PDEMOC6

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u_{x}+xu_{y}-u_{z}=u\,, u(x,y,1)=x+y\,,

Let the initial curve be parameterized by \Gamma (s_{1},s_{2},1)=s_{1}+s_{2}\,.


The characteristics are:

{\frac  {dx_{1}}{dt}}=1,{\frac  {dx_{2}}{dt}}=x,{\frac  {dx_{3}}{dt}}=-1,{\frac  {dz}{dt}}=z\,


{\frac  {dx_{1}}{dt}}=1\,

x_{1}(s,t)=t+c_{1}(s)\,

x_{1}(s,0)=c_{1}(s)=s_{1}\,

x_{1}=t+s_{1}\,

s_{1}=x_{1}-t\,


{\frac  {dx_{2}}{dt}}=x\,

x_{2}(s,t)={\frac  {1}{2}}t^{2}+c_{1}(s)t+c_{2}(s)\,

x_{2}(s,0)=c_{2}(s)=s_{2}\,

x_{2}={\frac  {1}{2}}t^{2}+s_{1}t+s_{2}\,

s_{2}=x_{2}-{\frac  {1}{2}}t^{2}-(x_{1}-t)t\,


{\frac  {dx_{3}}{dt}}=-1\,

x_{3}(s,0)=c_{3}(s)=1\,

x_{3}=1-t\,

t=1-x_{3}\,


{\frac  {dz}{dt}}=z\,

\ln z=t+c_{4}(s)\,

z(s,t)=c_{5}(s)e^{t}\,

z(s,0)=c_{5}(s)=s_{1}+s_{2}\,

z=(s_{1}+s_{2})e^{t}\,


Use these equations to get:

u(x,y)=z=\left[x_{1}+x_{2}+(x_{3}-1)\left[{\frac  {1}{2}}(x_{3}+1)+x_{1}\right]\right]e^{{1-x_{3}}}\,


Main Page : Partial Differential Equations : Method of Characteristics