I have the following PDE to solve with an initial dirac condition.
$$
\frac{\partial p(t,y)}{\partial t} = y \frac{\partial ^2 p}{\partial ^2 y} + (2-\alpha + y) \frac{\partial p}{\partial y} + p
$$
The simplest way I could think of was to Fourier transform it relative to the second variable y, and solve the first order PDE that comes using the method of characteristics. It looks pretty good except for one thing: I don't know how to express the following mixed boundary condition in the Fourier domain...
$$ s(t, y) = -\frac{\partial ( y p(t,y)}{\partial y} + (\alpha - y) p(t,y) $$ must be equal to zero at $y_{min}$ and $y_{max}$ for any $t$.
Is there a way?
First, I doubt that your Fourier transform is 100% accurate - since the right-hand side has terms with first order derivatives, there must be imaginary units somewhere.
Second, your condition $p(t,y_0)=0$ can be written as an application of a delta-distribution: $$\langle \delta_{y_0}, p(t,y)\rangle = 0.$$ Take a Fourier transform to obtain (up to some constants from the normalisation of Fourier transform) $$\langle \exp(i\xi y_0), \hat p(t,\xi)\rangle = 0 = \int_\Bbb R \hat p(t,\xi)\exp(i\xi y_0)d\xi.$$