×

Coupling technique of variational iteration and homotopy perturbation methods for nonlinear matrix differential equations. (English) Zbl 1267.65102

Summary: The variational iteration method proposed by Ji-Huan He is a new analytical method to solve nonlinear equations. This paper applies the method to search for exact analytical solutions of linear differential equations with constant coefficients. Furthermore, based on the precise integration method, a coupling technique of the variational iteration method and homotopy perturbation method is proposed to solve nonlinear matrix differential equations. A dynamic system and Burgers equation are taken as examples to illustrate its effectiveness and convenience.

MSC:

65L99 Numerical methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] He, J.-H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, International journal of non-linear mechanics, 35, 1, 37-43, (2000) · Zbl 1068.74618
[2] He, J.-H., New interpretation of homotopy perturbation method, International journal of modern physics B, 20, 18, 2561-2568, (2006)
[3] He, J.-H., Homotopy perturbation technique, Computer methods in applied mechanics and engineering, 178, 3-4, 257-262, (1999) · Zbl 0956.70017
[4] He, J.-H., Homotopy perturbation method: A new nonlinear technique, Applied mathematics and computation, 135, 73-79, (2003) · Zbl 1030.34013
[5] He, J.-H., Homotopy perturbation method: A new nonlinear analytical technique, Applied mathematics and computation, 135, 73-79, (2003) · Zbl 1030.34013
[6] He, J.-H., Limit cycle and bifurcation of nonlinear problems, Chaos, solitons and fractals, 26, 827-833, (2005) · Zbl 1093.34520
[7] He, J.-H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, solitons and fractals, 26, 695-700, (2005) · Zbl 1072.35502
[8] He, J.-H., Periodic solutions and bifurcations of delay-differential equations, Physics letters A, 347, 228-230, (2005) · Zbl 1195.34116
[9] He, J.-H., The homotopy-perturbation method for nonlinear oscillators with discontinuities, Applied mathematics and computation, 151, 287-292, (2004) · Zbl 1039.65052
[10] He, J.-H., Homotopy-perturbation method for bifurcation of nonlinear problems, International journal of nonlinear sciences and numerical simulation, 6, 207-208, (2005) · Zbl 1401.65085
[11] Cveticanin, L., Homotopy-perturbation method for pure nonlinear differential equation, Chaos, solitons and fractals, 30, 5, 1221-1230, (2006) · Zbl 1142.65418
[12] Abbasbandy, S., Application of he’s homotopy perturbation method for Laplace transform, Chaos, solitons and fractals, 30, 5, 1206-1212, (2006) · Zbl 1142.65417
[13] Rafei, M.; Ganji, D.D., Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method, International journal of nonlinear sciences and numerical simulation, 7, 3, 321-328, (2006) · Zbl 1160.35517
[14] Siddiqui, A.M.; Mahmood, R.; Ghori, Q.K., Thin film flow of a third grade fluid on a moving belt by he’s homotopy perturbation method, International journal of nonlinear sciences and numerical simulation, 7, 1, 7-14, (2006) · Zbl 1187.76622
[15] Siddiqui, A.M.; Ahmed, M.; Ghori, Q.K., Couette and Poiseuille flows for non-Newtonian fluids, International journal of nonlinear sciences and numerical simulation, 7, 1, 15-26, (2006) · Zbl 1401.76018
[16] He, J.-H., Variational iteration method: A kind of nonlinear analytical technique: some examples, International journal of nonlinear mechanics, 344, 699-708, (1999) · Zbl 1342.34005
[17] He, J.-H., Variational iteration method for autonomous ordinary differential systems, Applied mathematics and computation, 114, 2-3, 115-123, (2000) · Zbl 1027.34009
[18] He, Ji-H.; Wu, X.-H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, solitons and fractals, 29, 1, 108-113, (2006) · Zbl 1147.35338
[19] He, J.-H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer methods in applied mechanics and engineering, 167, 57-68, (1998) · Zbl 0942.76077
[20] He, J.-H., Approximate solution for nonlinear differential equations with convolution product nonlinearities, Computer methods in applied mechanics and engineering, 167, 69-73, (1998) · Zbl 0932.65143
[21] He, J.-H., Some asymptotic methods for strongly nonlinear equations, International journal of modern physics B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039
[22] Abdou, M.A.; Soliman, A.A., Variational iteration method for solving burgers’ and coupled burgers’ equations, Journal of computational and applied mathematics, 181, 245-251, (2005) · Zbl 1072.65127
[23] Soliman, A.A., A numerical simulation and explicit solutions of KdV-burgers’ and lax’s seventh-order KdV equations, Chaos, solitons and fractals, 29, 2, 294-302, (2006) · Zbl 1099.35521
[24] Abulwafa, E.M.; Abdou, M.A.; Mahmoud, A.A., The solution of nonlinear coagulation problem with mass loss, Chaos, solitons and fractals, 29, 2, 313-330, (2006) · Zbl 1101.82018
[25] Momani, S.; Abuasad, S., Application of he’s variational iteration method to Helmholtz equation, Chaos, solitons and fractals, 27, 5, 1119-1123, (2006) · Zbl 1086.65113
[26] 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, International journal of nonlinear sciences and numerical simulation, 7, 1, 65-70, (2006) · Zbl 1401.35010
[27] Odibat, Z.M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, International journal of nonlinear sciences and numerical simulation, 7, 1, 27-34, (2006) · Zbl 1401.65087
[28] Zhong, W.-X., On precise integration method, Journal of computational and applied mathematics, 163, 59-78, (2004) · Zbl 1046.65053
[29] Wan, D.-C.; Wei, G.-W., The study of quasi wavelets based numerical method applied to burgers’ equations, Applied mathematics and mechanics, 21, 1099-1100, (2000) · Zbl 1003.76070
[30] Wei, G.-W., Quasi wavelets and quasi interpolating wavelets, Chemical physics letters, 296, 3-4, 215-222, (1998)
[31] S.-L. Mei, C.-J. Du, S.-W. Zhang, Asymptotic numerical method for multi-degree-of-freedom nonlinear dynamic systems, Chaos, Solitons and Fractals (2006), doi:10.1016/j.chaos.2006.05.067
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.