I would like to know if there is any way to solve (both analytically and numerically) a 4th order PDE of the kind:
$$\phi_{xxxx}-\alpha(t)\ \phi_{xx} +\beta\ \phi\ - \gamma\ \phi_{t}=0$$
in the unknown $\phi(x,t)$; with suitable boundary conditions and where the apex indicates derivation with respect to $x$ and the subscript $(_{t})$ stands for derivation with respect to time. Is there a numerical procedure more suited for this kind of problem?