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I am trying for first time to solve a partial differential equation in Mathematica. Specifically the plate differential equation (see wikipedia https://en.wikipedia.org/wiki/Plate_theory#Transverse_loading).

I am interested on analytical solution, but first I try for numerical. Anyway, both are not give a result.

The problem is that I define the boundary conditions (at least I think so) but Mathematica says that "The spatial derivative order of the PDE may not exceed two."

The code is:

Clear[w, qD, W, L]
qD = 1; W = 1; L = 1;
NDSolve[{
   Laplacian[Laplacian[w[x, y], {x, y}], {x, y}] == qD,
   w[0, y] == 0,
   (D[w[x, y], x]) /. x -> 0) == 0,
   w[L, y] == 0,
   (D[w[x, y], x] /. x -> L) == 0,
   w[x, 0] == 0,
   (D[w[x, y], y]) /. y -> 0) == 0,
   w[x, W] == 0,
   (D[w[x, y], {y,2}]) /. y -> W) == 0
}, w, {x, 0, L}, {y, 0, W}]

So, boundary conditions means: 3 edges are fixed (no displacement, no rotation) and one has rotation allowed (but not displacement).

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