Solution of a nonlinear boundary value problem for a second-order ordinary differential equation by the method of finite differences

Abstract

THE solution of the boundary value problem (1.1) will be considered. This problem is approximated by the difference boundary value problem (1.2), represented by a nonlinear set of equations. A generalized Newton's method is described for solution of this latter problem (3.6). Convergence conditions for the method (verified in practice) are given in Theorem 3. When examining the convergence of the Newton's method we use a priori estimates for the solution of the linear difference boundary value problem, which are given in Theorem 1. Note that all the numerical quantities are assumed real.