# 特征线法

## 基本方法

${\displaystyle a_{1}\left({\boldsymbol {x}},u\right){\frac {\partial u}{\partial x_{1}}}+a_{2}\left({\boldsymbol {x}},u\right){\frac {\partial u}{\partial x_{2}}}+\cdots +a_{N}\left({\boldsymbol {x}},u\right){\frac {\partial u}{\partial x_{N}}}=b\left({\boldsymbol {x}},u\right)}$ (1)

${\displaystyle {\frac {du}{ds}}=\left({\frac {\partial x_{1}}{\partial s}}\right){\frac {\partial u}{\partial x_{1}}}+\left({\frac {\partial x_{2}}{\partial s}}\right){\frac {\partial u}{\partial x_{2}}}+\ldots +\left({\frac {\partial x_{N}}{\partial s}}\right){\frac {\partial u}{\partial x_{N}}}}$ (2)

${\displaystyle {\frac {du}{ds}}=a_{1}\left({\boldsymbol {x}},u\right){\frac {\partial u}{\partial x_{1}}}+a_{2}\left({\boldsymbol {x}},u\right){\frac {\partial u}{\partial x_{2}}}+\ldots +a_{N}\left({\boldsymbol {x}},u\right){\frac {\partial u}{\partial x_{N}}}=b\left({\boldsymbol {x}},u\right)}$ (3)

${\displaystyle \left\{{\begin{array}{rcl}{\frac {\partial x_{k}}{\partial s}}&=&a_{k}\left({\boldsymbol {x}},u\right)\\{\frac {du}{ds}}&=&b\left({\boldsymbol {x}},u\right)\end{array}}\right.}$ (4)

${\displaystyle g\left({\boldsymbol {x}},u\right)=0}$ (5)

${\displaystyle {\begin{array}{rcl}x_{1}\left(s=0\right)&=&h_{1}\left(t_{1},t_{2},\ldots ,t_{N-1}\right)\\x_{2}\left(s=0\right)&=&h_{2}\left(t_{1},t_{2},\ldots ,t_{N-1}\right)\\\vdots \\u\left(s=0\right)&=&v\left(t_{1},t_{2},\ldots ,t_{N-1}\right)\end{array}}}$ (6)

## 一阶偏微分方程的特征线法

${\displaystyle a(x,y,u){\frac {\partial u}{\partial x}}+b(x,y,u){\frac {\partial u}{\partial y}}=c(x,y,u).}$ (1)

${\displaystyle (u_{x}(x,y),u_{y}(x,y),-1).\,}$

${\displaystyle (a(x,y,z),b(x,y,z),c(x,y,z))\,}$

## 参考资料

• Courant, Richard; Hilbert, David, Methods of Mathematical Physics, Volume II, Wiley-Interscience, 1962
• Delgado, Manuel, The Lagrange-Charpit Method, SIAM Review, 1997, 39 (2): 298–304, doi:10.1137/S0036144595293534
• Evans, Lawrence C., Partial Differential Equations, Providence: American Mathematical Society, 1998, ISBN 0-8218-0772-2
• John, Fritz, Partial differential equations 4th, Springer, 1991, ISBN 978-0387906096
• Polyanin, A. D.; Zaitsev, V. F.; Moussiaux, A., Handbook of First Order Partial Differential Equations, London: Taylor & Francis, 2002, ISBN 0-415-27267-X
• Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Boca Raton: Chapman & Hall/CRC Press, 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.
• Streeter, VL; Wylie, EB, Fluid mechanics International ${\displaystyle 9^{th}}$ Revised, McGraw-Hill Higher Education, 1998 参数|edition=值左起第15位存在删除符 (帮助)