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mathmari
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Normal form of the hyperbolic equation
Hey!
I am looking at the following in my notes:
$$a(x,y) u_{xx}+2 b(x,y) u_{xy}+c(x,y) u_{yy}=d(x,y,u,u_x,u_y)$$
$$A u_{\xi \xi}+ 2B u_{\xi \eta}+C u_{\eta \eta}=D$$
$$A=a \xi_x^2+2b \xi_y \xi_x+c \xi_y^2 \ \ \ (*)$$
$$B=a \xi_x \eta_x +b \eta_x \xi_y+b\eta_y \xi_x+c \xi_y \eta_y$$
$$C=a \eta_x^2 +2b \eta_x \eta_y+c \eta_y^2 \ \ \ (**)$$
Hyperbolic case:
We will show that at the hyperbolic case we can set $A=C=0$.
The equations $(*)$ and $(**)$ have the general form:
$$a \Phi_x^2+2b \Phi_x \Phi_y+c \Phi_y^2=0$$
$$\Phi=\{ \xi, \eta \}$$
$$a (\frac{\Phi_x}{\Phi_y})^2+2b (\frac{\Phi_x}{\Phi_y})+c=0$$
$$\frac{\Phi_x}{\Phi_y}=\frac{1}{a} ( -b \pm \sqrt{b^2-ac})$$
We are looking for characteristics equations for which $\Phi (x,y)= \text{ constant }$.
$$\frac{\partial{\Phi}}{\partial{x}} dx+\frac{\partial{\Phi}}{\Phi{y}} dy=0$$
$$\frac{dy}{dx}=- \frac{\Phi_x}{\Phi_y}$$
So
$$\frac{dy}{dx}=\frac{1}{a}(b \pm \sqrt{b^2-ac})$$
The canonical form of the hyperbolic equation is:
$$u_{\xi \eta}=D'(x,y,u,u_x,u_y)$$Why are we looking for characteristics equations for which $\Phi (x,y)= \text{ constant }$?? (Wondering)
Hey!
I am looking at the following in my notes:
$$a(x,y) u_{xx}+2 b(x,y) u_{xy}+c(x,y) u_{yy}=d(x,y,u,u_x,u_y)$$
$$A u_{\xi \xi}+ 2B u_{\xi \eta}+C u_{\eta \eta}=D$$
$$A=a \xi_x^2+2b \xi_y \xi_x+c \xi_y^2 \ \ \ (*)$$
$$B=a \xi_x \eta_x +b \eta_x \xi_y+b\eta_y \xi_x+c \xi_y \eta_y$$
$$C=a \eta_x^2 +2b \eta_x \eta_y+c \eta_y^2 \ \ \ (**)$$
Hyperbolic case:
We will show that at the hyperbolic case we can set $A=C=0$.
The equations $(*)$ and $(**)$ have the general form:
$$a \Phi_x^2+2b \Phi_x \Phi_y+c \Phi_y^2=0$$
$$\Phi=\{ \xi, \eta \}$$
$$a (\frac{\Phi_x}{\Phi_y})^2+2b (\frac{\Phi_x}{\Phi_y})+c=0$$
$$\frac{\Phi_x}{\Phi_y}=\frac{1}{a} ( -b \pm \sqrt{b^2-ac})$$
We are looking for characteristics equations for which $\Phi (x,y)= \text{ constant }$.
$$\frac{\partial{\Phi}}{\partial{x}} dx+\frac{\partial{\Phi}}{\Phi{y}} dy=0$$
$$\frac{dy}{dx}=- \frac{\Phi_x}{\Phi_y}$$
So
$$\frac{dy}{dx}=\frac{1}{a}(b \pm \sqrt{b^2-ac})$$
The canonical form of the hyperbolic equation is:
$$u_{\xi \eta}=D'(x,y,u,u_x,u_y)$$Why are we looking for characteristics equations for which $\Phi (x,y)= \text{ constant }$?? (Wondering)
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