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I have a heat-type PDE in 2+1 dimensions with two modified Bessel operators, i.e.

\begin{equation} \frac{\partial}{\partial t} P = \mathcal{L}_x P + \mathcal{L}_y P \end{equation}

with \begin{align} \mathcal{L}_x &= x^2 \partial_x^2 + x \partial_x - x^2 \\ \nonumber \mathcal{L}_y &= y^2 \partial_y^2 + y \partial_y - y^2 \\ \nonumber \end{align}

This equation separates into three eigenvalue equations, with the two spatial equations becoming modified Bessel equations.

I would like to find the Green's function for these equations, but the boundary conditions for my problem require P vanishing on the boundaries $x=0$ and $x = \infty$, as well as $y=0$ and $y=\infty$. This comes about because $P$ is, in the original problem a probability density, and must be integrable under the measure $(dx/x) (dy/y)$. As far as I can see, there are no modified Bessel functions at all that satisfy these boundary conditions, even though the equation seems perfectly well defined with these boundary conditions, and I am able to find numerical solutions for it. How can I find the Green's function for this PDE, with these boundary conditions?

It really seems to me that, although the problem is separable, the separation of variables ansatz is nevertheless false.

Jyrki Lahtonen
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D. Eliezer
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    I do see your point, and considered doing that, but this is the kind of mathematics that physicists do, much more often than mathematicians. Greens functions are a standard part of every physicists education. – D. Eliezer Jul 11 '17 at 03:09

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