1 Week 02 : 27 January 2009 2 Systems in one dimension

Let u : [a, b] → R s be the solution of the well-posed first-order linear two-point boundary value problem Lu := u ′ + Qu = f a < x < b with homogeneous boundary conditions Au(a) − Bu(b) = 0, Thus u is a vector function from [a, b] to s-dimensional Euclidean space R s , and Q is an n × n matrix function. (A necessary but not sufficient condition for well-posedness is that the s × 2s partitioned matrix C = [A|B] has rank s. The example of computing periodic indefinite integrals where Q = 0, A = B = I, shows that the rank condition is not sufficient in general.) We begin by converting the equation to a " dual " form. There are several ways to do this, leading to different formulations and numerical methods. Assume the right-hand side f is in the space L 1 of absolutely integrable functions on [a, b]: