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_{1}e^{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_{2}e^{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}=he^{z}+e^{z}h'\cdot (-xe^{{-z}}-ye^{{-z}})=he^{z}-h'\cdot (x+y)\,



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