ON THE NUMERICAL-SOLUTION OF 2-POINT BOUNDARY-VALUE-PROBLEMS .2.

In a recent paper (L. Greengard and V. Rokhlin, On the Numerical Solut ion of Two-Point Boundary Value Problems, in Communications on Pure an d 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 pap er, we extend the method to systems of ordinary differential equations . After reducing the system of differential equations to a system of s econd kind integral equations, we discretize the latter via a high-ord er Nystrom scheme. A somewhat involved analytical apparatus is then co nstructed 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 t he system. Thus, the advantages of the integral equation formulation ( small condition number, insensitivity to boundary layers, insensitivit y to endpoint singularities, etc.) are retained, while achieving a com putational efficiency previously available only to finite difference o r finite element methods. We in addition present a Newton method for s olving boundary value problems for nonlinear first-order systems in wh ich each Newton iterate is the solution of a second kind integral equa tion; the analytical and numerical advantages of integral equations ar e thus obtained for nonlinear boundary value problems. (C) 1994 John W iley & Sons, Inc.