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Singularity Subtraction for Nonlinear Weakly Singular Integral Equations of the Second Kind

  • Mario AhuesEmail author
  • Filomena D. d’Almeida
  • Rosário Fernandes
  • Paulo B. Vasconcelos
Chapter

Abstract

The singularity subtraction technique for computing an approximate solution of a linear weakly singular Fredholm integral equation of the second kind is generalized to the case of a nonlinear integral equation of the same kind. Convergence of the sequence of approximate solutions to the exact one is proved under mild standard hypotheses on the nonlinear factor, and on the sequence of quadrature rules used to build an approximate equation. A numerical example is provided with a Hammerstein operator to illustrate some practical aspects of effective computations.

Notes

Acknowledgements

The third author was partially supported by CMat (UID/MAT/00013/2013), and the second and fourth authors were partially supported by CMUP (UID/ MAT/ 00144/ 2013), which are funded by FCT (Portugal) with national funds (MCTES) and European structural funds (FEDER) under the partnership agreement PT2020.

References

  1. [AhEtAl01]
    Ahues, M.; Largillier, A. and Limaye B.: Spectral Computations for Bounded Operators, Chapman & Hall/CRC, Boca Raton, FL (2001).Google Scholar
  2. [An71]
    Anselone, P.: Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall, Englewoodcliffs, NJ (1971).zbMATHGoogle Scholar
  3. [An81]
    Anselone, P.: Singularity subtraction in the numerical solution of integral equations, J. Austral. Math. Soc. Ser. B, 22, 408–418 (1981).MathSciNetCrossRefGoogle Scholar
  4. [At92]
    Atkinson, K.: A survey of numerical methods for solving nonlinear integral equations, Journal of Integral Equations, 4, 1, 15–46 (1992).MathSciNetCrossRefGoogle Scholar
  5. [Sh08]
    Shampine, L. F.: Vectorized Adaptive Quadrature in MATLAB, J. Comput. Appl. Math., 211, 131–140 (2008).MathSciNetCrossRefGoogle Scholar
  6. [XiEtAl12]
    Xiang S. and Bornemann, F.: On the Convergence Rates of Gauss and Clenshaw–Curtis Quadrature for Functions of Limited Regularity, SIAM J. on Numer. Anal., 50, 5, 2581–2587 (2012).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Mario Ahues
    • 1
    Email author
  • Filomena D. d’Almeida
    • 2
  • Rosário Fernandes
    • 3
  • Paulo B. Vasconcelos
    • 2
  1. 1.Université de LyonLyonFrance
  2. 2.Universidade do PortoPortoPortugal
  3. 3.Universidade do MinhoBragaPortugal

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