Numerical approximations of first kind Volterra convolution equations with discontinuous kernels

Penny J. Davies, Dugald B. Duncan

Research output: Contribution to journalArticle

1 Citation (Scopus)
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Abstract

The cubic "convolution spline'" method for first kind Volterra convolution integral equations was introduced in [Convolution spline approximations of Volterra integral equations, J. Integral Equations Appl., 26:369--410, 2014]. Here we analyse its stability and convergence for a broad class of piecewise smooth kernel functions and show it is stable and fourth order accurate even when the kernel function is discontinuous. Key tools include a new discrete Gronwall inequality which provides a stability bound when there are jumps in the kernel function, and a new error bound obtained from a particular B-spline quasi-interpolant.
Original languageEnglish
Pages (from-to)41-73
Number of pages33
JournalJournal of Integral Equations and Applications
Volume29
Issue number1
DOIs
Publication statusPublished - 28 Feb 2017

Keywords

  • convolution spline
  • Volterra convolution integral equations
  • kernel functions
  • discontinuous kernal
  • time delay

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