PDEMOC8

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

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

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

For $x\,$ ,

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

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

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

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

For $y\,$ ,

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

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

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

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

For $z\,$ ,

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

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

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

For $u\,$ ,

$\ln u = t + c_6\,$

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

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

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

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

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

Check:

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

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

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

Main Page : Partial Differential Equations : Method of Characteristics

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