Solving Spline Collocation Approximations to Nonlinear Two-point Boundary Value Problems by a Homotopy Method

Abstract

The Chow-Yorke algorithm is a homotopy method that has been proved globally convergent for Brouwer fixed point problems, certain classes of zero linding and nonlinear programming problems, and two-point boundary value approximations based on shooting and finite differences. The method is numerically stable and has been successfully applied to a wide range of practical engineering problems. Here the Chow-Yorke algorithm is proved globally convergent for a class of spline collocation approxlmetions to nonlinear two-point boundary value problems. Several numerical implementations of the algorithm are briefly described. and computational results are presented for a fairly difficult hid dynamics boundary value problem.