The author uses a polynomial spline collocation method to solve the problem \(y^{(r)}(t)=f(t,y(t),...,y^{(r- 1)}(t))+\int^{t}_{0}k(t,s,y(s),...,y^{(r-1)}(s))ds,\) \(t\in [0,T]\) with given initial values \(y^{(j)}(0)=y_{0j}\), \(0\leq j\leq r-1\). Global convergence and local superconvergence results are given.