Abstract
The variational iteration method was proposed by J. H. He in 1999 [153–155]. The method introduces a reliable and efficient process for a wide variety of scientific and engineering applications, linear and nonlinear, homogeneous or inhomogeneous, equations and systems of equations as well. The variational iteration method has no specific requirements, such as linearization, small variations, etc. for nonlinear operators. The power of the method gives it a wider applicability in handling a huge number of analytical and numerical applications.
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References
X.M. Tian, L.J. Feng, C. Delin, Non-linear vibration by a new method. J. Sound Vib. 215, 475–487 (1998)
J.H. He, Variational iteration method-some recent results and new interpretations. J. Comput. Appl. Math. 207(1), 3–17 (2007)
J.H. He, X.H. Wu, Variational iteration method: new developments and applications. Comput. Math. Appl. 54, 881–894 (2007)
J.H. He, G.-C. Wu, F. Austin, The variational iteration method which should be followed. Nonlin. Sci. Lett. A 1(1), 1–30 (2010)
A.A. Soliman, Numerical simulation and explicit solutions of KdV-Burgers and Lax’s seventh-order KdV equations. Chaos solitons Fract 29(2), 294–302 (2006)
E.M. Abulwafa, M.A. Abdou, A.M. Mahmood, The solution of nonlinear coagulation problem with mass loss. Chaos Solitons Fract 29(2), 313–330 (2006)
D. Slota, Direct and inverse one-plane Stefan problem solved by the variational iteration method. Comput. Math. Appl. 54, 1139–1146 (2007)
S. Momani, S. Abuasad, Application of He’s variational iteration method to Helmholtz equation. Chaos Solitons Fract 27(5), 1119–1123 (2006)
N. Bildik, A. Konuralp, The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int. J. Nonlin. Sci. Numer. Simul. 7(1), 65–70 (2006)
G.E. Drăgănescu, V. Căpălnăşan, Nonlinear relaxation phenomena in polycristaline solids. Int. J. Nonlin. Sci. Numer. Simul. 4(3), 219–225 (2003)
V. Marinca, N. Herişanu, Periodic solutions for some strongly nonlinear oscillation by He’s variational iteration method. Comput. Math. Appl. 54, 1188–1196 (2007)
R.E. Mickens, Oscillations in Planar Dynamic Systems (World Scientific, Singapore, 1966)
L. Elsgolts, Differential Equations and the Calculus of Variations (Mir Publishers, Moscow, 1980)
V. Marinca, N. Herişanu, D. Bălă, Some optimal approximate methods with application to thin film flow. WSEAS Trans. Syst. 7(9), 744–753 (2010)
V. Marinca, N. Herişanu, Analytical approach to the dynamic analysis of a rotating electric machine. Comput. Math. Appl. 58, 2320–2324 (2009)
H.P.W. Gottlieb, Frequencies of oscillators with fractional-power nonlinearities. J. Sound Vib. 261, 557–566 (2003)
H.K. Kuiken, On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small. IMA J. Appl. Math. 27, 387–405 (1991)
L. Xu, He’s homotopy perturbation method for boundary layer equation in unbounded domain. Comput. Math. Appl. 54, 1067–1070 (2007)
A.M. Wazwaz, The modified decomposition method and Pade approximants for a boundary layer equation in unbounded domain. Appl. Math. Comput. 177, 737–744 (2006)
S.J. Liao, On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004)
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Marinca, V., Herisanu, N. (2012). The Optimal Variational Iteration Method. In: Nonlinear Dynamical Systems in Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22735-6_8
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DOI: https://doi.org/10.1007/978-3-642-22735-6_8
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