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Variational iteration method – some recent results and new interpretations. (English) Zbl 1119.65049
Summary: This paper is an elementary introduction to the concepts of variational iteration method. First, the main concepts in variational iteration method, such as general Lagrange multiplier, restricted variation, correction functional, are explained heuristically. Subsequently, the solution procedure is systematically addressed, in particular, for nonlinear oscillators. Particular attention is paid throughout the paper to give an intuitive grasp for the method. The main motivation is to put things together in a convenient form for later reference and systematic use.

MSC:
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
65H05 Numerical computation of solutions to single equations
47J25 Iterative procedures involving nonlinear operators
34L05 General spectral theory of ordinary differential operators
34A34 Nonlinear ordinary differential equations and systems, general theory
65K10 Numerical optimization and variational techniques
49J15 Existence theories for optimal control problems involving ordinary differential equations
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[1] Abdou, M.A.; Soliman, A.A., New applications of variational iteration method, Physica D, 211, 1-2, 1-8, (2005) · Zbl 1084.35539
[2] 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
[3] Abulwafa, E.M.; Abdou, M.A.; Mahmoud, A.A., The solution of nonlinear coagulation problem with mass loss, Chaos solitons fractals, 29, 2, 313-330, (2006) · Zbl 1101.82018
[4] E.M. Abulwafa, M.A. Abdou, A.A. Mahmoud, Nonlinear fluid flows in pipe-like domain problem using variational-iteration method, Chaos Solitons Fractals, in press, doi:10.1016/j.chaos.2005.11.050. · Zbl 1128.76019
[5] 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, Internat. J. nonlinear sci. numer. simulation, 7, 1, 65-70, (2006)
[6] Draganescu, G.E.; Capalnasan, V., Nonlinear relaxation phenomena in polycrystalline solids, Internat. J. nonlinear sci. numer. simulation, 4, 3, 219-225, (2003)
[7] Draganescu, Gh.E.; Cofan, N.; Rujan, D.L., Nonlinear vibration of a nano sized sensor with fractional damping, J. optoelectron. adv. mater., 7, 2, 877-884, (2005)
[8] He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. methods appl. mech. eng., 167, 1-2, 57-68, (1998) · Zbl 0942.76077
[9] He, J.H., Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. methods appl. mech. eng., 167, 1-2, 69-73, (1998) · Zbl 0932.65143
[10] He, J.H., Variational iteration method—a kind of non-linear analytical technique: some examples, Internat. J. nonlinear mech., 34, 4, 699-708, (1999) · Zbl 1342.34005
[11] He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl. math. comput., 114, 2-3, 115-123, (2000) · Zbl 1027.34009
[12] J.H. He, Generalized Variational Principles in Fluids, Science & Culture Publishing House of China, 2003 ( in Chinese). · Zbl 1054.76001
[13] He, J.H., Some asymptotic methods for strongly nonlinear equations, Internat. J. modern phys. B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039
[14] He, J.H., Non-perturbative methods for strongly nonlinear problems, (2006), dissertation.de-Verlag im Internet GmbH Berlin
[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] Inokuti, M.; Sekine, H.; Mura, T., General use of the Lagrange multiplier in nonlinear mathematical physics, (), 156-162
[17] Marinca, V., An approximate solution for one-dimensional weakly nonlinear oscillations, Internat. J. nonlinear sci. numer. simulation, 3, 2, 107-120, (2002) · Zbl 1079.34028
[18] M. Moghimi, F.S.A. Hejazi, Variational iteration method for solving generalized Burger-Fisher and Burger equations, Chaos Solitons Fractals, in press, doi:10.1016/j.chaos.2006.03.0331. · Zbl 1138.35398
[19] Momani, S.; Abuasad, S., Application of He’s variational iteration method to Helmholtz equation, Chaos solitons fractals, 27, 5, 1119-1123, (2006) · Zbl 1086.65113
[20] S. Momani, Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons Fractals, in press, doi:10.1016/j.chaos.2005.10.068. · Zbl 1137.65450
[21] S. Momani, Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A 355 (4-5) (2006) 271-279. · Zbl 1378.76084
[22] Odibat, Z.M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Internat. J. nonlinear sci. numer. simulation, 7, 1, 27-34, (2006)
[23] Soliman, A.A., Numerical simulation of the generalized regularized long wave equation by He’s variational iteration method, Math. comput. simulation, 70, 2, 119-124, (2005) · Zbl 1152.65467
[24] Soliman, A.A., A numerical simulation and explicit solutions of KdV-burgers’ and Lax’s seventh-order KdV equations, Chaos solitons fractals, 29, 2, 294-302, (2006) · Zbl 1099.35521
[25] N.H. Sweilam, M.M. Khader, Variational iteration method for one dimensional nonlinear thermoelasticity, Chaos Solitons Fractals, in press, doi:10.1016/j.chaos.2005.11.028. · Zbl 1131.74018
[26] M. Tatari, M. Dehghan, He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, Chaos Solitons Fractals, in press, doi:10.1016/j.chaos.2006.01.059. · Zbl 1131.65084
[27] Wazwaz, A.M., A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems, Comput. math. appl., 41, 10-11, 1237-1244, (2001) · Zbl 0983.65090
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