Numerical investigation of the pantograph equation Original Research Article

This paper presents some numerical examples concerning the pantograph equation y′(t) = ay(t) + by(qt) for different values of the parameters a, b, q, satisfying the conditions |a| + b < 0, 0 < 1 − q ⪡ 1. “Naive” interpretation of these examples could lead to wrong conclusion on the asymptotic behaviour of the exact solutions. Using a perturbation method and a recent result of Kuruklis, we analyze a simple numerical discretization of the pantograph equation. The main result of this paper is that in order to see the correct asymptotic behaviour of the exact solution our numerical calculation has to go far beyond a certain critical point t∗, which depends on the parameters a, b and is inversely proportional to 1 − q.