Solving 4th order PDE

I am trying to solve this equation when t= constant time, how the x and h looks. I did try PDEPE but it can only solve 2nd order PDE. Can anyone give me some advice? THanks!

Answer 1

[Bill Greene]

You may be able to solve this in pdepe with two equations and two dependent variables h and P. The first equation would be your equation (1) and the second would be your last equation for P. Whether you can do this depends on your boundary conditions, which you did not show.