Rational approximation method for delay differential equations with proportional delay

Abstract

Consider the pantograph differential equation y′( t) = ay( t) + by( qt) + f( t), 0 ⩽ t ⩽ T, y(0) = y 0, with proportional delay qt, 0 < q < 1. In this paper, we propose a piecewise (2 m, m)-rational approximation with “quasi-uniform meshes” which corresponds to mth collocation method, and establish the global error analysis of O( h 2 m ) on successive mesh points. This method is useful than the known collocation method in case that a long time integration is needed, that is, if T is large, then the number of steps in our method is less than that of the collocation method. Numerical experiments are also contained.