Higher order discretization methods for \(y''=f(x,y,y')\) (Q1115122)

High order discretization methods are presented for a two-point boundary value problem \(y''=f(x,y,y'),\) \(y(a)=y_ a,\) \(y(b)=y_ b\) where \(f_ y>0\). Applying second and third degree B-splines on the equation, two methods are given. The former is a four-step one with the error of fourth order, the latter is a six-step one with the error of sixth order. Global error analysis is also given. As a by-product these results are shown to be applicable to find the necessary conditions to determine the extremal value of \(\int^{b}_{a}g(x,y,y')dx\). Starting procedure required to these algorithms is also considered. Some numerical examples are shown for the illustration.