Rational approximation method for delay differential equations with proportional delay

Consider the pantograph differential equation y′(t) = ay(t) + by(qt) + f(t), 0 ⩽ t ⩽ T, y(0) = y0, with proportional delay qt, 0 < q < 1. In this paper, we propose a piecewise (2m, m)-rational approximation with “quasi-uniform meshes” which corresponds to mth collocation method, and establish the global error analysis of O(h2m) 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.