On the numerical solution of two‐point boundary value problems II

In a recent paper (L. Greengard and V. Rokhlin, On the Numerical Solution of Two‐Point Boundary Value Problems. in Communications on Pure and Applied Mathematics, Volume XLIV, 1991, pages 419‐452). L. Greengard and V. Rokhlin introduce a numerical technique for the rapid 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 Nyström scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O(N.p².n³) 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 endpoint singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or 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. © 1994 John Wiley & Sons. Inc.