Multi-point Taylor approximations in one-dimensional linear boundary value problems

We consider second-order linear differential equations in a real interval $I$ with mixed Dirichlet and Neumann boundary data. We consider a representation of its solution by a multi-point Taylor expansion. The number and location of the base points of that expansion are conveniently chosen to guarantee that the expansion is uniformly convergent $\forall x\in I$. We propose several algorithms to approximate the multi-point Taylor polynomials of the solution based on the power series method for initial value problems.