# PDEMOC4

$\displaystyle u_x + 2u_y = u^2, u(x,0)=h(x)\,$

The parameterized curve at $\displaystyle t=0\,$ is $\displaystyle \Gamma(s,0,h(s))\,$ . This comes from the symbols $\displaystyle x,0,h(x)\,$ in the IC.

The characteristics are $\displaystyle \frac{dx}{dt}=1\,$ , $\displaystyle \frac{dy}{dt}=2\,$ , $\displaystyle \frac{dz}{dt}=z^2\,$ .

These differential equations give $\displaystyle x(s,t)=t+c_1(s)\,$ , $\displaystyle y(s,t)=2t+c_2(s)\,$ , $\displaystyle -z(s,t)^{-1}=t-c_3(s)^{-1}\,$ .

At $\displaystyle t=0\,$ , $\displaystyle x=c_1(s)=s\,$ , $\displaystyle y=c_2(s)=0\,$ , and $\displaystyle z=-c_3(s)^{-1}=h(s)\,$ .

So the solutions are $\displaystyle x=t+s\,$ , $\displaystyle y=2t\,$ , and $\displaystyle z=-1/(t-h(s)^{-1})\,$ .

Therefore $\displaystyle s=x-t\,$ , $\displaystyle t=y/2\,$ , and $\displaystyle z=u(x,y)=\frac{h(x-y/2)}{1-\frac{1}{2} y h(x-y/2)}\,$ .