By Shijun Liao

Fixing nonlinear difficulties is inherently tough, and the more suitable the nonlinearity, the extra intractable options turn into. Analytic approximations frequently holiday down as nonlinearity turns into powerful, or even perturbation approximations are legitimate just for issues of susceptible nonlinearity.This ebook introduces a robust new analytic approach for nonlinear problems-homotopy analysis-that continues to be legitimate in spite of powerful nonlinearity. partly I, the writer starts off with a very easy instance, then offers the elemental principles, specified tactics, and the benefits (and obstacles) of homotopy research. half II illustrates the applying of homotopy research to many fascinating nonlinear difficulties. those variety from basic bifurcations of a nonlinear boundary-value challenge to the Thomas-Fermi atom version, Volterra's inhabitants version, Von K?rm?n swirling viscous stream, and nonlinear revolutionary waves in deep water.Although the homotopy research technique has been tested in a couple of prestigious journals, it has but to be absolutely designated in publication shape. Written by means of a pioneer in its improvement, past Pertubation: advent to the Homotopy research process is your first chance to discover the main points of this necessary new process, upload it on your analytic toolbox, and maybe make a contribution to a couple of the questions that stay open.

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**Example text**

Similarly, the so-called homotopy-Pad´e technique can be applied to accelerate the convergence of the related series. For example, to accelerate the series of V (0) and V (0), we ﬁrst apply the traditional Pad´e technique to the series +∞ ∂ 2 Φ(t; q) = Vn (0) q n ∂ t2 t=0 n=0 and ∂ 3 Φ(t; q) ∂ t3 +∞ = t=0 Vn (0) q n n=0 to get their [m, n] Pad´e approximants about the embedding parameter q, respectively, and then set q = 1 to obtain the corresponding [m, n] homotopyPad´e approximants. 10. 11. In both cases, the homotopy-Pad´e technique greatly accelerates the convergence of V (0) and V (0).

The rule of solution expression, the rule of coeﬃcient ergodicity, and the rule of solution existence are proposed to direct the choice of the initial guess, the auxiliary linear operator, and the auxiliary function. These rules greatly simplify the application of the homotopy analysis method. We demonstrate that the convergence region and rate of the solution series may be adjusted and controlled by means of the auxiliary parameter . By plotting the socalled -curves, it is easy to ﬁnd out a proper value of to ensure that the solution series converge.

1 + + The sum of the ﬁrst m terms of above series is given by m (− )−α (−1)n α n (−1)n α n n=0 m = (− )−α n=0 m = (− )−α m tj j=0 (−1)n n=j tj j=0 i=0 m m−j = (− )−α tj j=0 © 2004 CRC Press LLC n j=0 i=0 n j α n m−j m = (− )−α (1 + + t)n (1 + )n−j n j (−1)i+j α i+j (−1)i+j α j j tj (1 + )n−j j i+j j α−j i (1 + )i (1 + )i j j m α j = j=0 m m−j tj i=0 µm,n α ( ) = α−j i (−1)i n=0 α n (1 + )i (− )j−α tn , where m−n n−α µm,n α ( ) = (− ) α−n j (−1)j j=0 (1 + )j . This ends the proof. 91) is deduced for real numbers −∞ < α < +∞ except integers α = 0, 1, 2, 3, · · ·, it is valid for all real numbers.