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.

Some asymptotic methods for strongly nonlinear equations

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.

Application of homotopy perturbation method to nonlinear wave equations

This review is an elementary introduction to the concepts of the recently developed asymptotic methods and new developments. Particular attention is paid throughout the paper to giving an intuitive grasp for Lagrange multiplier, calculus of variations, optimization, variational iteration method, parameter-expansion method, exp-function method, homotopy perturbation method, and ancient Chinese mathematics as well. Subsequently, nanomechanics in textile engineering and E-innit y theory in high energy physics, Kleiber’s 3/4 law in biology, possible mechanism in spider-spinning process and fractal approach to carbon nanotube are briey introduced. Bubble-electrospinning for mass production of nanob ers is illustrated. There are in total more than 280 references.

An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering

We propose a novel variational approach for limit cycles of a kind of nonlinear oscillators Some examples are given to illustrate the effectiveness and convenience of the method The obtained results are valid for the whole solution domain with high accuracy

Variational approach for nonlinear oscillators

In this paper a simple but effective iteration method is proposed to search for limit cycles or bifurcation curves of nonlinear equations. Some examples are given to illustrate its convenience and effectiveness.

Limit cycle and bifurcation of nonlinear problems

An innovative approach to finding limit cycles is proposed and illustrated on the van der Pol equation. The technique developed in this Letter is similar to the Ritz's method in variational theory. The present theory can be applied to not only weakly nonlinear equations, but also strongly nonlinear ones, and the obtained results are valid for the whole solution domain.

Ji-Huan He

Papers

Determination of limit cycles for strongly nonlinear oscillators.