PDE involving BiLaplace

[mariuspop]

well,
i have a partial differentiation equation that look like this:

c_{1}\frac{\partial^{4}u}{\partial x^{4}}+c_{2}\frac{\partial^{4}u}{\partial x^{3}\partial y}+c_{3}\frac{\partial^{4}u}{\partial x^{2}\partial y^{2}}+c_{4}\frac{\partial^{4}u}{\partial x\partial y^{3}}+c_{5}\frac{\partial^{4}u}{\partial y^{4}} = s


as u can se we have the bilaplace operator in 2 directions and those two terms that makes the eq. a bit heavier
s - is the source, c 1...5 - constants,
also we have boundary conditions
but
at this moment i m interested in finding a numerical solution to the above eq.
i tried with polynomial approximation, but i got stuck while concerning
\frac{\partial^{4}u}{\partial x^{3}\partial y}
and
\frac{\partial^{4}u}{\partial x\partial y^{3}}

can someone help?
or
if possible can u recommend some books
thanks in advance

ps: my approach was done considering a 4 degree polynomial eq:
u = a+bx+cx^{2}+dx^{3}+ex^{4}
in the u_{i} 's vicinity
therefore we have u_{i-2}, u_{i-1}, u_{i+1}, u_{i+2} as neighbors

u_{xxxx} = 24e
and
e = \frac{1}{24h^{4}}(u_{i+2}-8u_{i+1}+14u_{i}-8u_{i-1}+u_{i-2})

so u_{xxxx} = \frac{1}{h^{4}}(u_{i+2}-8u_{i+1}+14u_{i}-8u_{i-1}+u_{i-2})

how can i find u_{xxxy}, u_{xyyy} & u_{xxyy} ???