DOI QR코드

DOI QR Code

EVALUATION OF SINGULAR INTEGRALS BY HYPERBOLIC TANGENT BASED TRANSFORMATIONS

  • Yun, Beong-In (DEPARTMENT OF INFORMATICS AND STATISTICS KUNSAN NATIONAL UNIVERSITY)
  • Received : 2009.04.30
  • Published : 2011.01.01

Abstract

We employ a hyperbolic tangent function to construct nonlinear transformations which are useful in numerical evaluation of weakly singular integrals and Cauchy principal value integrals. Results of numerical implementation based on the standard Gauss quadrature rule show that the present transformations are available for the singular integrals and, in some cases, give much better approximations compared with those of existing non-linear transformation methods.

Keywords

hyperbolic tangent;weakly singular integral;Cauchy principal value integral;non-linear transformation

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. M. Cerrolaza and E. Alarcon, A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary methods, Internat. J. Numer. Methods Engrg. 28 (1989), no. 5, 987-999. https://doi.org/10.1002/nme.1620280502
  2. M. Doblare and L. Gracia, On non-linear transformations for the integration of weakly-singular and Cauchy principal value integrals, Internat. J. Numer. Methods Engrg. 40 (1997), no. 18, 3325-3358. https://doi.org/10.1002/(SICI)1097-0207(19970930)40:18<3325::AID-NME215>3.0.CO;2-Q
  3. D. Elliott, The cruciform crack problem and sigmoidal transformations, Math. Methods Appl. Sci. 20 (1997), no. 2, 121-132. https://doi.org/10.1002/(SICI)1099-1476(19970125)20:2<121::AID-MMA840>3.0.CO;2-7
  4. D. Elliott, Sigmoidal transformations and the trapezoidal rule, J. Austral. Math. Soc. Ser. B 40 (1998/99), (E), E77-E137.
  5. D. Elliott, The Euler-Maclaurin formula revisited, J. Austral. Math. Soc. Ser. B 40 (1998/99), (E), E27-E76.
  6. D. Elliott and E. Venturino, Sigmoidal transformations and the Euler-Maclaurin expansion for evaluating certain Hadamard finite-part integrals, Numer. Math. 77 (1997), no. 4, 453-465. https://doi.org/10.1007/s002110050295
  7. P. R. Johnston, Application of sigmoidal transformations to weakly singular and near-singular boundary element integrals, Internat. J. Numer. Methods Engrg. 45 (1999), no. 10, 1333-1348. https://doi.org/10.1002/(SICI)1097-0207(19990810)45:10<1333::AID-NME632>3.0.CO;2-Q
  8. P. R. Johnston, Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals, Internat. J. Numer. Methods Engrg. 47 (2000), no. 10, 1709-1730. https://doi.org/10.1002/(SICI)1097-0207(20000410)47:10<1709::AID-NME852>3.0.CO;2-V
  9. P. R. Johnston and D. Elliott, Error estimation of quadrature rules for evaluating sin-gular integrals in boundary element problems, Internat. J. Numer. Methods Engrg. 48 (2000), no. 7, 949-962. https://doi.org/10.1002/(SICI)1097-0207(20000710)48:7<949::AID-NME905>3.0.CO;2-Q
  10. P. R. Johnston and D. Elliott, A generalisation of Telles' method for evaluating weakly singular boundary element integrals, J. Comput. Appl. Math. 131 (2001), no. 1-2, 223-241. https://doi.org/10.1016/S0377-0427(00)00273-9
  11. N. M. Korobov, Number-Theoretic Methods in Approximate Analysis, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963.
  12. G. Monegato and I. H. Sloan, Numerical solution of the generalized airfoil equation for an airfoil with a flap, SIAM J. Numer. Anal. 34 (1997), no. 6, 2288-2305. https://doi.org/10.1137/S0036142995295054
  13. S. Prossdorf and A. Rathsfeld, On an integral equation of the first kind arising from a cruciform crack problem, Integral equations and inverse problems (Varna, 1989), 210-219, Pitman Res. Notes Math. Ser., 235, Longman Sci. Tech., Harlow, 1991.
  14. T. W. Sag and G. Szekeres, Numerical evaluation of high-dimensional integrals, Math. Comput. 18 (1964), 245-253. https://doi.org/10.1090/S0025-5718-1964-0165689-X
  15. J. Sanz Serna, M. Doblare, and E. Alarcon, Remarks on methods for the computation of boundary-element integrals by co-ordinate transformation, Comm. Appl. Numer. Methods 6 (1990), no. 2, 121-123. https://doi.org/10.1002/cnm.1630060208
  16. M. Sato, S. Yoshiyoka, K. Tsukui, and R. Yuuki, Accurate numerical integration of singular kernels in the two-dimensional boundary element method, in: C.A. Brebbia(Eds.), Boundary Elements X Vol.1, Springer, Berlin, 1988, pp. 279-296.
  17. A. Sidi, A new variable transformation for numerical integration, Numerical integration, IV (Oberwolfach, 1992), 359-373, Internat. Ser. Numer. Math., 112, Birkhauser, Basel, 1993.
  18. K. M. Singh and M. Tanaka, On non-linear transformations for accurate numerical evaluation of weakly singular boundary integrals, Internat. J. Numer. Methods Engrg. 50 (2001), no. 8, 2007-2030. https://doi.org/10.1002/nme.117
  19. J. C. F. Telles, A self-adaptive co-ordinate transformation for efficient numerical evalu-ation of general boundary element integrals, Internat. J. Numer. Meth. Eng. 24 (1987), 959-973. https://doi.org/10.1002/nme.1620240509
  20. B. I. Yun, An extended sigmoidal transformation technique for evaluating weakly singu-lar integrals without splitting the integration interval, SIAM J. Sci. Comput. 25 (2003), no. 1, 284-301. https://doi.org/10.1137/S1064827502414606
  21. B. I. Yun, A compositie transformation for numerical integration of singular integrals in the BEM, Internat. J. Numer. Methods Engrg. 57 (2003), no. 13, 1883-1898. https://doi.org/10.1002/nme.748
  22. B. I. Yun, A generalized non-linear transformation for evaluating singular integrals, Internat. J. Numer. Methods Engrg. 65 (2006), no. 12, 1947-1969. https://doi.org/10.1002/nme.1529
  23. B. I. Yun and P. Kim, A new sigmoidal transformation for weakly singular integrals in the boundary element method, SIAM J. Sci. Comput. 24 (2003), no. 4, 1203-1217. https://doi.org/10.1137/S1064827501396191