**Preprint series:** 05-15, Reports on Numerical Mathematics

I. P. Gavrilyuk, M. Hermann, M. V. and Makarov Kutniv

**MSC:**- 65L10 Boundary value problems
- 65L12 Finite difference methods
- 65L20 Stability and convergence of numerical methods
- 65L50 Mesh generation and refinement
- 34B15 Nonlinear boundary value problems
- 34L30 Nonlinear ordinary differential operators

**Abstract:** Two-point boundary value problems for a system of nonlinear first order 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 possesses a prescribed order of accuracy O(|h|^m) w.r.t. 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 a m-TDS are discussed. Examples are given which illustrate the theorems proved in this paper.

**Keywords:** *systems of nonlinear ordinary differential equations, nonlinear boundary-value problem, two-point difference scheme, exact difference scheme, truncated two-point difference scheme of an arbitrary given accuracy order, and fixed point iteration*

**Upload:** 2005-11-10

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