By Shijun Liao

ISBN-10: 0203491165

ISBN-13: 9780203491164

ISBN-10: 158488407X

ISBN-13: 9781584884071

Fixing nonlinear difficulties is inherently tricky, and the better the nonlinearity, the extra intractable strategies develop into. Analytic approximations frequently holiday down as nonlinearity turns into robust, or even perturbation approximations are legitimate just for issues of vulnerable nonlinearity.This e-book introduces a robust new analytic technique for nonlinear problems-homotopy analysis-that is still legitimate regardless of robust nonlinearity. partially I, the writer begins with a very easy instance, then offers the elemental principles, certain methods, and the benefits (and obstacles) of homotopy research. half II illustrates the appliance of homotopy research to many attention-grabbing nonlinear difficulties. those diversity from uncomplicated bifurcations of a nonlinear boundary-value challenge to the Thomas-Fermi atom version, Volterra's inhabitants version, Von K?rm?n swirling viscous movement, and nonlinear revolutionary waves in deep water.Although the homotopy research strategy has been established in a few prestigious journals, it has but to be absolutely particular in publication shape. Written by way of a pioneer in its improvement, past Pertubation: creation to the Homotopy research procedure is your first chance to discover the main points of this useful new procedure, upload it in your analytic toolbox, and maybe contribute to a few of the questions that stay open.

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Fixing nonlinear difficulties is inherently tricky, and the improved the nonlinearity, the extra intractable options develop into. Analytic approximations usually holiday down as nonlinearity turns into powerful, or even perturbation approximations are legitimate just for issues of vulnerable nonlinearity. This e-book introduces a robust new analytic procedure for nonlinear problems-homotopy analysis-that continues to be legitimate despite robust nonlinearity.

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**Additional info for Beyond Perturbation (Modern Mathematics and Mechanics)**

**Example text**

77) when = −1. t 5th-order 10th-order 15th-order 20th-order exact approx. approx. approx. approx. 84) when = −1. t 10th-order 20th-order 40th-order 50th-order exact approx. approx. approx. approx. 100) when = −1 and α = ±1/2, ±π/4. 101) when = −1/2 and α = ±1/2, ±π/4. 69). 78). 57). 57) when = −1/10. 69) when H(t) = 1/(1 + t). Dash-dotted line: 20th-order approximation of V (0); solid line: 20th-order approximation of V (0). 77) when H(t) = exp(−t). Dash-dotted line: 10th-order approximation of V (0); solid line: 20th-order approximation of V (0); dashed lined: 10th-order approximation of V (0); dash-dot-dotted line: 20th-order approximation of V (0).

T 10th-order 20th-order 40th-order 60th-order exact approx. approx. approx. approx. 77) for diﬀerent values of . 77) for diﬀerent values of . 77) when = −1. t 5th-order 10th-order 15th-order 20th-order exact approx. approx. approx. approx. 84) when = −1. t 10th-order 20th-order 40th-order 50th-order exact approx. approx. approx. approx. 100) when = −1 and α = ±1/2, ±π/4. 101) when = −1/2 and α = ±1/2, ±π/4. 69). 78). 57). 57) when = −1/10. 69) when H(t) = 1/(1 + t). Dash-dotted line: 20th-order approximation of V (0); solid line: 20th-order approximation of V (0).

Approx. 100) when = −1 and α = ±1/2, ±π/4. 101) when = −1/2 and α = ±1/2, ±π/4. 69). 78). 57). 57) when = −1/10. 69) when H(t) = 1/(1 + t). Dash-dotted line: 20th-order approximation of V (0); solid line: 20th-order approximation of V (0). 77) when H(t) = exp(−t). Dash-dotted line: 10th-order approximation of V (0); solid line: 20th-order approximation of V (0); dashed lined: 10th-order approximation of V (0); dash-dot-dotted line: 20th-order approximation of V (0). 8). 8). 99) at the 31st-order of approximation when α = π/4.

### Beyond Perturbation (Modern Mathematics and Mechanics) by Shijun Liao

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