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

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?

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There shouldn't be any epsilon in the equations.

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).

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.

Have you tried a numerical approach?

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.