We see often the equation for a clamped plate at 4 edges and we have a sort of closed form here (reduction to n coupled ODE).
But I am interested now in another practical example: the solution of the clamped plate at 4 points (like screws in the plate) instead of 4 edges submitted to a point load at $\mathbf{\rho_0} = (x_0,y_0)$ and I need only one more particular solution (found in the Polyanin's LPDE book).
This is the problem on the domain $[0,a] \times [0,b]$:
$$ \Delta^2 w(x,y) = \delta(x-x_0) \delta(y-y_0) $$
and the point conditions (not boundary: points):
$$ w = \partial_x w = \partial_y w = 0 $$ at points $(0,0)$, $(0,b)$, $(a,0)$ and $(a,b)$
This means: 12 point conditions so 12 coefficients to discover (it is not necessarily a series because it is only point conditions). Then I express simply the deflection $w$ as:
$$w = w_0 + w_p $$
with $w_0 = \frac{1}{8 \pi} \theta(\mathbf{\rho} - \mathbf{\rho_0}) |\mathbf{\rho} - \mathbf{\rho_0}|^2 \ln ( | (\mathbf{\rho} - \mathbf{\rho_0}) | ) $, the fundamental solution (with $\rho = \sqrt{x^2 + y^2}$ and $\theta(\mathbf{\rho})$ the Heaviside step function in 2-dimensions) and only 11 independent solutions inside $w_p$:
$$w_p = c_1 x^3 + c_2 y^3 + c_3 x^2 y + c_4 x y^2 + c_5 x^2 + c_6 y^2 + c_7 x y + c_8 x + c_9 y + c_{10} + c_{11} (x^2 + y^2) \ln (x^2 + y^2) $$
You see the problem: only 11 coefficients. Note that the functions should be continuous on the domain and the same for its first derivatives. So the other solutions $\ln \rho$, $x \ln \rho$ and $y \ln \rho$ cannot be use here.
Do you know what is (if known) the missing function ?