# How to handle the Dirac delta function as a boundary condition

1. Jul 6, 2009

### mathy_girl

Using perturbation theory, I'm trying to solve the following problem

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

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

Obviously, this gives $$P_0 = \alpha^2~\delta(f-K).$$

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

Does anyone of you know how to handle the Dirac Delta function in the initial condition and O(1) solution here?

Last edited: Jul 6, 2009
2. Jul 6, 2009

### AiRAVATA

There shouldn't be any epsilon in the equations.

3. Jul 6, 2009

### Bob_for_short

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. Jul 6, 2009

### mathy_girl

Sorry, that's my mistake.. think it's a copy-paste error. I corrected it in the previous message.

5. Jul 6, 2009

### CFDFEAGURU

Have you tried a numerical approach?

6. Jul 6, 2009

### mathy_girl

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.