Linear/linear rational spline collocation for linear boundary value problems

We investigate the collocation method with linear/linear rational spline S of smoothness class C-1 for the numerical solution of two-point boundary value problems if the solution y of the boundary value problem is a strictly monotone function. We show that for the linear/linear rational splines on a uniform mesh it holds vertical bar vertical bar S '' - y ''vertical bar vertical bar infinity = O(h). Established bound of error for the collocation method gives a dependence on the solution of the boundary value problem and its coefficients. We prove also convergence rates vertical bar vertical bar S '' - y ''vertical bar vertical bar infinity = O(h(2)), vertical bar vertical bar S '' - y ''vertical bar vertical bar infinity = O(h) and the superconvergence of order h(2) for the second derivative of S in certain points. Numerical examples support the obtained theoretical results. (C) 2013 Elsevier B.V. All rights reserved.