Numerical Solution of Two-Point Boundary Value Problems II

摘要

In a recent paper Greengard and Rokhlin introduce a numerical technique for therapid solution of integral equations resulting from linear two-point boundary value problems for second order ordinary differential equations. In this paper, we extend the method to systems of ordinary differential equations. After reducing the system of differential equations to a system of second kind integral equations, we discretize the latter via a high order Nystrom scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O(N . p squared . n cubed) operations, with N the number of nodes on the interval, p the desired order of convergence, and n the number of equations in the system. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to end-point singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference of finite element methods. We in addition present a Newton method for solving boundary value problems for nonlinear first order systems in which each Newton iterate is the solution of a second kind integral equation; the analytical and numerical advantages of integral equations are thus obtained for nonlinear boundary value problems. (kr)