特征线法

设所需求解的准线性偏微分方程为

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

(1)

其中 ${\displaystyle u\left({\boldsymbol {x}}\right)=u\left(x_{1,}x_{2},\ldots ,x_{N}\right)}$。

取某变量 ${\displaystyle s}$，令 ${\displaystyle u({\boldsymbol {x}})}$ 对 ${\displaystyle s}$ 求导數，可得

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

(2)

若定义 ${\displaystyle {\frac {\partial x_{k}}{\partial s}}=a_{k}\left({\boldsymbol {x}},u\right)}$，可知

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

(3)

即，求解的偏微分方程(1)的过程变作对联立的常微分方程组作积分

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

(4)

积分过程需要给定初始条件。一般初始条件给定的形式为 ${\displaystyle x}$ 空间中的流形

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

(5)

将此曲面对应为 ${\displaystyle s=0}$。

设想 ${\displaystyle {\boldsymbol {x}}}$ 和 ${\displaystyle u}$ 依赖于变量 ${\displaystyle \{s,t_{1},t_{2},...,t_{N-1}\}}$，则 ${\displaystyle \{t_{1},t_{2},...,t_{N-1}\}}$ 可作方程(5)中的初始值，即

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

(6)

从方程组(4)和初始条件(6)确定 ${\displaystyle {\boldsymbol {x}}}$ 和 ${\displaystyle u}$ 后，可以得到解的隐式形式。如果可以解析消掉 ${\displaystyle s,t_{1},t_{2},...,t_{N-1}}$，则可获得显式形式的解。

沿着一阶偏微分方程的特征线, 偏微分方程简化为一个常微分方程. 沿着特征线求出对应常微分方程的解就可以得到偏微分方程的解.

为了更好地解释这一方法, 考虑具有以下形式的偏微分方程

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

(1)

假设解 u 已知, 考虑R³中的曲面 z = u(x,y). 曲面的法向量为

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

那么,[1] 方程 (1) 表示向量场

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

与曲面 z = u(x,y) 在任意点处相切. 换句话说, 解函数的图像必定是该向量场的积分曲线的并. 这些积分曲线被称作偏微分方程的特征线.

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• Prof. Scott Sarra tutorial on Method of Characteristics （页面存档备份，存于）
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