Radial basis functions for solving differential equations: Ill-conditioned matrices and numerical stability

High-order numerical methods for solving differential equations are, in general, fairly sensitive to perturbations in their data. A previously proposed radial basis function (RBF) method, namely an integrated multiquadric scheme (IMQ), is applied to two-point boundary value problems whose solutions exhibit thin boundary layers. As frequently observed among RBF methods, the matrices arising are ill-conditioned, in this paper to the point of numerical singularity. The sensitivity of the method to perturbations and round-off error is investigated, and evidence is provided that perturbations are not nearly as strongly amplified as suggested by the large condition numbers of the matrices used in the computation.