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

The homotopy perturbation method is extremely accessible to non-mathematicians and engineers. The method decomposes a complex problem under study into a series of simple problems that are easy to be solved. This note gives an elementary introduction to the basic solution procedure of the homotopy perturbation method. Particular attention is paid to constructing a suitable homotopy equation.

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Recent development of the homotopy perturbation method

https://hal.archives-ouvertes.fr/hal-00315288/document

A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions

Hamiltonian approach to nonlinear oscillators

/pdf/applied-mathematical-sciences-2hr8rqfzub.pdf

Applied mathematical sciences

This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is different from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75p (135o) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency w in terms of the complete elliptic integral of the first kind. We can conclude that for amplitudes lower than 135o the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude). We believe that present study may be a suitable and fruitful exercise for teaching and better understanding the behavior of the nonlinear pendulum in advanced undergraduate courses on classical mechanics.

/pdf/exact-solution-for-the-nonlinear-pendulum-4fv0m9nvvv.pdf

Exact solution for the nonlinear pendulum

http://rua.ua.es/dspace/bitstream/10045/14978/2/MCM_vol52_n3-4_pp637-641_2010.pdf

An accurate closed-form approximate solution for the quintic Duffing oscillator equation

https://rua.ua.es/dspace/bitstream/10045/11906/2/PLA_v373_n32_p2805_2009pre.pdf

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

He's homotopy perturbation method is used to solve the nonlinear differential equation that governs the oscillations of a relativistic oscillator for which the nonlinearity (anharmonicity) is a relativistic effect due to the time line dilation along the world line. Unlike the usual non-relativistic harmonic oscillator, the velocity of the relativistic oscillator is bounded because this velocity must be less than the speed of light. Therefore, the phase space has a \"strip\" structure and a change of variable must to be made before application of the homotopy perturbation method. We find that this method, which does not need small parameters, works very well for the whole range of large values of oscillation amplitude, and excellent agreement of the approximate frequency with the exact one has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions and the maximal relative error for the approximate frequency is less than 3.3% not only for small but also for large values of oscillation amplitude. In addition, we show that the product of the amplitude and the exact frequency, Aa>ex(A), reaches π /2 when the amplitude tends to infinity.

Application of He's Homotopy Perturbation Method to the Relativistic (An)harmonic Oscillator. I: Comparison between Approximate and Exact Frequencies

http://rua.ua.es/dspace/bitstream/10045/9179/1/JSV_v314_n3-5_p775_2008.pdf

D. I. Méndez

Papers

Exact solution for the nonlinear pendulum

An accurate closed-form approximate solution for the quintic Duffing oscillator equation

An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method

Application of He's Homotopy Perturbation Method to the Relativistic (An)harmonic Oscillator. I: Comparison between Approximate and Exact Frequencies

Harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable