Interpretation of a PDE

Question

Let us say that I have the partial differential equation,

\begin{align} \frac{\partial W}{\partial t}=(b-a)\partial_p\partial_qW -(\dot{a}\partial_p^2+\dot{b}\partial_q^2)W\,, \end{align}

where $a=a(t)$, $b=b(t)$, and $W=W(q,p,t)$. As this looks similar to the diffusion equation, my guess is that the last term represents the diffusion with respect to $p$ and $q$. Is there a physical meaning for the $(b-a)\partial_p\partial_qW$ term? Alternatively, is this differential equation known by a specific name?

Answer 1

I don't think the $\partial_p\partial_qW$ term can be interpreted in isolation. The entire PDE is an anisotropic diffusion equation $$\frac{\partial W}{\partial t} = \nabla\cdot(\mathbf D\,\nabla W)$$ where $\mathbf D$ is the diffusion coefficient matrix, in this case $$\mathbf D=\begin{bmatrix}-\dot b & \frac12(b-a) \\ \frac12(b-a) & -\dot a\end{bmatrix}.$$ I believe the matrix $\mathbf D$ needs to be positive definite for the equation to be well-posed.

Answer 2

If we look carefully and replace

$p \implies X$

$q \implies Y$.

So that we can relate X- and Y- with space co-ordinates.

Then,

$W = f(X, Y, t)$

This makes the things easy and the given equation is comparable to damped wave equation.