# PDEMOC8

$\displaystyle xu_x + yu_y + u_z = u, u(x,y,0) = h(x,y)\,$

The characteristics are $\displaystyle \frac{dx}{dt}=x,\frac{dy}{dt}=y,\frac{dz}{dt}=1,\frac{du}{dt}=u\,$ .

The inital data curve at $\displaystyle t=0\,$ is $\displaystyle \Gamma(s_1,s_2,0,h(s_1,s_2))\,$ .

For $\displaystyle x\,$ ,

$\displaystyle \ln x = t+c_1(s_1,s_2)\,$

$\displaystyle x(s_1,s_2,t) = c_2(s_1,s_2)e^t\,$

$\displaystyle x(s_1,s_2,0) = c_2(s_1,s_2) = s_1\,$

$\displaystyle x(s_1,s_2,t) = s_1e^t\,$

For $\displaystyle y\,$ ,

$\displaystyle \ln y = t+c_3(s_1,s_2)\,$

$\displaystyle y(s_1,s_2,t) = c_4(s_1,s_2)e^t\,$

$\displaystyle y(s_1,s_2,0) = c_4(s_1,s_2) = s_2\,$

$\displaystyle y(s_1,s_2,t) = s_2e^t\,$

For $\displaystyle z\,$ ,

$\displaystyle z = t+c_5(s_1,s_2)\,$

$\displaystyle z(s_1,s_2,0) = c_5(s_1,s_2)=0\,$

$\displaystyle z(s_1,s_2,t) = t\,$

For $\displaystyle u\,$ ,

$\displaystyle \ln u = t + c_6\,$

$\displaystyle u(s_1,s_2,t) = c_6(s_1,s_2)e^t\,$

$\displaystyle u(s_1,s_2,0) = c_6(s_1,s_2) = h(s_1,s_2)\,$

$\displaystyle u(s_1,s_2,t) = h(s_1,s_2)e^t\,$

$\displaystyle u(s_1,s_2,t) = h(s_1,s_2) e^t\,$

$\displaystyle u(x,y,z) = h(xe^{-z},ye^{-z})e^z\,$

Check:

$\displaystyle u_x = h' e^{-z} e^z = h'\,$

$\displaystyle u_y = h' e^{-z} e^z = h'\,$

$\displaystyle u_z = h e^z + e^z h'\cdot (-xe^{-z}-ye^{-z}) = he^z - h'\cdot(x+y)\,$