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Journal of Scientific Computing

, Volume 81, Issue 3, pp 2162–2187 | Cite as

A Müntz-Collocation Spectral Method for Weakly Singular Volterra Integral Equations

  • Dianming Hou
  • Yumin Lin
  • Mejdi Azaiez
  • Chuanju XuEmail author
Article

Abstract

In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel \((x-s)^{-\mu },0<\mu <1\). First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both \(L^{\infty }\)- and weighted \(L^{2}\)-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change \(x\rightarrow x^{1/\lambda }\) for a suitable real number \(\lambda \). Finally a series of numerical examples are presented to demonstrate the efficiency of the method.

Keywords

Müntz-collocation spectral method Volterra integral equations Weakly singular Exponential convergence 

Mathematics Subject Classification

65N35 65M70 45D05 45Exx 41A10 41A25 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Dianming Hou
    • 1
    • 2
  • Yumin Lin
    • 2
  • Mejdi Azaiez
    • 2
    • 3
  • Chuanju Xu
    • 2
    • 3
    Email author
  1. 1.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouChina
  2. 2.School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific ComputingXiamen UniversityXiamenChina
  3. 3.Laboratoire I2M UMR 5295Bordeaux INPPessacFrance

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