Numerical analysis: approximation of boundary conditions (with fictive nodes)

[Thorra]

Help!

I need to know how to approximate this Laplace equation:


\(\displaystyle (\partial^2 u)/(\partial x^2) + (\partial^2 u)/(\partial y^2) = 0 \;\; \text{with boundary conditions:}\)


\(\displaystyle u(x,0) = 0\)


\(\displaystyle \dfrac{\partial u(x,1)}{\partial y}+\dfrac{2u(x,1)}{(1+x^2)+1} = \dfrac{1}{(1+x^2)+1}\)

\(\displaystyle u(0,y) = \dfrac{y}{1+y^2}\)

\(\displaystyle \dfrac{\partial u(1,y)}{\partial x} = - \dfrac{4y}{(4+y^2)^2}\)


I would REALLY appreciate some help.

 

[Deleted member 4993]

Help!

I need to know how to approximate this Laplace equation:


\(\displaystyle (\partial^2 u)/(\partial x^2) + (\partial^2 u)/(\partial y^2) = 0 \;\; \text{with boundary conditions:}\)


\(\displaystyle u(x,0) = 0\)


\(\displaystyle \dfrac{\partial u(x,1)}{\partial y}+\dfrac{2u(x,1)}{(1+x^2)+1} = \dfrac{1}{(1+x^2)+1}\)

\(\displaystyle u(0,y) = \dfrac{y}{1+y^2}\)

\(\displaystyle \dfrac{\partial u(1,y)}{\partial x} = - \dfrac{4y}{(4+y^2)^2}\)


I would REALLY appreciate some help.

Are you planning to use difference equations to approximate the PDE?

 

[Thorra]

Are you planning to use difference equations to approximate the PDE?

Thanks for correcting my equations and making the latex work! I always need to have someone do that for me on new math forums.

I want to approximate them with the differential analogue to an analytical differential equation.

I need to use fictive nodes, meaning I have to take a step out of the actual model, at least on both the x and y axis upper boundaries. My situation happens to look like this:

But I have no idea how to properly approximate the end points with that nor how to make a coefficient matrix (which I also have to do).
All I know is that the difference analogue is this, pretty much:
\(\displaystyle \dfrac{u(x+h,y)-2u(x,y)+u(x-h,y)}{h^2}+\dfrac{u(x,y+g)-2u(x,y)+u(x,y-g)}{g^2} + \mathcal O(h^2+g^2)\)

How do I involve fictive values? Here's my take for the fourth boundary bondition (BC):

\(\displaystyle \dfrac{\partial u(1,y)}{\partial x}=-\dfrac{4y}{(4+y^2)^2}\)
It's a neuman's equation and the boundary is basically this \(\displaystyle y'=\alpha\) or the differential analogue \(\displaystyle \dfrac{u_{N-1}^j-u_{N+1}^j}{2h}=-\dfrac{4y}{(4+y^2)^2}\) where \(\displaystyle u_{N+1}\) is meant to be the fictious node.
right? See I'm already confused here and I cannot really wrap my head around what I'm supposed to be doing and if I'm even on the right track here.

So yeah, I'm stuck in the woods here. Do I use \(\displaystyle u_{N+1}=u_{N-1}+\dfrac{8yh}{(4+y^2)^2}\) for something? Do I have to use the undertermined coefficient method somehow?
I keep reading books but can't really find anything that would help me with this.