PDE involving BiLaplace

mariuspop

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

[tex]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[/tex]


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
[tex]\frac{\partial^{4}u}{\partial x^{3}\partial y}[/tex]
and
[tex]\frac{\partial^{4}u}{\partial x\partial y^{3}}[/tex]

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:
[tex]u = a+bx+cx^{2}+dx^{3}+ex^{4}[/tex]
in the [tex]u_{i}[/tex] 's vicinity
therefore we have [tex]u_{i-2}, u_{i-1}, u_{i+1}, u_{i+2}[/tex] as neighbors

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

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

how can i find [tex]u_{xxxy}, u_{xyyy} & u_{xxyy} [/tex] ???