Solving two-point boundary value problems with spline functions

We consider a collocation method for the approximation of thesolution of the nonlinear two-point boundary value problem y'(x)=f(x,y(x)), y(a)=A, y(b)=B, using splines of degree m3. The methodwhich we shall use leads to a system of recurrence relationswhich can be solved by Newton's method. By obtaining asymptotic error bounds we verify a conjectureof Khalifa & Eilbeck, i.e. splines of even degree can giveeven better solutions than splines of odd degree in certaincases.