I understand that a circle in cartesian coordinates is not a rectangular domain and using polar coordinates gives us our required domain to solve a PDE by separation of variables. However I do not understand very well on how to change certain boundary and initial conditions from the cartesian problem to the polar form. For example :
$u_{xx}+u_{yy}=0$, $x^2 + y^2<1$, $x>0$
$u(0,y)=1$
$u(x,y)=1+x^2$, for $x^2 +y^2=1, x>0$
I understand that our equation that we need to solve is the laplacian in polar coordinates and that $0<r<1$ and $-\pi/2<\theta<\pi/2$ but what happens to the initial and boundary conditions?