Chebyshev collocation for linear, periodic ordinary and delay differential equations: a posteriori estimates

Abstract

We present a Chebyshev collocation method for linear ODE and DDE problems. We first give a posteriori estimates for the accuracy of the approximate solution of a scalar ODE initial value problem. Examples of the success of the estimate are given. For linear, periodic DDEs with integer delays we define and discuss the monodromy operator U as our main goal is reliable estimation of the stability of such DDEs. We prove a theorem which gives a posteriori estimates for eigenvalues of U, our main result. This result is based on a generalization to operators on Hilbert spaces of the Bauer-Fike theorem for (matrix) eigenvalue perturbation problems. We generalize these results to systems of DDEs. A delayed, damped Mathieu equation example is given. The computation of good bounds on ODE fundamental solutions is an important technical issue; an a posteriori method for such bounds is given. Certain technical issues are also addressed, namely the evaluation of polynomials and the estimation of L^\infty norms of analytic functions. Generalization to the non-integer delays case is also considered.