On the attainable order of collocation methods for pantograph integro-differential equations

For the pantograph integro-differential equation (PIDE) with nonhomogeneous term: y′(t)=ay(t)+∫01y(σ(q)t) dμ(q)+∫01y′(ρ(q)t) dν(q)+f(t), t>0, y(0)=y0, with proportional delays σ(q)t and ρ(q)t, 0<σ(q), ρ(q)⩽1, 0<q⩽1, we consider the attainable order of m-stage implicit (collocation-based) Runge–Kutta methods at the first mesh point t=h, and give conditions on the collocation polynomials Mm(t) of degree m to v(th), t∈[0,1] such that |v(h)−y(h)|=O(h2m+1), where y(t) is the solution and v(t) is the collocation solution of PIDE. If m=2 or f(t) is a polynomial of t whose degree is less than or equal to m, then such conditions of Mm(t) are simplified. A numerical example is also included.