Solution of Nonlinear Two-Point Boundary Value Problems by Linear Programming

Abstract

A system of n nonlinear ordinary differential equations is considered on the interval [a, b] with at least one of the n boundary conditions specified at each end of the interval. In addition, any available a priori bounds on the solution vector may be imposed. An iterative method for solution is described which is essentially a Newton-Raphson method with a linear programming solution at each iteration. Every iterate is a minimax solution to a linearized finite difference approximation to the original system, and also satisfies the boundary conditions and the a priori bounds. The method will always converge at least as fast as Newton-Raphson, and may converge when Newton-Raphson fails. A number of computational examples are described.