Table of Contents
ISRN Applied Mathematics
Volume 2011 (2011), Article ID 571583, 21 pages
http://dx.doi.org/10.5402/2011/571583
Research Article

An Algorithm for the Numerical Evaluation of Certain Finite Part Integrals

1Department of Mathematics, Faculty of Sciences, Cairo University, Giza, Egypt
2Department of Mathematics, Faculty of Education, King Abdulaziz University, Jeddah 2356, Saudi Arabia
3Department of Mathematics, Faculty of Industrial Education, Helwan University, Cairo, Egypt
4Department of Mathematics, Faculty of Sciences, Shaqra University, Shaqra, Saudi Arabia

Received 14 March 2011; Accepted 19 April 2011

Academic Editor: Z. Huang

Copyright © 2011 Samir A. Ashour and Hany M. Ahmed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. Criscuolo and G. Mastroianni, β€œFormule gaussiane per il calcolo di integrali a valor principale secondo Cauchy e loro convergenza,” Calcolo, vol. 22, no. 3, pp. 391–411, 1985. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  2. D. Elliott, β€œOn the convergence of Hunter's quadrature rule for Cauchy principal value integrals,” BIT, vol. 19, no. 4, pp. 457–462, 1979. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. D. Elliott and D. F. Paget, β€œGauss type quadrature rules for Cauchy principal value integrals,” Mathematics of Computation, vol. 33, no. 145, pp. 301–309, 1979. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. D. B. Hunter, β€œSome Gauss-type formulae for the evaluation of Cauchy principal values of integrals,” Numerische Mathematik, vol. 19, pp. 419–424, 1972. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. N. I. Ioakimidis, β€œFurther convergence results for two quadrature rules for Cauchy type principal value integrals,” Aplikace Matematiky, vol. 27, no. 6, pp. 457–466, 1982. View at Google Scholar Β· View at Zentralblatt MATH
  6. N. I. Ioakimidis, β€œOn the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives,” Mathematics of Computation, vol. 44, no. 169, pp. 191–198, 1985. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. G. Tsamasphyros and P. S. Theocaris, β€œOn the convergence of some quadrature rules for Cauchy principal-value and finite-part integrals,” Computing, vol. 31, no. 2, pp. 105–114, 1983. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  8. G. Criscuolo and G. Mastroianni, β€œSulla convergenza di alcune formule di quadratura per il calcolo di integrali a a valor principale secondo Cauchy,” L'Analyse Numérique et la Théorie de l'Approximation, vol. 14, no. 2, pp. 109–116, 1985. View at Google Scholar Β· View at Zentralblatt MATH
  9. D. Elliott and D. F. Paget, β€œOn the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals,” Numerische Mathematik, vol. 23, pp. 311–319, 1975. View at Google Scholar Β· View at Zentralblatt MATH
  10. D. Elliott and D. F. Paget, β€œOn the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals : an addendum,” Numerische Mathematik, vol. 25, no. 3, pp. 287–289, 1976. View at Publisher Β· View at Google Scholar
  11. D. F. Paget and D. Elliott, β€œAn algorithm for the numerical evaluation of certain Cauchy principal value integrals,” Numerische Mathematik, vol. 19, pp. 373–385, 1972. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. P. Rabinowitz, β€œOn an interpolatory product rule for evaluating Cauchy principal value integrals,” BIT, vol. 29, no. 2, pp. 347–355, 1989. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  13. M. A. Sheshko, β€œOn the convergence of quadrature processes for a singular integral,” Izvestija Vysših Učebnyh Zavedeniĭ Matematika, vol. 20, pp. 108–118, 1976. View at Google Scholar
  14. K. Diethelm, β€œGaussian quadrature formulae of the third kind for Cauchy principal value integrals: basic properties and error estimates,” Journal of Computational and Applied Mathematics, vol. 65, no. 1–3, pp. 97–114, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  15. G. Monegato, β€œOn the weights of certain quadratures for the numerical evaluation of Cauchy principal value integrals,” Computing, vol. 29, pp. 337–354, 1982. View at Google Scholar
  16. C. W. Clenshaw, β€œA note on the summation of Chebyshev series,” Mathematical Tables and Other Aids to Computation, vol. 9, pp. 118–120, 1955. View at Google Scholar Β· View at Zentralblatt MATH
  17. C. Y. Hui and D. Shia, β€œEvaluations of hypersingular integrals using Gaussian quadrature,” International Journal for Numerical Methods in Engineering, vol. 44, no. 2, pp. 205–214, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH