Tough PDE

Hi group,

I'm still racking my brain trying to find the cylindrically symmetric
solutions to this PDE

Laplacian f(r,z) = f(r,z) - e^(-a r^2 - b r^2 )

where (r,z) are cylindrical coordinates. Writing the Laplacian out in
full gives

(1 / r) f_r + f_rr + f_zz = f - e^(- a r^2 - b z^2)

where f_r = @f/@r etc.

I applied fourier transform along the cylindrical symmetry axis z,
resulting in the following ODE

fhat_rr + (1 / r) fhat_r - (k^2 + 1) fhat = - F{e^(- a r^2 - b z^2)}

where F is the forward fourier transform.

The solution to the homogenous part, according to mathematica is

fhat = A J_0[ i * r * sqrt(k^2 + 1)] + B Y_0[- i * r * sqrt(k^2 +
1)]

for arbitrary constants A,B, where J_0 and Y_0 are Bessel functions of
the first and second kind, respectively.

But I have no idea how to evaluate the inverse fourier transform of
this expression. Is it even possible?

Thanks in advance.

James

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