(adsbygoogle = window.adsbygoogle || []).push({}); How to use Greens functions to solve variation of Helmholtz equation !?

Hey I'm trying to solve the following equation:

d^{2}/dx^{2}f(x,y) - d^{2}/dy^{2}f(x,y) + λf(x,y)=δ(x)δ(y) .

The right hand side seems to indicate that Greens functions will be useful here but the solution f(x,y) isn't axisymmetric when it's transformed to polars, therefore you're left with a PDE. I also thought about (d/dx -d/dy)(d/dx+d/dy) and substituting in X_{1}=x+y and X_{2}=x-y, however assuming separation of variables for f(X_{1},X_{2})=A(X_{1})B(X_{2}) gives:

d/dX_{1}A(X_{1}) d/dX_{2}B(X_{2})+ λA(X_{1})B(X_{2})=δ(0.5(X_{1}+X_{2}))δ(0.5(X_{1}-X_{2}))

which doesn't lend itself to be entirely separable. Any advice would be much appreciated !

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# How to use Greens functions to solve variation of Helmholtz equation ?

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