Skip to main content
SpringerLink
Log in

Singular integral equations — The convergence of the Nyström interpolant of the Gauss-Chebyshev method

  • Part II Numerical Mathematics
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

Nyström's interpolation formula is applied to the numerical solution of singular integral equations. For the Gauss-Chebyshev method, it is shown that this approximation converges uniformly, provided that the kernel and the input functions possess a continuous derivative. Moreover, the error of the Nyström interpolant is bounded from above by the Gaussian quadrature errors and thus convergence is fast, especially for smooth functions. ForC ∞ input functions, a sharp upper bound for the error is obtained. Finally numerical examples are considered. It is found that the actual computational error agrees well with the theoretical derived bounds.

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. K. Atkinson,A survey of Numerical Methods for the solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, 1976.

    Google Scholar 

  2. F. Erdogan and G. D. Gupta,On the numerical solution of singular integral equations, Quart. Appl. Math., 30 (1972), 525–534.

    Google Scholar 

  3. Davis and Rabinowitz,Numerical Integration, Blaisdell Publ. Company, 1967.

  4. A. Gerasoulis,On the existence of approximate solutions for singular integral equations of Cauchy type discretized by Gauss-Chebyshev quadrature formulae, BIT, 21 (1981), No. 3.

  5. N. Ioakimidis,On the natural interpolation formula for Cauchy type singular integral equations of the first kind, Computing, 26 (1981), 73–77.

    Google Scholar 

  6. N. Ioakimidis and P. S. Theocaris,On convergence of two direct methods for solution of Cauchy type singular integral equations of the first kind, BIT, 20 (1980), 83–87.

    Google Scholar 

  7. S. Krenk,On the use of the interpolation polynomial for solutions of singular integral equations, Q. Applied Math., 32 (1975), 479–484.

    Google Scholar 

  8. N. Muskhelishvili,Singular Integral Equations, P. Noordhoff, Groningen, 1953.

    Google Scholar 

  9. G. Szegö,Orthogonal Polynomials, Fourth Ed., Amer. Math. Soc., Providence, R. I., 1975.

  10. P. S. Theocaris and N. I. Ioakimidis,Numerical integration methods for the solution of singular integral equations, Q. Appl. Math., 35 (1977), 173–182.

    Google Scholar 

  11. P.S. Theocaris and N. I. Ioakimidis,A remark on the numerical solution of singular integral equations and the determination of stress-intensity factors, J. Eng. Math., 18 (1979), 213–222.

    Google Scholar 

  12. G. Tsamasphyros and P. S. Theocaris,Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations, Computing, 27 (1981), 71–80.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Rutgers University, 08903, New Brunswick, N.J., U.S.A.

    Apostolos Gerasoulis

Authors
  1. Apostolos Gerasoulis
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research has been partially supported by a grant from the Rutgers Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gerasoulis, A. Singular integral equations — The convergence of the Nyström interpolant of the Gauss-Chebyshev method. BIT 22, 200–210 (1982). https://doi.org/10.1007/BF01944477

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01944477

Keywords

Search

Navigation