I'm trying to numerically solve the three coupled PDEs;
$\frac{\partial\theta}{\partial t} = w + \nabla^2 \theta, \ \ \ \ \ (1)$
$\frac{\partial Q}{\partial t} = -RaPr\nabla^2_H\theta + \nabla^2 Q, \ \ \ \ \ (2)$
$\nabla^2 w = Q. \ \ \ \ \ (3)$
Which represent a linear stability analysis for temperature ($\theta$) and z-component velocity ($w$) perturbations for a fluid initially at rest between two infinite horizontal plates. I'm looking for a particular form of periodic solution such that;
$\nabla^2 = -a^2 + \frac{\partial^2}{\partial z^2}, \ \ \ \ \ \ (4)$
$\nabla^2_H = -a^2. \ \ \ \ \ \ (5)$
$Ra$, $Pr$, and $a$ are parameters that do not depend on space or time. The domain is $z\in[0,1]$ and $t\in[0,\infty)$. I have the following boundary conditions: $\theta = w = \partial{w}/\partial{z} = 0 $. I am trying to convert these to obtain a boundary condition on $Q$. I have tried substituting $(4)$ into $(3)$ and taking $\partial/\partial z$ but this does not tell me anything about $Q$ and I know nothing about $\partial ^3 w/\partial z^3$. I am unsure on what I can try. Any help/advice would be greatly appreciated.