特征线法

基本方法

$a_1 \left( \boldsymbol{x}, u \right) \frac{\partial u}{\partial x_1} + a_2 \left( \boldsymbol{x}, u \right) \frac{\partial u}{\partial x_2} + \cdots + a_N \left( \boldsymbol{x}, u \right) \frac{\partial u}{\partial x_N} = b \left( \boldsymbol{x}, u \right)$ (1)

$\frac{d u}{d s} = \left( \frac{\partial x_1}{\partial s} \right) \frac{\partial u}{\partial x_1} + \left( \frac{\partial x_2}{\partial s} \right) \frac{\partial u}{\partial x_2} + \ldots + \left( \frac{\partial x_N}{\partial s} \right) \frac{\partial u}{\partial x_N}$ (2)

$\frac{d u}{d s} = a_1 \left( \boldsymbol{x}, u \right) \frac{\partial u}{\partial x_1} + a_2 \left( \boldsymbol{x}, u \right) \frac{\partial u}{\partial x_2} + \ldots + a_N \left( \boldsymbol{x}, u \right) \frac{\partial u}{\partial x_N} = b \left( \boldsymbol{x}, u \right)$ (3)

$\left\{ \begin{array}{rcl} \frac{\partial x_k}{\partial s} &=& a_k \left( \boldsymbol{x}, u \right)\\ \frac{d u}{d s} &=& b \left( \boldsymbol{x}, u \right) \end{array} \right.$ (4)

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

$\begin{array}{rcl} x_1 \left( s = 0 \right) &=& h_1 \left( t_1, t_2, \ldots, t_{N - 1} \right)\\ x_2 \left( s = 0 \right) &=& h_2 \left( t_1, t_2, \ldots, t_{N - 1} \right)\\ \vdots \\ u \left( s = 0 \right) &=& v \left( t_1, t_2, \ldots, t_{N - 1} \right) \end{array}$ (6)

一阶偏微分方程的特征线法

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

(1)

$(u_x(x,y),u_y(x,y),-1).\,$

$(a(x,y,z),b(x,y,z),c(x,y,z))\,$

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 $9^{th}$ Revised, McGraw-Hill Higher Education, 1998