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

In summary, the conversation discusses a differential equation involving a Dirac Delta function and Robin boundary conditions. The equation is being used to model a 1D diffusion system with a point scatterer at a certain point. There is a difficulty in solving the equation, but it is suggested to split it into two coupled DEQs with shared boundary conditions. The conversation also touches on the geometrical interpretation and handling of the problem.
  • #1
rynlee
45
0
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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  • #2
Properly I should title this more like

"diffusion-reaction DEQ with delta reaction term in steady state with homogenous generation"
 
  • #3
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?
 
  • #4
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.
 
  • #5
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?
 
  • #6
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.
 
  • #7
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.
 

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

What is a Dirac delta function?

The Dirac delta function, denoted as δ(x), is a mathematical function that is used to represent an infinitely narrow, infinitely tall spike at a specific point in a graph. It is also known as the impulse function because it can be thought of as an instantaneous "impulse" of force at a single point.

How is the Dirac delta function used to solve differential equations?

The Dirac delta function can be used to represent a point source of force or energy in a differential equation. This allows us to mathematically model physical systems that involve point forces or impulses. By incorporating the Dirac delta function into the differential equation, we can solve for the behavior of the system at the specific point where the force or impulse occurs.

Can the Dirac delta function be integrated or differentiated?

Yes, the Dirac delta function can be integrated and differentiated, but it requires the use of special techniques such as the sifting property and the delta function derivative rule. These techniques allow us to manipulate the Dirac delta function in order to solve for unknown variables in a differential equation.

Are there any limitations to using the Dirac delta function in solving differential equations?

Yes, there are limitations to using the Dirac delta function. It can only be used to model point forces or impulses in a system, and cannot be used to represent distributed forces or continuous functions. Additionally, the Dirac delta function is not defined at x=0 and its integral is undefined. Therefore, it should be used with caution and with a thorough understanding of its properties.

Can the Dirac delta function be used in real-world applications?

Yes, the Dirac delta function has many practical applications in physics, engineering, and other scientific fields. It is commonly used in the modeling of electrical circuits, fluid dynamics, and signal processing. It is also used in solving boundary value problems and initial value problems in differential equations.

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