Numerische Mathematik

, Volume 19, Issue 3, pp 248–259 | Cite as

The numerical solution of Fredholm integral equations of the second kind with singular kernels

  • Kendall Atkinson


A numerical method is given for integral equations with singular kernels. The method modifies the ideas of product integration contained in [3], and it is analyzed using the general schema of [1]. The emphasis is on equations which were not amenable to the method in [3]; in addition, the method tries to keep computer running time to a minimum, while maintaining an adequate order of convergence. The method is illustrated extensively with an integral equation reformulation of boundary value problems forΔuP(r2)u=0; see [9].


Integral Equation Mathematical Method General Schema Fredholm Integral Equation Product Integration 
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  1. 1.
    Anselone, P. M.: Convergence and error bounds for approximate solutions of integral and operator equations. In: Error in digital computation, vol. II, ed. by L. B. Rall, 231–252. New York: John Wiley and Sons 1965.Google Scholar
  2. 2.
    Anselone, P. M.: Collectively compact operator approximation theory. Prentice-Hall, 1971.Google Scholar
  3. 3.
    Atkinson, K.: Extensions of the Nystrom method for the numerical solution of linear integral equations of the second kind. Math. Research Center Technical Rep. #686, Univ. of Wisconsin, Madison, Wis. 1966.Google Scholar
  4. 4.
    Atkinson, K.: The numerical solution of Fredholm integral equations of the second kind. SIAM Num. Anal. Jour.4, 337–348 (1967).Google Scholar
  5. 5.
    Brakhage, H.: Über die numerische Behandlung von Integralgleichungen nach der Quadraturformelmethode. Numerische Math.2, 183–196 (1960).Google Scholar
  6. 6.
    Bückner, H.: Numerical methods for integral equations. In: Survey of numerical analysis, 439–467, ed. by John Todd. New York: McGraw-Hill 1962.Google Scholar
  7. 7.
    Dejon, B., Walther, A.: General report on the numerical treatment of integral and integro-differential equations. Symposium on the numerical treatment of ordinary differential equations, integral equations, and integro-differential equations, 647–671. Rome 1960.Google Scholar
  8. 8.
    Garwick, J. V.: On the numerical solution of integral equations. Den 11 te Skandinaviske Matimatiker Kongress, 1949, 113–121. Oslo: Johan Grundt Tanums Forlag 1952.Google Scholar
  9. 9.
    Gilbert, R. P.: The construction of solutions for boundary value problems by function theoretic methods. SIAM Jour. of Math. Anal.1, 96–114 (1970).Google Scholar
  10. 10.
    Gilbert, R. P.: Integral operator methods for approximating solutions of Dirichlet problems, to appear in the proceedings of the conference on “Numerische Methoden der Approximationstheorie”. Oberwolfach, Germany 1969. Appendix by K. Atkinson.Google Scholar
  11. 11.
    Hillstrom, K. E.: Comparison of several adaptive Newton-Cotes quadrature routines in evaluating definite integrals with peaked integrands. Argonne National Lab. Tech. Rep. ANL-7511, 1968.Google Scholar
  12. 12.
    Kadner, H.: Die numerische Behandlung von Integralgleichungen nach der Kollokationsmethode. Numerische Math.10, 241–260 (1967).Google Scholar
  13. 13.
    Kantorovich, L. V., Akilov, G. P.: Functional analysis in normal spaces. Transl. by D. E. Brown. New York: Pergamon Press 1964.Google Scholar
  14. 14.
    Kontorovich, L. V., Krylov, V. I.: Approximate methods of higher analysis. Transl. by C. D. Benster. New York: Interscience 1958.Google Scholar
  15. 15.
    Kussmaul, R.: Ein numerisches Verfahren zur Lösung des Neumannschen Außenraumproblems für die Helmholtzsche Schwingungsgleichung. Computing4, 246–273 (1969).Google Scholar
  16. 16.
    Kussmaul, R., Werner, P.: Fehlerabschätzung für ein numerisches Verfahren zur Auflösung linearer Integralgleichungen mit schwachsingulären Kernen. Computing3, 22–46 (1968).Google Scholar
  17. 17.
    Lynn, M. S., Timlake, W. P.: The numerical solution of singular integral equations of potential theory. Numerische Math.11, 77–98 (1968).Google Scholar
  18. 18.
    Taylor, A. E.: Introduction to functional analysis. New York: John Wiley and Sons 1958.Google Scholar
  19. 19.
    Young, A.: The application of approximate product integration to the numerical solution of integral equations. Proc. Roy. Soc. London (A)224, 561–573 (1954).Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • Kendall Atkinson
    • 1
  1. 1.Mathematics DepartmentIndiana UniversityBloomingtonUSA

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