Collocation methods for third-kind VIEs

Sonia Seyed Allaei*, Zhan Wen Yang, Hermann BRUNNER

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)

Abstract

The spline collocation method in piecewise polynomial spaces is applied to a class of third-kind Volterra integral equations (VIEs). Under certain conditions the operator associated with the equivalent secondkind VIE is compact, and this guarantees that the resulting algebraical system is uniquely solvable for all sufficiently small mesh diameters. However, the solvability of this system is not ensured, both on uniform and classical graded meshes, when the operator is noncompact (which typically is the case). Hence, we introduce a modified graded mesh to overcome the solvability problem. For such meshes we establish results on the optimal order of global convergence of the collocation solutions for third-kind VIEs. Numerical tests confirm the validity of these results.

Original languageEnglish
Pages (from-to)1104-1124
Number of pages21
JournalIMA Journal of Numerical Analysis
Volume37
Issue number3
DOIs
Publication statusPublished - 1 Jul 2017

Scopus Subject Areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • collocation methods
  • cordial Volterra integral equations
  • global convergence order
  • noncompact integral operators
  • Volterra integral equations of the third kind
  • weakly singular kernel

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