Optimal Superconvergence Results for Delay Integro‐Differential Equations of Pantograph Type

Hermann Brunner, Qiya Hu

Research output: Contribution to journalJournal articlepeer-review

28 Citations (Scopus)
53 Downloads (Pure)


We analyze the optimal (global and local) orders of superconvergence of collocation solutions Uh on uniform meshes Ih for delay Volterra integro-differential equations with proportional delay functions given by θ(t) = qt (0 < q < 1, t € [0, T]). In particular, we show that if Uh is a continuous piecewise polynomial of degree m > 2, and if collocation is at the Gauss ( Legendre) points, then the (optimal) order of local superconvergence on Ih is p* = m + 2. It turns out that the same order p* holds for nonlinear (strictly increasing) delay functions vanishing at t = 0. However, on judiciously chosen geometric meshes, collocation at the Gauss points yields the order 2m - εN, where εN → 0 as the number N of mesh points tends to infinity. Optimal local superconvergence results for the pantograph delay differential equation are obtained as special cases of our general analysis.

Original languageEnglish
Pages (from-to)986-1004
Number of pages19
JournalSIAM Journal on Numerical Analysis
Issue number3
Publication statusPublished - 4 May 2007

Scopus Subject Areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Collocation solutions
  • Optimal order of superconvergence
  • Pantograph equation
  • Proportional delays
  • Vanishing delays
  • Volterra integro-differential equation


Dive into the research topics of 'Optimal Superconvergence Results for Delay Integro‐Differential Equations of Pantograph Type'. Together they form a unique fingerprint.

Cite this