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Variational iteration method for singular perturbation initial value problems. (English) Zbl 1205.65210
Summary: The variational iteration method (VIM) is applied to solve singular perturbation initial value problems (SPIVPs). The obtained sequence of iterates is based on the use of Lagrange multipliers. Some convergence results of VIM for solving SPIVPs are given. Moreover, the illustrative examples show the efficiency of the method.

MSC:
65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
34E15 Singular perturbations, general theory for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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[1] Wazwaz, A.M., The variational iteration method for analytic treatment for linear and nonlinear odes, Appl. math. comput., 212, 120-134, (2009) · Zbl 1166.65353
[2] A. Saadatmandi, M. Dehghan, Variational iteration method for solving a generalized pantograph equation, Comput. Math. Appl., in press, doi:10.1016/j.camwa.2009.03.017 · Zbl 1189.65172
[3] Salkuyeh, D.K., Convergence of the variational iteration method for solving linear systems of ODEs with constant coefficients, Comput. math. appl., 56, 2027-2033, (2008) · Zbl 1165.65376
[4] Dejager, E.M.; Jiang, F.R., The theory of singular perturbation, (1996), Elsevier Science B.V. The Netherlands
[5] He, J.H.; Wu, X.H., Variational iteration method: new development and applications, Comput. math. appl., 54, 881-894, (2007) · Zbl 1141.65372
[6] He, J.H., Variational iteration method: some recent results and new interpretations, J. comput. appl. math., 207, 3-17, (2007) · Zbl 1119.65049
[7] He, J.H.; Wu, X.H., Variational iteration method for autonomous ordinary differential systems, Appl. math. comput., 114, 115-123, (2000) · Zbl 1027.34009
[8] He, J.H., Variational iteration method for delay differential equations, Commun. nonlinear sci. numer. simul., 2, 4, 235-236, (1997)
[9] He, J.H., Some applications of nonlinear fractional differential equation and their approximations, Bull. sci. technol., 15, 12, 86-90, (1999)
[10] He, J.H., A new approach to linear partial differential equations, Commun. nonlinear sci. numer. simul., 2, 4, 230-235, (1997)
[11] He, J.H., Approximate solution of nonlinear differential equation with convolution product nonlinearities, Comput. methods appl. mech. engrg., 167, 69-73, (1998) · Zbl 0932.65143
[12] He, J.H.; Wu, G.C.; Austin, F., The variational iteration method which should be followed, Nonlinear sci. lett. A, 1, 1, 1-30, (2010)
[13] Lu, J.F., Variational iteration method for solving two point boundary value problems, J. comput. appl. math., 207, 92-95, (2008) · Zbl 1119.65068
[14] Xu, L., Variational iteration method for solving integral equations, Comput. math. appl., 54, 1071-1078, (2007) · Zbl 1141.65400
[15] Rafei, M.; Ganji, D.D.; Daniali, H.; Pashaei, H., The variational iteration method for nonlinear oscillators with discontinuities, J. sound vibration, 30, 6, 1338-1360, (1992)
[16] Darvishi, M.T.; Khani, F.; Soliman, A.A., The numerical simulation for stiff systems of ordinary differential equations, Comput. math. appl., 54, 1055-1063, (2007) · Zbl 1141.65371
[17] Tatari, M.; Dehghan, M., On the convergence of He’s variational iteration method, J. comput. appl. math., 207, 121-128, (2007) · Zbl 1120.65112
[18] M. Mamode, Variational iteration method and initial value problems, Appl. Math. Comput., in press, doi:10.1016/j.amc.2009.05.008 · Zbl 1175.65118
[19] O’Malley, R.E., Singular perturbation methods for ordinary differential equations, (1990), Springer-Verlag New York
[20] R. Saadati, M. Dehghan, S.M. Vaezpour, M. Saravi, The convergence of He’s variational iteration method for solving integral equations, Comput. Math. Appl., in press, doi:10.1016/j.camwa.2009.03.008 · Zbl 1189.65312
[21] Yu, Z.H., Variational iteration method for solving the multi-pantograph delay equation, Phys. lett. A, 372, 6475-6479, (2008) · Zbl 1225.34024
[22] Marinca, V.; Herisanu, N.; Bota, C., Application of the variational iteration method to some nonlinear one dimensional oscillations, Meccanica, 43, 75-79, (2008) · Zbl 1238.70018
[23] Xiao, A.G.; Zhao, Y.X., Convergence of parallel multistep hybrid methods for singular perturbation problems, Appl. math. comput., 215, 2139-2148, (2009) · Zbl 1178.65092
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