Stability of Runge-Kutta methods for the generalized pantograph equation

This paper deals with stability properties of Runge-Kutta (RK) methods applied to a non-autonomous delay differential equation (DDE) with a constant delay which is obtained from the so-called generalized pantograph equation, an autonomous DDE with a variable delay by a change of the independent variable. It is shown that in the case where the RK matrix is regular stability properties of the RK method for the DDE are derived from those for a difference equation, which are examined by similar techniques to those in the case of autonomous DDEs with a constant delay. As a result, it is shown that some RK methods based on classical quadrature have a superior stability property with respect to the generalized pantograph equation. Stability of algebraically stable natural RK methods is also considered.