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

by DoctorDeath64
Tags: green's function, helmholtz, wave
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May10-12, 12:35 PM
P: 2
Hey I'm trying to solve the following equation:

d2/dx2 f(x,y) - d2/dy2 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 X1=x+y and X2=x-y, however assuming separation of variables for f(X1,X2)=A(X1)B(X2) gives:

d/dX1 A(X1) d/dX2B(X2)+ λA(X1)B(X2)=δ(0.5(X1+X2))δ(0.5(X1-X2))

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