Can you give here the explicit form of the equation that you want to solve along with your boundary conditions?

Here is the equation-

(∂^2/(∂x^2 )+∂^2/(∂y^2 ))(∂^2/(∂x^2 )+∂^2/(∂y^2 ))w=p/D

and the general boundary condition is- w=0 at x=0 and w= 0 at x=l and same for the y axis.

Is p/D just a constant? If yes, your problem admits a particular solution that reads w=(1/8) x(x-l) y(y-l) p/D. However, this is not the most general solution: the latter is obtained by adding to the above particular solution a general solution of the homogeneous problem : (∂^2/(∂x^2 )+∂^2/(∂y^2 ))(∂^2/(∂x^2 )+∂^2/(∂y^2 ))w=0 with the boundary conditions w=0 at x=0, w= 0 at x=l, w=0 at y=0, w= 0 at y=l. The general solution in question is easily constructed using e.g. the Fourier series expansion in both variables x and y, or, more precisely, in (x/l) and (y/l), assuming that the solution is extended periodically beyond the square 0

Thanks Artur

How did you get the final answer? Can you show me the procedure Artur?

This is the biharmonic equation. You do not have enough boundary conditions (w specified on boundary) to uniquely specify the solution. You need either an extra essential boundary condition (e.g., dw/dn on boundary = 0) or an extra natural boundary condition (e.g., no traction condition: Laplace(w) = 0 on boundary). The latter can be handled by Laplace(u) = p/D, Laplace(w) = u, but the former requires a different treatment. There are C^1 finite element methods that are applicable for the biharmonic equation, whatever the boundary conditions are.

The weak formulation, by the way, is \int_\Omega [Laplace(w)-p/D].Laplace(v).dx = 0 for all smooth v satisfying the homogeneous essential boundary conditions (so far, just v = 0 on boundary).

If you are using Biharmonic plate equation, subjected to Electrostatic force. You can use Galerkin method to solve it.

Refer to the book " MEMS linear and nonlinear statics and dynamics" by Mohammad Younis.

How can I clasified the equation u_{xxxx}-u_{xxyy}+u_{yyyy}=0?

Thanks