Advertisement

BIT Numerical Mathematics

, Volume 15, Issue 2, pp 136–143 | Cite as

On the approximate solution of the abel integral equation with discontinuous solution

  • H. Brunner
Article

Abstract

The classical Abel integral equation with discontinuous solution is solved numerically by replacing the inhomogeneous termg(x) (known on some finite subsetZ N of the interval of integration) by a suitable linear familyψ(β, x) of continuous functions. The choice of these functions will be governed by two criteria: they are to reflect the discontinuous behavior of the exact solution of the integral equation, and they are to be such that the problem of finding a best (discrete) Chebyshev approximation tog onZ N possesses a unique solution.

Keywords

Continuous Function Integral Equation Exact Solution Unique Solution Approximate Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Bôcher,An Introduction to the Study of Integral Equations, Cambridge Tracts in Mathematical Physics, No, 10 (2nd ed.), Cambridge University Press, 1913.Google Scholar
  2. 2.
    J. Burlak,A further note on certain integral equations of Abel type, Proc. Edinburgh Math. Soc. (Ser. 2), 14 (1965), 255–256.Google Scholar
  3. 3.
    E. W. Cheney,Introduction to Approximation Theory, McGraw-Hill, New York, 1966.Google Scholar
  4. 4.
    H. E. Fettis,On the numerical solution of the Abel equation, Math. Comp., 18 (1964), 491–496.Google Scholar
  5. 5.
    J. Friedrich, Bemerkung zur Abelschen Integralgleichung, Z. Angew. Math. Phys., 11 (1960), 191–197.Google Scholar
  6. 6.
    E. Goursat,Cours d'analyse mathématique, Tome III (3ème éd.), Gauthier-Villars, Paris, 1923.Google Scholar
  7. 7.
    E. Isaacson and H. B. Keller,Analysis of Numerical Methods, Wiley, New York, 1966.Google Scholar
  8. 8.
    P. Linz,Numerical Methods for Volterra Integral Equations, Dissertation (Part B), University of Wisconsin, Madison, 1968.Google Scholar
  9. 9.
    R. Piessens and P. Verbaeten,Numerical solution of the Abel integral equation, BIT, 13 (1973), 451–457.Google Scholar
  10. 10.
    G. Pólya and G. Szegö,Aufgaben und Lehrsätze aus der Analysis, Band II, Springer, Berlin, 1925.Google Scholar
  11. 11.
    R. Rothe, Zur Abelschen Integralgleichung, Math. Z., 33 (1931), 375–387.Google Scholar
  12. 12.
    W. Schmeidler,Integralgleichungen mit Anwendungen in Physik und Technik, Akad. Verlagsgesellschaft Geest & Portig, Leipzig, 1950.Google Scholar
  13. 13.
    G. Y. Shilov,Mathematical Analysis, Pergamon Press, Oxford, 1965.Google Scholar
  14. 14.
    I. H. Sneddon,The Use of Integral Transforms, McGraw-Hill, New York, 1972.Google Scholar
  15. 15.
    H. Werner and R. Schaback,Praktische Mathematik II:Methoden der Analysis, Springer-Verlag, Berlin, 1972.Google Scholar

Copyright information

© BIT Foundations 1975

Authors and Affiliations

  • H. Brunner
    • 1
  1. 1.Department of MathematicsDalhousie UniversityHalifaxCanada

Personalised recommendations