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How to use Greens functions to solve variation of Helmholtz equation ? 
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#1
May1012, 12:35 PM

P: 2

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}=xy, 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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