Skip to main content
SpringerLink
Log in

A fully discrete Galerkin method for Abel-type integral equations

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we present a Galerkin method for Abel-type integral equation with a general class of kernel. Stability and quasi-optimal convergence estimates are derived in fractional-order Sobolev norms. The fully-discrete Galerkin method is defined by employing simple tensor-Gauss quadrature. We develop a corresponding perturbation analysis which allows to keep the number of quadrature points small. Numerical experiments have been performed which illustrate the sharpness of the theoretical estimates and the sensitivity of the solution with respect to some parameters in the equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev spaces. In: Vol. 140 of Pure and Applied Mathematics (Amsterdam). 2nd edn. Elsevier/Academic Press, Amsterdam (2003)

  2. Bieniasz, L.K.: An adaptive Huber method with local error control for the numerical solution of the first kind Abel integral equations. Computing 83(1), 25–39 (2008)

    Article  MathSciNet  Google Scholar 

  3. Brezis, H., Mironescu, P.: Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces. J. Evol. Equ. 1(4), 387–404 (2001)

    Article  MathSciNet  Google Scholar 

  4. Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations, vol. 15 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  5. Brunner, H., Davies, P.J., Duncan, D.B.: Discontinuous Galerkin approximations for Volterra integral equations of the first kind. IMA J. Numer. Anal. 29(4), 856–881 (2009)

    Article  MathSciNet  Google Scholar 

  6. Cameron, R.F., McKee, S.: Product integration methods for second-kind Abel integral equations. J. Comput. Appl. Math. 11(1), 1–10 (1984)

    Article  MathSciNet  Google Scholar 

  7. Carstensen, C., Praetorius, D.: Averaging techniques for the effective numerical solution of symm’s integral equation of the first kind. SIAM J. Sci. Comput. 27(4), 1226–1260 (2006)

    Article  MathSciNet  Google Scholar 

  8. Dahmen, W., Faermann, B., Graham, I., Hackbusch, W., Sauter, S.A.: Inverse inequalities on non-qasiuniform meshes and applications to the mortar element method. Math. Comput. 73(247), 1107–1138 (2003)

    Article  Google Scholar 

  9. Davis, P.: Interpolation and Approximation. Blaisdell Publishing Co., New York (1963)

    MATH  Google Scholar 

  10. Eggermont, P.P.B.: A new analysis of the trapezoidal-discretization method for the numerical solution of abel-type integral equations. J. Integ. Equa. Appl. 3, 317–332 (1981)

    MathSciNet  MATH  Google Scholar 

  11. Eggermont, P.P.B.: Stability and robustness of collocation methods for Abel-type integral equations. Numer. Math. 45(3), 431–445 (1984)

    Article  MathSciNet  Google Scholar 

  12. Eggermont, P.P.B.: On Galerkin methods for Abel-type integral equations. SIAM J. Numer. Anal. 25(5), 1093–1117 (1988)

    Article  MathSciNet  Google Scholar 

  13. Gautschi, W.: Numerical Analysis, 2nd edn. Birkhäuser Boston, Inc, Boston (1997)

    MATH  Google Scholar 

  14. Gorenflo, R., Vessella, S.: Abel Integral Equations Vol. 1461 of Lecture Notes in Mathematics. Springer, Berlin (1991). Analysis and applications

    Google Scholar 

  15. Gorenflo, R., Yamamoto, M.: Operator-theoretic treatment of linear Abel integral equations of first kind. Jpn. J. Ind. Appl. Math. 16(1), 137–161 (1999)

    Article  MathSciNet  Google Scholar 

  16. Heimgartner, L.: The Efficient Solution of Abel-type Integral Equations by Hierarchial Matrix Techniques. Master’s thesis, Inst. f. Mathematik. Unversitȧt Zu̇rich. http://www.math.uzh.ch/compmath/index.php?id=dipl (2018)

  17. Plato, R.: The regularizing properties of the composite trapezoidal method for weakly singular Volterra integral equations of the first kind. Adv. Comput. Math. 36 (2), 331–351 (2012)

    Article  MathSciNet  Google Scholar 

  18. Sauter, S.A., Schwab, C.: Boundary Element Methods vol. 39 of Springer Series in Computational Mathematics. Springer, Berlin (2011)

    Google Scholar 

  19. Tot, T. (ed.): Breast Cancer: A Lobar Disease. Springer, London (2001)

  20. Trefethen, L.N.: Approximation Theory and Approximation Practice. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2013)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zürich, Switzerland

    Urs Vögeli & Stefan A. Sauter

  2. Institute for Advanced Studies in Basic Sciences, Zanjan, Iran

    Khadijeh Nedaiasl

Authors
  1. Urs Vögeli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Khadijeh Nedaiasl
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan A. Sauter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan A. Sauter.

Additional information

Communicated by: Leslie Greengard

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vögeli, U., Nedaiasl, K. & Sauter, S.A. A fully discrete Galerkin method for Abel-type integral equations. Adv Comput Math 44, 1601–1626 (2018). https://doi.org/10.1007/s10444-018-9598-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-018-9598-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation