Increased Accuracy Cubic Spline Solutions to Two-Point Boundary Value Problems

Abstract

The relation between finite difference approximation and cubic spline solutions of a two-point boundary value problem for the differential equation y″ +f(x)y'+g(x)y = r(x) has been considered in a previous paper. The present paper extends the analysis to the integral equation formulation of the problem. It is shown that an improvement in accuracy (local truncation error O(h6) rather than O(h4)) now results from a cubic spline approximation and that for the particular case f(x) 0 the resulting recurrence relations have a form and accuracy similar to the well-known Numerov formula. For this case also a formula with local truncation error O(h8) is derived.