In a paper I read, an elliptic boundary value problem on a bounded domain D x (0,T) is solved by first transforming it in a set of equations on half-spaces R^n and then applying partial Fourier transform. Eventually one single ODE is obtained. It is not explained how this transformation is done and it seems very mysterious to me. Can someone please give references for this approach? I am also interested if it is possible in general for a single evolution equation to be transformed in system of equations on half spaces. Thank you for all replies.