How to handle the Dirac delta function as a boundary condition

In summary, the conversation is discussing the use of perturbation theory to solve a problem involving a partial differential equation with initial and boundary conditions. The problem is expanded into a series and the equations for the first two terms are given. The speaker is looking for a way to handle the Dirac Delta function in the initial condition and considers using a simpler 2D equation for analysis. There is also a discussion about using a numerical approach versus finding an analytical solution.
  • #1
mathy_girl
22
0
Using perturbation theory, I'm trying to solve the following problem

[tex]\frac{\partial P}{\partial \tau} = \frac{1}{2}\varepsilon^2 \alpha^2 \frac{\partial^2 P}{\partial f^2} + \rho \varepsilon^2 \nu \alpha^2 \frac{\partial^2 P}{\partial f \partial \alpha} + \frac{1}{2}\varepsilon^2 \nu^2 \alpha^2 \frac{\partial^2 P}{\partial \alpha^2}, \quad \mbox{for } \tau>0,
[/tex]
with initial condition [tex]P = \alpha^2~\delta(f-K), \quad \mbox{for } \tau=0. [/tex]

Expanding [tex] P_\varepsilon=P_0 + \varepsilon^2 P_1 + \ldots[/tex] the [tex]\mathcal{O}(1)[/tex] equation is given by
[tex]\frac{\partial P_0}{\partial \tau} = 0, \quad \mbox{for } \tau>0,[/tex]
with boundary condition [tex]P_0 = \alpha^2~\delta(f-K) \mbox{ for } \tau=0[/tex].

Obviously, this gives [tex]P_0 = \alpha^2~\delta(f-K).[/tex]

Now I would like to solve the [tex]\mathcal{O}(\varepsilon^2)[/tex] problem
[tex]\frac{\partial P_1}{\partial \tau} = \frac{1}{2} \alpha^2 \frac{\partial^2 P_0}{\partial f^2} + \rho \nu \alpha^2 \frac{\partial^2 P_0}{\partial f \partial \alpha} + \frac{1}{2} \nu^2 \alpha^2 \frac{\partial^2 P_0}{\partial \alpha^2}, \quad \mbox{for } \tau>0[/tex]
with initial condition [tex]P_1 = 0 \mbox{ for } \tau=0[/tex].

Does anyone of you know how to handle the Dirac Delta function in the initial condition and O(1) solution here?
 
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  • #2
There shouldn't be any epsilon in the equations.
 
  • #3
It is kind of 2D diffusion (or heat conduction) equation with an initial condition, not boundary.

Physically the exact solution should describe the "relaxation" of initial non uniformity of P.

I am afraid it cannot be solved by the perturbation theory in powers of epsilon - you neglect the derivative terms that are responsible for the space relaxation.

Consider a simpler 2D equation - with constant coefficients and analyse the exact solution, if it is expandable (analytical in epsilon at epsilon=0).
 
  • #4
AiRAVATA said:
There shouldn't be any epsilon in the equations.

Sorry, that's my mistake.. think it's a copy-paste error. I corrected it in the previous message.
 
  • #5
Have you tried a numerical approach?
 
  • #6
CFDFEAGURU said:
Have you tried a numerical approach?

That's always possible, but the assignment here is to do it analytically... Tomorrow I'll ask my supervisor if he thinks there's another way to solve this analytically.
 

1. What is the Dirac delta function?

The Dirac delta function is a mathematical concept that represents an infinitely narrow and tall spike at a specific point. It is often used in physics and engineering to model point sources of energy or mass.

2. How is the Dirac delta function used as a boundary condition?

The Dirac delta function can be used as a boundary condition in mathematical models to represent a concentrated or localized input or output of a system. It can also be used to enforce specific constraints on a solution.

3. How do you handle the Dirac delta function as a boundary condition in calculations?

To handle the Dirac delta function as a boundary condition, it is typically approximated by a sequence of functions that approach the delta function as the sequence becomes infinitely dense. This allows for the delta function to be used in calculations and simulations.

4. Are there any special rules for handling the Dirac delta function as a boundary condition?

Yes, there are some special rules for handling the Dirac delta function as a boundary condition. For example, when using the delta function to enforce a constraint, the integral of the delta function must equal the value of the constraint. Additionally, when approximating the delta function, the sequence of functions must be chosen carefully to ensure convergence to the delta function.

5. Can the Dirac delta function be used as a boundary condition in all types of problems?

No, the Dirac delta function may not be applicable as a boundary condition in all types of problems. It is most commonly used in problems involving point sources or concentrated inputs/outputs. In other types of problems, alternative boundary conditions may be more appropriate.

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