A New Collocation Scheme Using Non-polynomial Basis Functions

Abstract

In this paper, we construct a set of non-polynomial basis functions from a generalised Birkhoff interpolation problem involving the operator: \({\mathscr {L}}_\lambda ={d^2}/{dx^2}-\lambda ^2 \) with constant \(\lambda .\) With a direct inverting the operator, the basis can be pre-computed in a fast and stable manner. This leads to new collocation schemes for general second-order boundary value problems with (i) the matrix corresponding to the operator \({\mathscr {L}}_\lambda \) being identity; (ii) well-conditioned linear systems and (iii) exact imposition of various boundary conditions. This also provides efficient solvers for time-dependent nonlinear problems. Moreover, we can show that the new basis has the approximability to general functions in Sobolev spaces as good as orthogonal polynomials.