Information

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微扰求解 Hard Problem 的步骤

1. 使某一项成为微扰 $\epsilon$ 2. 将无微扰的解加上修正后代回方程, 逐项对应求解 3. 将所有的项加进来.

1. Convert the original problem into a perturbation problem by introducing the small parameter $\epsilon$ . 2. Assume an expression for the answer in the form of a perturbation series and compute the coefficients of that series. 3. Recover the answer to the original problem by summing the perturbation series for the appropriate value of $\epsilon$ .

Symbols

- $\sim$ : 'is asympotic to' \begin{align*} f(x) \sim g(x) \quad (x \to x_0) \end{align*} means \begin{align*} \lim_{x\to x_0} \frac{f(x)}{g(x)} = 1 \end{align*}
- $\ll$ : 'is negligible compared with' \begin{align*} f(x) \ll g(x) \quad (x \to x_0) \end{align*} means \begin{align*} \lim_{x\to x_0} \frac{f(x)}{g(x)} = 0 \end{align*}

Method of Dominant Balance

求方程

$$\begin{align*} \epsilon x^5 +x = 1 \end{align*}$$

的解在 $\epsilon \to 0$ 时, 行为.

  • Neglect 1
$$\begin{align*} \epsilon x^5 \sim -x \end{align*}$$

得到

$$\begin{align*} x \sim \frac{(-1)^{1/4}}{\epsilon^{1/4}} \end{align*}$$

是复平面上四个在无穷远的解. $1$ 相比无穷大, 可以忽略, 不矛盾.

  • Neglect $x$
$$\begin{align*} \epsilon x^5 \sim 1 \end{align*}$$

得到

$$\begin{align*} x \sim \frac{1^{1/5}}{\epsilon^{1/5}} \end{align*}$$

这里无穷大的 $x$ 相比于 $1$ 是不可忽略的. 矛盾

  • Neglect $\epsilon x^5$
$$\begin{align*} x \sim 1 \end{align*}$$

那么

$$\begin{align*} \epsilon x^5 \sim \epsilon\cdot 1 \ll 1 \end{align*}$$

可以忽略.