What is correct plot of Laplace 2d equation

Question

I have to plot distribution of 2D Laplace equation: $\Delta^2\psi\ (x,y)=0$

$\psi\ (x, 0) = \psi\ (x, 1) = \sin{x}$

$\psi\ (0, y) = \psi\ (1, y) = 0$

So I have these graphs:

Is there correct one?

Answer 1

On the first two images, the boundary effects on the left look bad; they should not be there. The third image looks exactly right for the boundary value problem with $$\psi(x,0)=-\sin x,\quad \psi(x,1)=\sin x$$ $$\psi(0,y)=0=\psi(1,y)$$ Indeed, its graph shows anti-symmetry with respect to the horizontal line $y=1/2$: namely, $\psi(x,1-y)=-\psi(x,y)$. The boundary conditions you describe are symmetric: $\psi(x,1-y)=\psi(x,y)$. None of the plots have this symmetry.

So, my suggestion is: look for a sign error in the script that produced the third plot.