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

  • Authors:

  • Yoshiaki Muroya;Emiko Ishiwata;Hermann Brunner

  • Affiliations:

  • Department of Mathematical Sciences, Waseda University, Tokyo 169-8555, Japan;Department of Mathematical Information Science, Tokyo University of Science, Tokyo 162-8601, Japan;Department of Statistics and Information Sciences, Newfoundland University, St. John's Newfoundland, A1C5S7, Canada

  • Venue:
  • Journal of Computational and Applied Mathematics - Proceedings of the international conference on recent advances in computational mathematics
  • Year:
  • 2003

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Abstract

For the pantograph integro-differential equation (PIDE) with nonhomogeneous term: y'(t) = ay(t)+ ∫01 y(σ(q)t)dµ(q)+ ∫01 y'(ρ(q)t)dv(q)+ f(t), t 0, y(0)= y0, with proportional delays σ(q)t and ρ(q)t, 0 (q), ρ(q) ≤ 1, 0 q 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 u(th), t ∈ [0, 1] such that |u(h)- y(h)| = O(h2m+1), where y(t) is the solution and u(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.