I came across the following Laplace bvp:

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




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 $\ 0<x+y<1,\ 0<x-y<2$




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) $.


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