Characterization of local strict convexity preserving interpolation methods by C1 functions
Journal of Approximation Theory
B-Spline Linear Multistep Methods and their Continuous Extensions
SIAM Journal on Numerical Analysis
The continuous extension of the B-spline linear multistep methods for BVPs on non-uniform meshes
Applied Numerical Mathematics
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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 @?S-y@?"~=O(h^2). 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 @?S^'-y^'@?"~=O(h^2), @?S^''-y^''@?"~=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.