Advances in Difference Equations 
Volume 2006 (2006), Article ID 12167, 29 pages
doi:10.1155/ADE/2006/12167

Difference schemes for nonlinear BVPs using Runge-Kutta IVP-solvers

I. P. Gavrilyuk,1 M. Hermann,2 M. V. Kutniv,3 and V. L. Makarov4

 1Berufsakademie Thüringen, Staatliche Studienakademie, Am Wartenberg 2, Eisenach 99817, Germany
2Institute of Applied Mathematics, Friedrich Schiller University, Ernst-Abbe-Platz 1-4, Jena 07740, Germany
3Lviv Polytechnic National University, 12 St. Bandery Street, Lviv 79013, Ukraine
4Department of Numerical Analysis, Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka Street, Kyiv-4 01601, Ukraine
Received 11 November 2005; Revised 1 March 2006; Accepted 2 March 2006

Abstract

Difference schemes for two-point boundary value problems for systems of first-order nonlinear ordinary differential equations are considered. It was shown in former papers of the authors that starting from the two-point exact difference scheme (EDS) one can derive a so-called truncated difference scheme (TDS) which a priori possesses an arbitrary given order of accuracy 𝒪(|h|m) with respect to the maximal step size |h|. This m-TDS represents a system of nonlinear algebraic equations for the approximate values of the exact solution on the grid. In the present paper, new efficient methods for the implementation of an m-TDS are discussed. Examples are given which illustrate the theorems proved in this paper.