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Calcolo

, 56:7 | Cite as

Analysis of collocation methods for nonlinear Volterra integral equations of the third kind

  • Huiming Song
  • Zhanwen YangEmail author
  • Hermann Brunner
Article
  • 28 Downloads

Abstract

We study the approximation of solutions of a class of nonlinear Volterra integral equations (VIEs) of the third kind by using collocation in certain piecewise polynomial spaces. If the underlying Volterra integral operator is not compact, the solvability of the collocation equations is generally guaranteed only if special (so-called modified graded) meshes are employed. It is then shown that for sufficiently regular data the collocation solutions converge to the analytical solution with the same optimal order as for VIEs with compact operators. Numerical examples are given to verify the theoretically predicted orders of convergence.

Keywords

Nonlinear Volterra integral equations of the third kind Noncompact Volterra integral operator Collocation methods Solvability of collocation equations Convergence order 

Mathematics Subject Classification

65R20 45G10 

Notes

Acknowledgements

This research was supported by the National Natural Science Foundation of China (NSFC 11771128 and NSFC 11771111) and the Hong Kong Research Grants Councial (GRF Grant HKBU 12300014). Part of this work was carried out during a visit of the first two authors to the Department of Mathematics at Hong Kong Baptist University. We also thank Prof. Xiao Yu for his support. The authors would like to express their gratitude to the reviewers: their valuable comments and suggestions led to a greatly improved version of the paper.

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Copyright information

© Istituto di Informatica e Telematica (IIT) 2019

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinPeople’s Republic of China
  2. 2.Department of MathematicsHong Kong Baptist UniversityKowloon TongChina
  3. 3.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada

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