# 一阶偏微分方程

${\displaystyle F(x_{1},\ldots ,x_{n},u,u_{x_{1}},\ldots u_{x_{n}})=0.\,}$

## 通解及全積分

${\displaystyle u=\phi (x_{1},x_{2},\dots ,x_{n},a_{1},a_{2},\dots ,a_{n})}$

## 波方程的特徵曲面

${\displaystyle u_{t}^{2}=c^{2}\left(u_{x}^{2}+u_{y}^{2}+u_{z}^{2}\right).\,}$

${\displaystyle u_{x}^{2}+u_{y}^{2}+u_{z}^{2}={\frac {1}{c^{2}}}.\,}$

${\displaystyle {\vec {x}}=(x,y,z)\quad {\hbox{and}}\quad {\vec {p}}=(u_{x},u_{y},u_{z}).\,}$

${\displaystyle u({\vec {x}})={\vec {p}}\cdot ({\vec {x}}-{\vec {x_{0}}}),\,}$

${\displaystyle |{\vec {p}}\,|={\frac {1}{c}},\quad {\text{and}}\quad {\vec {x_{0}}}\quad {\text{is arbitrary}}.\,}$

xx0不變，此解的包絡線可以由找到半徑1/c圓球上的點，且u值為定值的點來求得。若${\displaystyle {\vec {p}}}$ 平行${\displaystyle {\vec {x}}-{\vec {x_{0}}}}$ ，此條件會成立。因此，包絡線為

${\displaystyle u({\vec {x}})=\pm {\frac {1}{c}}|{\vec {x}}-{\vec {x_{0}}}\,|.}$

${\displaystyle {\frac {1}{c}}|{\vec {x}}-{\vec {x_{0}}}\,|\quad {\hbox{is stationary for}}\quad {\vec {x_{0}}}\in S.\,}$

${\displaystyle |{\vec {x}}-{\vec {x_{0}}}\,|}$ S垂直，上式就會成立，因此包絡線對應和S垂直，速度為c的運動，這也就是Huygens波前建立法：S上的每一點在t=0時發射一個球狀波，較晚時間t的波前就是這些球狀波的包絡線。S的法向量即為光線。

## 參考資料

1. ^ P.R. Garabedian, "Partial differential equations" , Wiley (1964)

## 相關書目

• R. Courant]and D. Hilbert, Methods of Mathematical Physics, Vol II, Wiley (Interscience), New York, 1962.
• L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
• A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
• Sarra, Scott The Method of Characteristics with applications to Conservation Laws, Journal of Online Mathematics and its Applications, 2003.