Quintic C² -spline integration methods for solving second-order ordinary initial value problems

In this paper we propose a global collocation method for the integration of the special second-order ordinary initial value problem (IVP) y″=f(x,y). The presented method is based on quintic C²-splines s(x) as an approximation to the exact solution y(x) of the (IVP). Analysis of stability shows that the method possesses (0,36)∪(54,110.2) as interval of periodicity and absolute stability. Moreover, the method has phase-lag of order four with actual phase-lag H⁴/18(6!). Error bounds, in the uniform norm, for ∥s⁽ⁱ⁾-y⁽ⁱ⁾∥=O(h⁴),i=0(1)2, if y∈C⁶ [0,b], together with illustrative test examples will also be considered.