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)

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

(2)

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

(3)

${\displaystyle \left\{{\begin{array}{rcl}{\dfrac {\partial x_{k}}{\partial s}}&=&a_{k}\left({\boldsymbol {x}},u\right)\\[1em]{\dfrac {{\text{d}}u}{{\text{d}}s}}&=&b\left({\boldsymbol {x}},u\right)\end{array}}\right.}$

(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)

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

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