JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 154,529-542 (1991) Numerical Methods for Discontinuous Linear Boundary Value Problems with Deviation Arguments BARUCH CAHLON AND LOUIS J. NACHMAN Department qf Mathematical Sciences, Oaklund University, Rochester, Michigan 48309 Submitted by Stavros N. Busenberg Received April 27, 1987 Numerical methods for the solution of discontinuous linear boundary value problems with deviation arguments are developed. ‘7) 1991 Academic Press. Inc. We consider the linear boundary value problem of differential equations with-deviation arguments and integrable coefftcients. Relatively few papers on numerical methods for boundary value problems of differential equa- tions with deviation arguments exist. Shooting methods are used in [3] and iterative methods are used in [4, 5, 61. For other methods, see [7, 8, 9, lo]. Using K. E. Atkinson’s product integration technique and the methods developed in [ 1] we give practical methods for the solution of the problem. These methods converge rapidly and do not require large blocks of computer memory in their implementation. We also consider a boundary value problem of a differential equation with deviation argument with a regular singularity at 0, and a boundary value problem with singularity at the boundary points. Methods are developed for converting these problems into problems which can be solved numerically. Numerical examples are given. I. THE PROBLEM We develop a practical numerical method for solving the following boundary value problem with deviation argument (BVPD) Ly = y” + a,(x) y’ + q(x) y =fi(X)+.fZ(X)Y(x-~(x))+f3(X)Y’(x-~(X)), a<x<b (1) 529 0022-247X/91 $3.00 CopyrIght ( I 1991 by Academic Press. Inc All nghts of reproductxon ,n any form reserved