# How to solve Linear 4th order PDE

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?

• Does the PDE mean $\phi_{xxxx}-\alpha(t)\phi_{xx}+\beta\phi-\gamma\phi_{t}=0$ ? – doraemonpaul Jul 18 '18 at 12:46
• yes, it does. I have updated the PDE – Riccardo Jul 19 '18 at 13:39