How can boundary conditions be written for a DEQ with Dirac delta?

[rynlee]

Hi All,

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!

 

[rynlee]

Properly I should title this more like

"diffusion-reaction DEQ with delta reaction term in steady state with homogenous generation"

 

[Coelum]

Rynlee,
can you see the geometrical meaning of your ODE in a small neighborhood of a? Do you understand why you have 4 BCs for a second order ODE?

 

[rynlee]

don't I have 2 BCs in a second order DEQ?

If you stick with the original 2D problem I have 2BCs (those) and in the steady state assumption no longer need an initial conditions since I eliminate t, leaving me with the 2nd order DEQ and two robin BCs.

For a simpler problem Neumann BCs could be taken,
f'[0] == 0, f'[c]==0
But the difficulty remains.

 

[Coelum]

You are right, I misinterpreted your statements. Hint: you have two problems, one before and one after a. Do you know how to handle them?

 

[rynlee]

You are right, I misinterpreted your statements. Hint: you have two problems, one before and one after a. Do you know how to handle them?

That's a good point, really this could be viewed as two coupled DEQs, one defined on [0,a] and the other defined on [a,c], each with a set of Robin BCs, with one of them shared (at a).

The two DEQs aren't independent though, since the BCs are Robin not Neumann. If we instead had
f'[0]=f'[a]=f'[c]=0, then I could split this into two DEQs. Since that's not the case though, the distribution on each side of a effects the other side.

 

[Coelum]

Rynlee, you got the point: do you know how to write the BCs in a? Hint: integrate the ODE in [a-delta,a+delta] and compute the limit when delta->0.