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so I'm trying to tackle this DEQ:

f''[x] = f[x] DiracDelta[x - a] - b,

with robin boundary conditions

f'[0] == f[0], f'[c] == f[c]

where a,b, and c are constants.

If you're curious, I'm getting this because I'm trying to treat steady state in a 1D diffusion system where I have homogenous generation along the length (b, in 1/(length-time) units), f(x) is the population distribution, and I have a point scatterer at x=a consuming population at a rate proportional to the concentration there (f(x)). i.e.

f=f(x,t)

df/dt = D*(d^2/dx^2)f + b - f*DiracDelta(x-a) = 0

I tried to take a laplace transform approach but couldn't hack it, if someone has another idea on how to approach this I'd appreciate it!

Thanks!