Numerische Lösung gewöhnlicher Differentialgleichungen mit Splinefunktionen

In this paper a general procedure to obtain spline approximations for the solutions of initial value problems for ordinary differential equations is presented. Several well-known spline approximation methods are included as special cases. It is common practice to partition the interval for which the initial value problem is defined into equidistant subintervals and to construct successively the spline approximation; thereby the spline function has to satisfy certain conditions at the knots. In the general procedure presented here additional knots are admitted in every subinterval. At these points which need not be equally spaced the spline approximation has to fulfill analogous conditions as at the original knots. Convergence and divergence theorems are proved; especially the influence of the additional knots on convergence and divergence of the method is investigated.