Sinc-Galerkin Methods

In this chapter we consider an FK2 of the form (1.2.2): ϕ(x) — λ ∫ a b k(x,s)ϕ(s)ds = f (s), where a ≤ x,s ≤ b, and the kernel k(x,s) has a weak singularity at an endpoint. In numerical approximations, whether in quadrature, finite differences,finite elements, and the like, the computational methods generally use polynomials as basis functions to obtain approximate solutions that are sufficiently accurate in a region where the function to be approximated is ‘smooth’ (or analytic). However,such methods fail significantly in a neighborhood of singularities of the function.An analytic function ϕ has a singularity at a point at which ϕ does not exist and at the endpoints of an interval or a contour. On the other hand, the numerical approximation obtained by using Whittaker’s cardinal function C(ϕ, h, x) yields much better results than those obtained by polynomial methods in the case when singularities are present at an endpoint of an interval. This method, however,may or may not yield better results in the absence of singularities. The function C(ϕ, h, x) is called the cardinal interpolant to ϕ(x).