Difference schemes for nonlinear BVPs on the half-axis

The scalar boundary value problem (BVP) du dx2 −mu = −f (x, u) , x ∈ (0,∞) , u (0) = μ1, lim x→∞ u (x) = 0, for a second order differential equation 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 of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number and [·] denotes the entire part of the expression in brackets. The n-TDS has the order of accuracy n̄ = 2[(n+1)/2], i.e., the global error is of the form O(|h|), where |h| is the maximum step size. The n-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. The theoretical and practical results are used to develop a new algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm. COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol. ? (200?), No. ?, pp. 1–3