mathematics

Application of He’s variational iteration method to nonlinear Jaulent-Miodek equations and comparing it with ADM. (English) Zbl 1120.65107
Summary: Instead of finding a small parameter for solving nonlinear problems through perturbation method, a new analytical method called J.-H. He’s variational iteration method [Int. J. Non-Linear Mech. 34, No. 4, 699–708 (1999; Zbl 1342.34005)] is introduced to be applied to solve nonlinear Jaulent-Miodek, coupled Korteweg-de Vries (KdV) and coupled modified KdV equations. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers can be identified optimally via the variational theory. The results are compared with exact solutions.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI
[1] 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, (2005) · Zbl 1072.65127
[2] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Boston · Zbl 0802.65122
[3] 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
[4] Ganji, D.D.; Rajabi, A., Assessment of homotopy-perturbation and perturbation methods in heat radiation equations, Internat commun. heat mass transfer, 33, 3, 391-400, (2006)
[5] 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
[6] He, J.H., Approximate solution for nonlinear differential equations with convolution product nonlinearities, Comput. methods appl. mech. eng., 167, 69-73, (1998) · Zbl 0932.65143
[7] He, J.H., Homotopy perturbation technique, Comput. methods appl. mech. eng., 178, 257-262, (1999) · Zbl 0956.70017
[8] He, J.H., Variational iteration method: a kind of nonlinear analytical technique: some examples, Internat nonlinear mech., 344, 699-708, (1999) · Zbl 1342.34005
[9] He, J.H., A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Internat J. nonlinear mech., 35, 37-43, (2000) · Zbl 1068.74618
[10] He, J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, solitons fractals, 26, 3, 695-700, (2005) · Zbl 1072.35502
[11] He, J.H., Limit cycle and bifurcation of nonlinear problems, Chaos, solitons fractals, 26, 3, 827-833, (2005) · Zbl 1093.34520
[12] He, J.H., Homotopy perturbation method for bifurcation of nonlinear problems, Internat J. nonlinear sci. numer. simul., 6, 2, 207-208, (2005)
[13] He, J.H., Non-perturbative methods for strongly nonlinear problems, (2006), Dissertation.de-Verlag im Internet GmbH Berlin
[14] He, J.H., Some asymptotic methods for strongly nonlinear equations, Internat J. mod. phys. B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039
[15] He, J.H.; Wu, X.H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, solitons fractals, 29, 1, 108-113, (2006) · Zbl 1147.35338
[16] Kaya, D.; El-Sayed, S.M., A numerical method for solving jaulent – miodek equation, Phys. lett. A, 318, 345-353, (2003) · Zbl 1045.35065
[17] Liao, S.J., An approximate solution technique not depending on small parameters: a special; example, Internat J. nonlinear mech., 303, 371-380, (1995) · Zbl 0837.76073
[18] Wazwaz, A.M., Partial differential equations: methods and applications, (2002), Balkema Rottesdam · Zbl 0997.35083
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.