New methods for nonlinear BVPs on the half-axis using Runge-Kutta IVP-solvers

The scalar BVP du dx2 −mu = −f (x, u) , x ∈ (0,∞) , u (0) = μ1, lim x→∞ u (x) = 0, on the infinite interval [0,∞) is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme of which the solution coincides with the projection onto the grid of the exact solution of the corresponding differential equation. A constructive algorithm is proposed to derive from the EDS a so-called truncated difference scheme (TDS) of a given rank n̄ = 2[(n + 1)/2], provided that the righthand side possesses n continuous derivatives between a finite number of discontinuity points. Here [·] denotes the entire part of the expression in brackets. The n̄-TDS possesses the order of accuracy O(|h|) w.r.t. the maximal step size |h|. The n̄-TDS is represented by a system of nonlinear algebraic equations for the approximate values of the exact solution on the grid. Iterative methods for its numerical solution are discussed. Numerical examples are given which illustrate the theorems proved. ∗email:[email protected] †email:[email protected] ‡email:[email protected] §email:[email protected]