Numerical Solution of Nonlinear Two-Point Boundary Problems by Finite Differences Using Newton'S Method

摘要

One method of solving nonlinear two-point boundary problems is to replace the nonlinear differential equations with implicit finite difference equations, linearize the nonlinear simultaneous algebraic equations, and solve them by an iterative process using Newton's well-known rule. This report describes this method in complete detail, gives a set of finite-difference equations for a general class of nonlinear two-point boundary problems, discusses the storage and solution of the band matrix, and suggests techniques for overcoming the difficulties of obtaining good initial guesses for the iterative procedure. Two examples are given to clarify the use of the method. The first example has multiple solutions, and both examples have boundary conditions specified at infinity. (Author)