Journal ArticleDOI
Some asymptotic methods for strongly nonlinear equations
TLDR
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.read more
Citations
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Journal ArticleDOI
Variational iteration method-Some recent results and new interpretations
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.
Journal ArticleDOI
Variational iteration method: New development and applications
Ji-Huan He,Xu-Hong Wu +1 more
TL;DR: The basic conceptual framework of variational iteration technique with application to nonlinear problems is outlined and a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested.
Journal ArticleDOI
The application of homotopy analysis method to nonlinear equations arising in heat transfer
TL;DR: In this paper, the homotopy analysis method (HAM) is compared with the numerical and HPM in the heat transfer file and the auxiliary parameter ℏ, which provides a simple way to adjust and control the convergence region of solution series.
Journal ArticleDOI
Addendum:. New Interpretation of Homotopy Perturbation Method
TL;DR: In this article, a guided tour through the mathematics needed for a proper understanding of homotopy perturbation method as applied to various nonlinear problems is presented, and a new interpretation of the concept of constant expansion is given.
Journal ArticleDOI
An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering
TL;DR: In this article, an elementary introduction to the concepts of the recently developed asymptotic methods and new developments is given, giving an intuitive grasp for Lagrange multiplier, calculus of variations, optimization, variational iteration method, parameterexpansion method, exp-function method, homotopy perturbation method, and ancient Chinese mathematics as well.
References
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Journal ArticleDOI
Homotopy perturbation technique
TL;DR: In this paper, the homotopy perturbation technique does not depend upon a small parameter in the equation and can be obtained uniformly valid not only for small parameters, but also for very large parameters.
Journal ArticleDOI
Variational iteration method – a kind of non-linear analytical technique: some examples
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.
Book
Asymptotic Methods in the Theory of Nonlinear Oscillations
TL;DR: In this article, a wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering.
Journal ArticleDOI
Homotopy perturbation method: a new nonlinear analytical technique
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.
Journal ArticleDOI
A review of the decomposition method in applied mathematics
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.