Computational Mechanics

, Volume 7, Issue 1, pp 21–29 | Cite as

Numerical solution of Cauchy type singular integral equations with logarithmic weight, based on arbitrary collocation points

  • A. C. Chrysakis
  • G. Tsamasphyros
Article

Abstract

Singular integral equations with a Cauchy type kernel and a logarithmic weight function can be solved numerically by integrating them by a Gauss-type quadrature rule and, further, by reducing the resulting equation to a linear system by applying this equation at an appropriate number of collocation points xk. Until now these xk have been chosen as roots of special functions. In this paper, an appropriate modification of the original method permits the arbitrary choice of xk without any loss in the accuracy. The performance of the method is examined by applying it to a numerical example and a plane crack problem.

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References

  1. Danloy, B. (1973): Numerical construction of Gaussian quadrature formulas for 29-1. Math. Comput. 27, 861–869Google Scholar
  2. Erdogan, F.; Gupta, G. D.; Cook, T. S. (1973): Numerical solution of singular integral equations. In: Sih, G. C. (ed): Mechanics of fracture, vol. 1, Leyden: NoordhoffGoogle Scholar
  3. Erdogan, F.; Biricikoglu, V. (1973): Two bonded half planes with a crack going through the interface. Int. J. Eng. Sci. 11, 745–766Google Scholar
  4. Erdogan, F.; Cook, T. S. (1974): Antiplane shear crack terminating at and going through a bimaterial interface. Int. J. Fract. 10, 227–240Google Scholar
  5. Junghanns, P.; Silbermann, B. (1981): Zur Theorie der Näherung verfahren für singuläre Integralgleichungen auf Intervallen. Math. Nachr. 103, 199–244Google Scholar
  6. Ioakimidis, N. I. (1976): General methods for the solution of crack problems in the theory of plane elasticity. Doctoral Thesis at the National Technical University of Athens, Greece. Univ. Microfilms order no. 76-21, 056Google Scholar
  7. Ioakimidis, N. I. (1980): The numerical solution of crack problems in plane elasticity in the case of loading discontinuities. Eng. Fract. Mech. 13, 709–716Google Scholar
  8. Ioakimidis, N. I. (1981): On the natural interpolation formula for Cauchy type integral equations of the first kind. Computing 26, 73–77Google Scholar
  9. Ioakimidis, N. I. (1984): Application of interpolation formulas to the numerical solution of singular integral equations. Serdica 10, 78–87Google Scholar
  10. Ioakimidis, N. I.; Theocaris, P. S. (1980): On the selection of collocation points for the numerical solution of singular integral equations with generalized kernels appearing in elasticity problems. Comput. Struct. 11, 289–295Google Scholar
  11. Kulich, N. V. (1974): The computation of certain Cauchy-type integrals and singular integrals with logarithmic singularities. Versi Akademii Navuk BSSR-Seriya Fizika-Matematychnykh 1, 90–94Google Scholar
  12. Muskhelishvili, N. I. (1975): Some basic problems of the mathematical theory of elasticity. Leyden: NoordhoffGoogle Scholar
  13. Stroud, A. H.; Secrest D. (1966): Gaussian quadrature formulas. Englewood, Cliffs, NJ: Prentice-HallGoogle Scholar
  14. Theocaris, P. S.; Chrysakis, A. C.; Ioakimidis, N. I. (1979): Cauchy-type integrals and integral equations with logarithmic singularities. J. Eng. Math. 13, 63–74Google Scholar
  15. Tsamasphyros, G.; Theocaris, P. S. (1976): Sur une méthode génerale de quadrature des integrales du type Cauchy. Balkan Conference of Applied Mathematics in Salonica.Google Scholar
  16. Tsamasphyros, G.; Theocaris, P. S. (1979): Numerical solution of systems of singular equations with variable coefficients. Appl. Anal. 7, 37–52Google Scholar
  17. Tsamasphyros, G.; Theocaris, P. S. (1981a); Are special collocation points necessary for the numerical solution of singular integral equations? Int. J. Fract. 17, R21-R24.Google Scholar
  18. Tsamasphyros, G.; Theocaris, P. S. (1981b): Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations. Computing 27, 71–80Google Scholar
  19. Tsamasphyros, G. (1986): A study of factors influencing the solution of singular integral equations. Eng. Fract. Mech. 24, 567–578Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • A. C. Chrysakis
    • 1
  • G. Tsamasphyros
    • 1
  1. 1.Section of Mechanics, Department of Engineering ScienceThe National Technical University of AthensAthensGreece

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