Nyström type methods for Fredholm integral equations with weak singularities

Journal of Computational and Applied Mathematics

Volumn 234, Issue 9, 2010, Pages 2848-2858

  Orav Puurand, Kerli ^a   Pedas, Arvet ^a   Vainikko, Gennadi ^a  

^a UNIVERSITY OF TARTU   (Estonia)

Author keywords

Nystrm type method; Product integration; Smoothing transformation; Weakly singular integral equation

Indexed keywords

BOUNDARY SINGULARITIES; CHANGE OF VARIABLES; FREDHOLM INTEGRAL EQUATIONS; GLOBAL CONVERGENCE; NUMERICAL RESULTS; PRODUCT INTEGRATION; SMOOTHING TRANSFORMATION; TYPE METHODS; WEAK SINGULARITY; WEAKLY SINGULAR INTEGRAL EQUATIONS;

CONVERGENCE OF NUMERICAL METHODS; NUMERICAL ANALYSIS;

INTEGRAL EQUATIONS;

EID: 77955278082     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2010.01.033     Document Type: Conference Paper

References (34)

1. K.E. Atkinson The Numerical Solution of Integral Equations of the Second Kind 1997 Cambridge Univ. Press Cambridge
2. I.G. Graham Singularity expansions for solutions of second kind Fredholm integral equations with weakly singular convolution kernels J. Integral Equations 4 1982 1 30
3. G. Vainikko, and A. Pedas The properties of solutions of weakly singular integral equations J. Aust. Math. Soc. Ser. B 22 1981 419 430
4. C.T.H. Baker The Numerical Treatment of Integral Equations 1977 Clarendon Press Oxford
5. H. Brunner Collocation Methods for Volterra Integral and Related Functional Equations 2004 Cambridge Univ. Press Cambridge
6. H. Brunner, and P.J. van der Houwen The Numerical Solution of Volterra Equations 1986 North-Holland Amsterdam
7. W. Hackbusch Integral Equations 1995 Birkhuser Basel
8. C. Schneider Product integration for weakly singular integral equations Math. Comp. 36 1981 207 213
9. Y. Xu, and Y. Zhao Quadratures for improper integrals and their applications in integral equations Proc. Sympos. Appl. Math. 48 1994 409 413
10. P. Baratella, and A.P. Orsi A new approach to the numerical solution of weakly singular Volterra integral equations J. Comput. Appl. Math. 163 2004 401 418
11. G. Monegato, and L. Scuderi High order methods for weakly singular integral equations with nonsmooth input functions Math. Comp. 67 1998 1493 1515 (Pubitemid 128383316)
12. A. Pedas, and G. Vainikko Smoothing transformation and piecewise polynomial collocation for weakly singular Volterra integral equations Computing 73 2004 271 293
13. A. Pedas, and G. Vainikko Smoothing transformation and piecewise polynomial projection methods for weakly singular Fredholm integral equations Commun. Pure Appl. Anal. 5 2006 395 413
14. A. Pedas, and G. Vainikko Piecewise polynomial approximations for integral equations with diagonal and boundary singularities J. Anal. 14 2006 159 173
15. E. Tamme Numerical computation of weakly singular integrals Proc. Estonian Acad. Sci. Phys. Math. 49 2000 215 224
16. D. Elliott The cruciform crack problem and sigmoidal transformations Math. Methods Appl. Sci. 20 1997 121 132
17. D. Elliott Sigmoidal transformations and the trapezoidal rule J. Aust. Math. Soc. Ser. B 40 1998 E77 E137
18. P.R. Johnston Application of sigmoidal transformations to weakly singular and near singular boundary element integrals Internat. J. Numer. Methods Engrg. 45 1999 1333 1348
19. P.R. Johnston, and D. Elliott Error estimation of quadrature rules for evaluating singular integrals in boundary element Internat. J. Numer. Methods Engrg. 48 2000 949 962
20. H. Kaneko, and Y. Xu Gauss-type quadratures for weakly singular integrals and their applications to Fredholm integral equations of the second kind Math. Comp. 62 1994 739 753
21. P. Khler Order-preserving mesh spacing for compound quadrature formulas and functions with endpoint singularities SIAM J. Numer. Anal. 32 1995 671 686
22. D.P. Laurie Periodizing transformation for numerical integration J. Comput. Appl. Math. 66 1996 337 344
23. G. Monegato, and L. Scuderi Numerical integration of functions with boundary singularities J. Comput. Anal. Math. 112 1998 201 214
24. A. Sidi A new variable transformation for numerical integration H. Brass, G. Hmmerlin, Numerical Integration IV 1993 Birkhuser Basel 359 373
25. K.M. Singh, and M. Tanaka On non-linear transformations for accurate numerical evaluation of weakly singular integrals Internat. J. Numer. Methods Engrg. 50 2001 2007 2030
26. B.I. Yun An efficient transformation with Gauss quadrature rule for weakly singular integrals Comm. Numer. Methods Engrg. 17 2001 881 891
27. 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 1203 1217
28. E. Vainikko, and G. Vainikko A spline product quasi-interpolation method for weakly singular Fredholm integral equations SIAM J. Numer. Anal. 46 2008 1799 1820
29. R. Kress Linear Integral Equations 1989 Springer Berlin
30. A. Pedas, and G. Vainikko Integral equations with diagonal and boundary singularities of the kernel Z. Anal. Anwend. 25 2006 487 516
31. A. Pedas, and G. Vainikko On the regularity of solutions to integral equations with nonsmooth kernels on a union of open intervals J. Comput. Appl. Math. 229 2009 440 451
32. G. Vainikko Multidimensional Weakly Singular Integral Equations 1993 Springer-Verlag Berlin
33. P.M. Anselone Collectively Compact Operator Approximation Theory 1971 Prentice-Hall Englewood Cliffs, NJ
34. G. Vainikko Funktionalanalysis der Diskretisierungsmethoden 1976 Teubner Leipzig