Boundary lifting ($w=\ell +z$)

Subtract a known function with the required boundary trace so that the remaining unknown has homogeneous boundary conditions. For an interval with Dirichlet boundary conditions $g(t),h(t)$, $\ell =(1-x/L)g+(x/L)h$ is a simple lifting. In $w_t=w_{xx}-cw$, $z=w-\ell$ has forcing $-{\ell }_t-c\ell$ and zero endpoint values. This improves Fourier sine series convergence.

For the free Schrodinger equation with compatible smooth initial and boundary data, subtract $g(t)e^{-x}$ before taking a Fourier sine transform. The lifted initial datum vanishes at the endpoint, making its sine transform $O(k^{-2})$ if its second derivative is integrable. The transformed forcing is $k(ig-g^{\prime })/(1+k^2)$; time integration followed by integration by parts makes its solution contribution uniformly $O(k^{-3})$. An integrable spectral majorant proves uniform convergence even at the compatible initial-boundary corner.