How can I find numerical solutions to a PDE involving BiLaplace operator?

[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 $$u_{xxxy}, u_{xyyy} & u_{xxyy}$$ ?

 

[jvicens]

Thank you for sharing your partial differentiation equation and your approach to finding a numerical solution. It seems like you have a good understanding of the problem and have made some progress already.

To find the derivatives \frac{\partial^{4}u}{\partial x^{3}\partial y}, \frac{\partial^{4}u}{\partial x\partial y^{3}}, and \frac{\partial^{4}u}{\partial x^{2}\partial y^{2}}, you can use a similar approach to what you did for \frac{\partial^{4}u}{\partial x^{4}}. For example, to find \frac{\partial^{4}u}{\partial x^{3}\partial y}, you can use the following formula:

\frac{\partial^{4}u}{\partial x^{3}\partial y} = \frac{1}{h^{3}}(u_{i+2,j}-3u_{i+1,j}+3u_{i-1,j}-u_{i-2,j})

where h is the step size in the x-direction and j is the index in the y-direction. Similarly, you can find \frac{\partial^{4}u}{\partial x\partial y^{3}} and \frac{\partial^{4}u}{\partial x^{2}\partial y^{2}} using similar formulas.

As for books, there are many resources available for numerical methods for partial differential equations. Some good ones to start with are "Numerical Methods for Partial Differential Equations" by George F. Pinder and "Numerical Solution of Partial Differential Equations: An Introduction" by K. W. Morton and D. F. Mayers. These books provide a good overview of numerical methods for solving partial differential equations and can help you with your approach.

I hope this helps and good luck with your research!