By Shijun Liao
In contrast to different analytic innovations, the Homotopy research technique (HAM) is self sustaining of small/large actual parameters. along with, it presents nice freedom to settle on equation variety and resolution expression of similar linear high-order approximation equations. The HAM presents an easy approach to warrantly the convergence of answer sequence. Such specialty differentiates the HAM from all different analytic approximation equipment. moreover, the HAM will be utilized to unravel a few hard issues of excessive nonlinearity.
This booklet, edited by means of the pioneer and founding father of the HAM, describes the present advances of this strong analytic approximation technique for hugely nonlinear difficulties. Coming from varied nations and fields of study, the authors of every bankruptcy are best specialists within the HAM and its functions.
Readership: Graduate scholars and researchers in utilized arithmetic, physics, nonlinear mechanics, engineering and finance.
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Extra info for Advances in the Homotopy Analysis Method
J. Liao solitary waves. If the peaked solitary waves given by the exact wave equation exists mathematically but is impossible in physics, we had to check the physical validity of the peaked solitary waves. So, this is an interesting and valuable work, although with great challenge. For some attempts in this direction, please refer to Liao , who proposed a generalized wave model based on the symmetry and the fully nonlinear wave equations, which admits not only the traditional waves with smooth crest but also peaked solitary waves.
Math. Comput. 147: 499–513 (2004). J. Liao and Y. Tan, A general approach to obtain series solutions of nonlinear differential equations, Stud. Appl. Math. 119: 297–354 (2007). J. Liao, Notes on the homotopy analysis method: some definitions and theorems, Commun. Nonlinear Sci. Numer. Simulat. 14: 983–997 (2009). J. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simulat. 15: 2315– 2332 (2010). J. Liao, On the relationship between the homotopy analysis method and Euler transform, Commun.
1. Periodic solutions of chaotic dynamic systems It is well known that chaotic dynamic systems have the so-called “butterfly effect” [52, 53], say, the computer-generated numerical simulations have sensitive dependence to initial conditions (SDIC). 32) has chaotic solution in case of r = 28, b = 8/3 and σ = 10 for most of given initial conditions x0 , y0 , z0 of x, y, z at t = 0. 1345751139, z0 = 27, the above dynamic system of Lorenz equation has unstable periodic solutions, as reported by Viswanath .