I'm trying to solve the laplace equation on a rectangle domain with the following boundaries.
$U_{xx}+U_{yy}=0$
$1)\quad U_x(0,y)=0$
$2)\quad U_y(x,0)=0$
$3)\quad U(x,b)=0$
$4)\quad U_x(a,y)=f(y) \rightarrow f(y=b)\neq0$
the problem is that at the point $(a,b)$ I have a contradiction between the boundary conditions (3) and (4).
BC (3) immediately dictates that there is no current in the $x$ direction ($U_x(a,x)=0$) at the top BC. to be more specifically $U_x(a,b)=0$
However, from BC (4) I should get $U_x(a,b)=f(b)\neq0$. When I solve it analytically and ploted it. it shows that there is a discontinuity at this point from zero to a certain value, namely, between $U_x(x,b)$ and $Ux(z,b-\varepsilon)$
So my question are:
1) Is there a problem with the fact that there is one point that contradicts?
2) This problem need to describe a current source from the right side, then it makes a "turn" and go out from the upper side. meaning that at the corner $(a,b)$ the current immediately go out. is there another way to choose the BC?
3) and if it's Ok to have a problem with one problematic point, are there ways/ tricks to help the convergence of the fourier transform? I also tried a numerical approach (using finite difference), but it also converge slowly.
Thanks for the help.