Variational iteration method: New development and applications

By Ji-Huan He, Xu-Hong Wu. Computers and Mathematics with Applications, 2007

827 citations recorded by the source corpus.

Variational iteration method has been favourably applied to various kinds of nonlinear problems. The main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this paper recent trends and developments in the use of the method are reviewed. Major applications to nonlinear wave equation, nonlinear fractional differential equations, nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed. The confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods. Variational iteration method is uniquely qualified to address this challenge. The flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions. This paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems. Both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations. A new iteration formulation is suggested to overcome the shortcoming. A very useful formulation for determining approximately the period of a nonlinear oscillator is suggested. Examples are given to illustrate the solution procedure.

Subfields: Modeling and Simulation, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b9a067b66f7fe38536153a8718968d7b19311440

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Citations in the corpus

Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review (2012, methodology): “124 The VIM is useful to obtain exact and approximate solutions of linear and nonlinear differen125 tial equations [35, 57, 62, 99, 104, 117, 122, 136, 139, 153, 167, 177, 179, 184, 191, 202, 206].”
A Tutorial Review on Fractal Spacetime and Fractional Calculus (2014, positive): “The variational iteration method [47–51] has been shown to solve a large class of nonlinear differential problems effectively, easily, and accurately with the approximations converging rapidly to accurate solutions.”
Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation (2010, background): “A number of authors have investigated analytic solutions for α ≤ 1 [1, 16, 34, 35, 46].”
Free vibration analysis of rotating tapered Timoshenko beams via variational iteration method (2017, methodology): “Variational iteration method is a relatively new technique initially proposed by He (1999), which is capable of dealing with a number of linear and nonlinear problems in an easy and efficient way (He and Wu, 2007; He, 2007).”
Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations (2020, background): “[55,56], a more summarizing iteration formula can be made, known as variational iteration algorithm-II (VIA-II),”
Variational iteration method for the Burgers' flow with fractional derivatives—New Lagrange multipliers (2013, neutral): “(25a) in [25]) for the ordinary differential equation (ODE) du dt þ R1⁄2u þ N1⁄2u 1⁄4 f ðtÞ, G.”