A fully discrete Galerkin method for Abel-type integral equations


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 to check access.


  1. 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. 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)

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 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. 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)

    MathSciNet  Article  Google Scholar 

  6. 6.

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

    MathSciNet  Article  Google Scholar 

  7. 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)

    MathSciNet  Article  Google Scholar 

  8. 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. 9.

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

    Google Scholar 

  10. 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. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    Google Scholar 

  14. 14.

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

    Google Scholar 

  15. 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)

    MathSciNet  Article  Google Scholar 

  16. 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. 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)

    MathSciNet  Article  Google Scholar 

  18. 18.

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

    Google Scholar 

  19. 19.

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

  20. 20.

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

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Stefan A. Sauter.

Additional information

Communicated by: Leslie Greengard

Rights and permissions

Reprints and Permissions

About this article

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


  • Abel’s integral equation
  • Galerkin method
  • Tensor-Gauss quadrature

Mathematics Subject Classification (2000)

  • 45E10
  • 65R20
  • 65D32