# Laplace boundary value problem

I came across the following Laplace bvp:

$u_{xx}+u_{yy}=0\$ for $\ 0<x<1,\ 0<y<2$

$u(x,0)=u(x,2)=0$

$u(0,y)=0$

$u(1,y)=y(2-y)$

I didn't have any problems solving it. It was quite straightforward, following the general theory. However, the next exercise asked to solve the following bvp:

$u_{xx}+u_{yy}=0\$ for $\ 0<x+y<1,\ 0<x-y<2$

$u(x,x)=u(x,x-2)=0$

$u(x,-x)=0$

$u(x,1-x)=(3-2x)(2x-1)$

The exercise gave a hint stating that I should use an appropriate transformation in order to relate this problem with the previous one. However, I had no luck with it. My thought was to set: $\eta=x+y$ and $\xi=x-y$ but I can't handle the boundary conditions with this transformation! How can I solve this one? Any help would be appreciated. Thanks in advance!

Your transformation is the right thing. You can transform the boundary conditions in the following way: the inverse coordinate transformation is ( $x = \frac{\eta + \xi}{2}, y = \frac{\eta - \xi}{2}$ ). Now the boundary line $y = 1 - x$ with the non zero boundary condition is transformed to $\eta = 1$ in the $\eta-\xi$ plane, so on this line $x = \frac{1+\xi}{2}$. The boundary data $(3 - 2 x)(2 x -1 )$ then becomes $\xi (2 - \xi)$.