Variational iteration method – a kind of non-linear analytical technique: some examples
01 Jul 1999-International Journal of Non-linear Mechanics (INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS)-Vol. 34, Iss: 4, pp 699-708
TL;DR: In this paper, a variational iteration method for non-linear problems is proposed, where the problems are initially approximated with possible unknowns and a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.
Abstract: In this paper, a new kind of analytical technique for a non-linear problem called the variational iteration method is described and used to give approximate solutions for some well-known non-linear problems. In this method, the problems are initially approximated with possible unknowns. Then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory. Being different from the other non-linear analytical methods, such as perturbation methods, this method does not depend on small parameters, such that it can find wide application in non-linear problems without linearization or small perturbations. Comparison with Adomian’s decomposition method reveals that the approximate solutions obtained by the proposed method converge to its exact solution faster than those of Adomian’s method.
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TL;DR: In this paper, a survey of recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones, is presented.
Abstract: This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modied perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this eld in China so the references, therefore, are not exhaustive.
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TL;DR: The result reveals that the first order of approximation obtained by the proposed method is valid uniformly even for very large parameter, and is more accurate than the perturbation solutions.
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TL;DR: In this article, the homotopy perturbation method is applied to the search for traveling wave solutions of nonlinear wave equations and some examples are given to illustrate the determination of the periodic solutions or the bifurcation curves of the nonlinear Wave equations.
Abstract: The homotopy perturbation method is applied to the search for traveling wave solutions of nonlinear wave equations. Some examples are given to illustrate the determination of the periodic solutions or the bifurcation curves of the nonlinear wave equations.
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TL;DR: Wazwaz et al. as mentioned in this paper applied homotopy perturbation method to nonlinear boundary value problems and compared the result obtained by the present method with that obtained by Adomian method.
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TL;DR: The main concepts in variational iteration method, such as general Lagrange multiplier, restricted variation, correction functional, are explained heuristically and the solution procedure is systematically addressed, in particular, for nonlinear oscillators.
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Cites methods from "Variational iteration method – a ki..."
...Introduction The variational iteration method [8–11,15] has been extensively worked out over a number of years by numerous authors....
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References
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TL;DR: The decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions, closure approximations, perturbation theory, or restrictive assumptions on stochasticitiy as mentioned in this paper.
1,409 citations
"Variational iteration method – a ki..." refers methods in this paper
...Comparison with Adomian’s decomposition method [12, 13]...
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...In the Adomian’s method [12, 13], the linear term ( ̧ ) in Eq....
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01 Jan 1980
455 citations
Book•
07 Aug 1985
TL;DR: Algebraic and Transcendental Equations Integrals Conservative Equations with Odd Nonlinearities Free Oscillations of Positively Damped Systems Self-Excited Oscillators Free oscillations of Systems with Quadratic Nonlinearity General Systems with Odd nonlinearities Nonlinear Systems Subject to Harmonic Excitations Multifrequency Excitations Parametric Excitations Boundary-Layer Problems Linear Equation with Variable Coefficients Differential Equations and a Large Parameter Solvability Conditions Index
Abstract: Algebraic and Transcendental Equations Integrals Conservative Equations with Odd Nonlinearities Free Oscillations of Positively Damped Systems Self-Excited Oscillators Free Oscillations of Systems with Quadratic Nonlinearities General Systems with Odd Nonlinearities Nonlinear Systems Subject to Harmonic Excitations Multifrequency Excitations Parametric Excitations Boundary-Layer Problems Linear Equations with Variable Coefficients Differential Equations with a Large Parameter Solvability Conditions Index
429 citations
"Variational iteration method – a ki..." refers methods in this paper
...while the period obtained by perturbation method [9] reads 1"2n(1!5eA4/16) (32) and the exact one can be readily obtained as follows:...
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TL;DR: In this article, the variational iteration method is applied to delay analytical approximate solutions for analytical approximations to delay differential equations, and it is shown to be applicable to delay-sensitive problems.
375 citations
154 citations
"Variational iteration method – a ki..." refers background in this paper
...with u 0 (x ) as initial approximation with possible unknowns, and uJ n is considered as a restricted variation [3], i....
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