I would like to solve the advection diffusion equation $$ \partial_t P(x,t) = -V \partial_x P(x,t) + D \partial_x^2P(x,t) $$ for the conditions $P(x,0) = \delta(x)$, $P(x\rightarrow\infty,t) = 0$, and $\partial_t P(x,t)|_{t=0} = -V\delta'(x)$, but I am not sure how to proceed.
So far, I attempted to use separation of variables with $P(x,t) = X(x)T(t)$, finding $$ T' - \lambda T = 0$$ and $$ DX'' - VX' - \lambda X = 0.$$ I think this gives a product solution
$$P(x,t) = \int_0^\infty K(\lambda)e^{-\lambda t} \exp\Big[\frac{Vx}{2D}\bigg(1-\sqrt{1+\frac{4D\lambda}{V^2}}\bigg)\Big] d\lambda,$$ where $K$ is an unkown function of the parameter $\lambda$, but I am not sure if this is correct. In fact I think it is not, because there should be two free parameters remaining to fix by the conditions. Should it also involve a convolution over $x$? And if so, how would I know this from the mathematics?
Any advice here would be most welcome - I'm left with many questions. What is this combination of initial/boundary conditions called? Is there a reference for this solution? Is the product solution by separation of variables possible, and if so, where am I going wrong? Why does a simple Fourier-Laplace transform approach fail? Thanks for any guidance.