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Variational iteration method: New development and applications. (English) Zbl 1141.65372

Summary: The 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. The 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.

MSC:
65L05Initial value problems for ODE (numerical methods)
65-02Research monographs (numerical analysis)
34A34Nonlinear ODE and systems, general
45J05Integro-ordinary differential equations
65R20Integral equations (numerical methods)
45G10Nonsingular nonlinear integral equations