How to handle the Dirac delta function as a boundary condition

[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?

 

[AiRAVATA]

There shouldn't be any epsilon in the equations.

 

[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).

 

[mathy_girl]

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.

 

[CFDFEAGURU]

Have you tried a numerical approach?

 

[mathy_girl]

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.