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I am working on a quite difficult, though seemingly simple, non-homogeneous differential equation in cylindrical coordinates. The main equation is the non homogeneous modified Helmholtz Equation

[itex]\nabla^{2}\psi - k^{2}\psi = \frac{-1}{D}\frac{\delta(r-r')\delta(\theta-\theta')\delta(z-z')}{r}[/itex]

with Robin boundary condition

[itex]\psi - \kappa\hat{\Omega}_n\cdot\vec{\nabla}\psi = 0[/itex]

on [itex]r=a[/itex], the edge of a virtual infinitely long cylinder of radius [itex]r=a[/itex]. [itex]\hat{\Omega}_n[/itex] is a vector pointing out of the cylinder.

The solution [itex]\psi[/itex] must also vanish at infinity, i.e. [itex]\psi(r\rightarrow\infty,z\rightarrow\pm\infty) = 0[/itex], to satisfy the Sommerfeld Radiation Condition.

I have tried the Green's function approach in cartesian coordinates, though the Robin boundary condition makes it hard to easily solve. I have also tried it in polar coordinates, but I can't find any reference on how to use Green's function on periodic domains.

This problem arises from the diffusion approximation in biomedical imaging, and a solution would be of great help in my research.

Thanks a lot !

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# NH Modified Helmholtz Equation with Robin Boundary Condition

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