Interpolation and Integration of Initial Value Problems of Ordinary Differential Equations by Regular Splines

Abstract

We give the definition of regular splines, slightly improving the axioms stated earlier by R. Schaback. Considering splines that are twice continuously differentiable, we can prove that the problem of interpolation is solvable if the interpolation points are close enough together. The solution converges to the interpolated function with the fourth order of the maximal distance between adjacent points.The regular splines are then used to define an implicit scheme for the integration of initial value problems of ordinary differential equations, following Loscalzo–Talbot and Runge. Again fourth order convergence is established. The method may be particularly useful in treating solutions with movable singularities.