# How to solve fourth order pde similar to biharmonic equation.

I'm trying to solve a fourth order pde similar to the biharmonic equation

$0=\frac{\partial ^4}{\partial x^4}u(x,y)+Q\frac{\partial ^4}{\partial x^2 \partial y^2}u(x,y)+\frac{\partial ^4}{\partial y^4}u(x,y)$

Where Q is a constant. The boundary condition are the following,

$\frac{\partial^2}{\partial x^2}u(x,y)=0\quad \quad \frac{\partial^2}{\partial y^2}u(x,y)=0 \quad \textit{and} \quad \frac{\partial^2}{\partial x \partial y}u(x,y)=0$

while the boundary is a rectangle.

So far neither Fourier series nor separation of variables seems to work on this one and I'm no expert to the topic.

• Do you have boundary conditions? – EpicMochi Oct 21 '14 at 13:27
• Yes, I just edited the question accordingly. Thanks for pointing out. – Cedric Oct 21 '14 at 13:52
• Do you have some boundary conditions on $u$ itself? – spatially Oct 21 '14 at 13:57
• no, only on the derivatives. – Cedric Oct 21 '14 at 14:00
• $\dfrac{\partial^2}{\partial x^2}u(x,y)=0$ , $\dfrac{\partial^2}{\partial y^2}u(x,y)=0$ and $\dfrac{\partial^2}{\partial x\partial y}u(x,y)=0$ are not belonging to the boundary conditions since they are already restricted the form of $u(x,y)$ should be. – doraemonpaul Oct 22 '14 at 21:13