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I'm following the solution of a Klein-Gordon PDE in a textbook. The equation is

[tex]\begin{align}

k_{xx}(x,y) - k_{yy}(x,y) &= \lambda k(x,y) \\

k(x,0) &= 0 \\

k(x,x) &= - \frac{\lambda}{2} x

\end{align}

[/tex]

The book uses a change of variables

[tex]$\xi = x+y$, $\eta = x-y$[/tex]

to write

[tex]\begin{align}

k(x,y) &= G(\xi,\eta)\\

k_{xx} &= G_{\xi \xi} + 2G_{\xi \eta} + G_{\eta \eta}\\

k_{yy} &= G_{\xi \xi} - 2G_{\xi \eta} + G__{\eta \eta}

\end{align}[/tex]

and then they write the original PDE as

[tex]\begin{align}

G_{\xi \eta}(\xi,\eta) &= \frac{\lambda}{4} G(\xi,\eta),\\

G(\xi,\xi) &= 0,\\

G(\xi,0) &= - \frac{\lambda}{4} \xi

\end{align}

[/tex]

I'm fine with the first line in the new PDE, but the other two, the boundary conditions, i don't get how they arrive at.

Can somebody help me understand? I'll be much appreciative :-)

J.

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# Klein-Gordon PDE

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