# 匹配渐近展开法

## 示例

${\displaystyle \epsilon y''+(1+\epsilon )y'+y=0,}$

### 外解（t = O(1)）

${\displaystyle \epsilon }$十分小，故可以当作正则摄动问题处理。取${\displaystyle \epsilon =0}$，有

${\displaystyle y'+y=0.\,}$

${\displaystyle y=Ae^{-t}\,}$

### 内解（t = O(ε））

${\displaystyle {\frac {1}{\epsilon }}y''(\tau )+\left({1+\epsilon }\right){\frac {1}{\epsilon }}y'(\tau )+y(\tau )=0,\,}$

${\displaystyle y''+y'=0.\,}$

${\displaystyle y=B-Ce^{-\tau }\,}$

${\displaystyle y_{I}=B\left({1-e^{-\tau }}\right)=B\left({1-e^{-t/\epsilon }}\right).\,}$

### 合并

${\displaystyle \epsilon }$取不同值时的近似解

${\displaystyle y(t)=y_{I}+y_{O}-y_{\mathrm {overlap} }=e\left({1-e^{-t/\epsilon }}\right)+e^{1-t}-e=e\left({e^{-t}-e^{-t/\epsilon }}\right).\,}$

## 参考文献

1. ^ Verhulst, F. Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics. Springer. 2005. ISBN 0-387-22966-3.
2. ^ Nayfeh, A. H. Perturbation Methods. Wiley Classics Library. Wiley-Interscience. 2000. ISBN 978-0-471-39917-9.
3. ^ Kevorkian, J.; Cole, J. D. Multiple scale and singular perturbation methods. Springer. 1996. ISBN 0-387-94202-5.