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

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Discussion Overview

The discussion revolves around formulating boundary conditions for a differential equation involving a Dirac delta function, specifically in the context of a diffusion-reaction system. Participants explore the implications of the equation and the appropriate boundary conditions in both steady-state and transient scenarios.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a differential equation f''[x] = f[x] DiracDelta[x - a] - b, with Robin boundary conditions f'[0] == f[0] and f'[c] == f[c], arising from a steady-state analysis of a 1D diffusion system.
  • Another participant questions the number of boundary conditions required for a second-order differential equation, suggesting that typically only two are needed.
  • There is a suggestion that the problem can be viewed as two coupled differential equations, one defined on [0,a] and the other on [a,c], each with Robin boundary conditions that are interdependent.
  • A participant hints at the need to integrate the ODE around the point a to properly formulate the boundary conditions at that point.
  • Some participants discuss the implications of changing boundary conditions, such as switching to Neumann conditions, while noting that the underlying difficulty remains unchanged.

Areas of Agreement / Disagreement

Participants express differing views on the number and nature of boundary conditions required for the problem, indicating that there is no consensus on how to approach the boundary conditions at the point of the Dirac delta function.

Contextual Notes

The discussion highlights the complexity of handling boundary conditions in the presence of a Dirac delta function and the potential for multiple interpretations of the problem setup, particularly regarding the coupling of the two regions defined by the delta function.

rynlee
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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!
 
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Properly I should title this more like

"diffusion-reaction DEQ with delta reaction term in steady state with homogenous generation"
 
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?
 
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.
 
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?
 
Coelum said:
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.
 
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.
 

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