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imported>Pythagoras0 |
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| 26번째 줄: | 26번째 줄: | ||
===asymptotics=== | ===asymptotics=== | ||
* $f\sim g\, \quad (x\to x_0)$ iff $$\lim_{x\to x_0}\frac{f(x)}{g(x)}=1$$ | * $f\sim g\, \quad (x\to x_0)$ iff $$\lim_{x\to x_0}\frac{f(x)}{g(x)}=1$$ | ||
| − | * method of dominant balance | + | * apply the method of dominant balance to $\epsilon x^5+ x=1$ |
| + | * as $x^4\sim -1/\epsilon\, \quad (\epsilon \to 0)$ | ||
2014년 2월 27일 (목) 11:07 판
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lecture 1 perturbation method
- solve $x^5+x=1$===
method 1
- try $x^5+\epsilon x=1$
- find $x(\epsilon)$ satisfying $x(\epsilon)^5+\epsilon x(\epsilon)=1$
- answer
$$x(\epsilon)=1-\frac{\epsilon }{5}-\frac{\epsilon ^2}{25}-\frac{\epsilon ^3}{125}+\frac{21 \epsilon ^5}{15625}+\frac{78 \epsilon ^6}{78125}+\cdots$$
- Setting $\epsilon=1$ gives numerical value $0.75\cdots$
weak coupling approach
- use the similar idea to Feynman diagrams
- try $\epsilon x^5+ x=1$
- we get
$$ x(\epsilon)=1-\epsilon +5 \epsilon ^2-35 \epsilon ^3+285 \epsilon ^4-2530 \epsilon ^5+23751 \epsilon ^6+\cdots $$
- can we get a meaningful number out of this?
- yes, for example, Pade summation can be used
asymptotics
- $f\sim g\, \quad (x\to x_0)$ iff $$\lim_{x\to x_0}\frac{f(x)}{g(x)}=1$$
- apply the method of dominant balance to $\epsilon x^5+ x=1$
- as $x^4\sim -1/\epsilon\, \quad (\epsilon \to 0)$
books
- Bender, Carl M., and Steven A. Orszag. 1999. Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer.