Information

  • Vidoes: [[https://www.youtube.com/playlist?list=PLzcd6SoIscwjHuWRE38UXWG92uq0Sy4UF]]
  • Book: Carl M. Bender, Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I Asympotic Methods and Perturbation Theory, 1999

1

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.

#+BEGIN_QUOTE 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 $\epsilon$ . #+END_QUOTE

Symbols

#+BEGIN_QUOTE

  • $\sim$ : '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*} #+END_QUOTE

#+BEGIN_QUOTE

  • $\ll$ : '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*} #+END_QUOTE

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.