Multilevel correction for collocation solutions of Volterra integral equations with proportional delays

  • Authors:

  • Junmin Xiao;Qiya Hu

  • Affiliations:

  • LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematical and System Sciences, Chinese Academy of Sciences, Beijing, China 100190;LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematical and System Sciences, Chinese Academy of Sciences, Beijing, China 100190

  • Venue:
  • Advances in Computational Mathematics
  • Year:
  • 2013

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Abstract

In this paper, we propose a convergence acceleration method for collocation solutions of the linear second-kind Volterra integral equations with proportional delay qt$(0. This convergence acceleration method called multilevel correction method is based on a kind of hybrid mesh, which can be viewed as a combination between the geometric meshes and the uniform meshes. It will be shown that, when the collocation solutions are continuous piecewise polynomials whose degrees are less than or equal to ${m} (m \leqslant 2)$, the global accuracy of k level corrected approximation is $O(N^{-(2m(k+1)-\varepsilon)})$, where N is the number of the nodes, and $\varepsilon$ is an arbitrary small positive number.