# Analytical solve for biharmonic equation [duplicate]

I want to analytical Solve or numerical solve by Mathematica a biharmonic equation with homogeneous boundary conditions homogeneous?

I considered its solution using Fourier analysis, which is not responsive and its convergence speed is low and can not be used.

$$\nabla^4 U=f(x,y);\,\,U=\frac{\partial^2U}{\partial x^2}=0,\,\,\,at\,\, x=0,1;U=\frac{\partial^2U}{\partial y^2}=0,\,\,at\,\, y=0,1$$

## marked as duplicate by Kuba♦Aug 16 '18 at 6:17

• It seems you could expand both $U$ and $f$ as $\sin(\pi p x ) \sin(\pi q y)$. then the $\nabla^4 U_{pq}$ yields $\pi ^4 \left(p^2+q^2\right)^2 U_{pq}$ – chris Aug 13 '18 at 12:27