Collocation for linear problems

Collocation: the locations of the unknowns and approximations are at the same nodes.

Univariate linear TVBVP:

u

Matrix linear TVBVP with differencing:

\mathbf{Lu} = \mathbf{r}, \quad \mathbf{L} = \mathbf{D}_{xx} + \mathbf{PD}_{x} + \mathbf{Q}.

Then, add in boundary problems with the deletion matrix \mathbf{E}:

\left[\begin{array}{c} e_0^T\\ \mathbf{EL}\\ e_n^T \end{array}\right] \mathbf{u} = \left[\begin{array}{c} \alpha\\ \mathbf{Er}\\ \beta \end{array}\right].