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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] ???

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# PDE involving BiLaplace

Can you offer guidance or do you also need help?

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