Collocation for linear problems

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

Univariate linear TVBVP:

\[ u'' + p(x)u' + q(x)u = r(x), \quad u(a) = \alpha, u(b) = \beta. \]

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 \(\mathbf{E}\):

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