TABLE OF CONTENTS This chapter is different from the previous ones in that it deals with ODE for which boundary conditions, rather than initial conditions, are given. In 1 we survey the class of problems that will be discussed. Then in 2 we treat Fredholm s alternative for existence of solutions and also introduce Green s functions. The latter play an important role in 3 where we investigate the conditioning of boundary value problems (BVP); it is shown how an analogue of stability for IVP is given by the notion of dichotomy. A simple (but often naive) approach for solving BVP is guessing the missing part of the initial condition, solving the corresponding IVP and iteratively refining the solution through the boundary condition; this so-called shooting technique is considered in 4 for the linear case and in 5 for the nonlinear case. Single shooting may both lack sufficient numerical stability and suffer from convergence problems (in the nonlinear case for Newton s method). Therefore we consider an important improvement, multiple shooting, in 6.
Although the emphasis of this book is on initial value problems, it is useful to pay some attention to boundary value problems (BVP) as well. There are quite a lot of practical ODE problems that arise as BVP. As it will turn out there are some essential distinctions between BVP and IVP, the most important one being the fact that the former are not related to something like an evolution of a phenomenon; consequently space rather than time should be...
