Advances in Computational Mathematics

, Volume 44, Issue 5, pp 1601–1626 | Cite as

A fully discrete Galerkin method for Abel-type integral equations

  • Urs Vögeli
  • Khadijeh Nedaiasl
  • Stefan A. SauterEmail author


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.


Abel’s integral equation Galerkin method Tensor-Gauss quadrature 

Mathematics Subject Classification (2000)

45E10 65R20 65D32 


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  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)Google Scholar
  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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brezis, H., Mironescu, P.: Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces. J. Evol. Equ. 1(4), 387–404 (2001)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)CrossRefGoogle Scholar
  9. 9.
    Davis, P.: Interpolation and Approximation. Blaisdell Publishing Co., New York (1963)zbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Eggermont, P.P.B.: On Galerkin methods for Abel-type integral equations. SIAM J. Numer. Anal. 25(5), 1093–1117 (1988)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gautschi, W.: Numerical Analysis, 2nd edn. Birkhäuser Boston, Inc, Boston (1997)zbMATHGoogle Scholar
  14. 14.
    Gorenflo, R., Vessella, S.: Abel Integral Equations Vol. 1461 of Lecture Notes in Mathematics. Springer, Berlin (1991). Analysis and applicationsGoogle 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)MathSciNetCrossRefGoogle 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. (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)MathSciNetCrossRefGoogle 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)Google Scholar
  20. 20.
    Trefethen, L.N.: Approximation Theory and Approximation Practice. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2013)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZürichSwitzerland
  2. 2.Institute for Advanced Studies in Basic SciencesZanjanIran

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