Help with Equation involving BiLaplace

[mariuspop]

hello
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 $$u_{xxxx} = 24e$$

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

 

[Nino MMP]

?The best way to find the partial derivatives is to use a numerical method like finite differences. You can use a four-point stencil (using 4 points around the center point u_{i}) to approximate the derivatives. For example, for u_{xxxx} you can use the following formula:u_{xxxx} = \frac{1}{h^{4}}(-u_{i-2} + 16u_{i-1} -30u_{i} + 16u_{i+1} - u_{i+2}) You can also use the same formula to approximate u_{xxxy}, u_{xyyy} and u_{xxyy}, but with different coefficients. For example, for u_{xxxy} you can use the following formula:u_{xxxy} = \frac{1}{h^{4}}(-u_{i-2,j} + 16u_{i-1,j} -30u_{i,j} + 16u_{i+1,j} - u_{i+2,j}) where j is the index of the y-direction. You can find more information about numerical methods for partial differential equations in books like "Numerical Methods for Partial Differential Equations" by Kincaid and Cheney.