Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind

Authors: Hideaki Kaneko and Yuesheng Xu
Journal: Math. Comp. 62 (1994), 739-753
MSC: Primary 65D32; Secondary 45L10, 65R20
MathSciNet review: 1218345
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we establish Gauss-type quadrature formulas for weakly singular integrals. An application of the quadrature scheme is given to obtain numerical solutions of the weakly singular Fredholm integral equation of the second kind. We call this method a discrete product-integration method since the weights involved in the standard product-integration method are computed numerically.

References [Enhancements On Off] (What's this?)

  • [1] P. M. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice Hall, Englewood Cliffs, NJ, 1971. MR 0443383 (56:1753)
  • [2] K. E. Atkinson, An introduction to numerical analysis, 2nd ed., Wiley, New York, 1989. MR 1007135 (90m:65001)
  • [3] P. J. Davis and P. Rabinowitz, Methods of numerical integration, 2nd ed., Academic Press, Orlando, Florida, 1984. MR 760629 (86d:65004)
  • [4] I. G. Graham, Singularity expansions for the solutions of the second kind Fredholm integral equations with weakly singular convolution kernels, J. Integral Equations 4 (1982), 1-30. MR 640534 (83e:45006)
  • [5] E. Hopf, Mathematical problems of radiative equilibrium, Stechert-Hafner Service Agency, New York, 1964. MR 0183517 (32:997)
  • [6] H. Kaneko, R. Noren, and Y. Xu, On the regularity of the solution of the weakly singular Hammerstein equations, Integral Equations Operator Theory 13 (1990), 660-670. MR 1066128 (92a:45022)
  • [7] J. G. Kirkwood and J. Riseman, The intrinsic viscosities and diffusion constants of flexible macromolecules in solution, J. Chem. Phys. 16 (1948), 565-573.
  • [8] R. Kress, Linear integral equations, Appl. Math. Sci., vol. 82, Springer-Verlag, Berlin, 1989. MR 1007594 (90j:45001)
  • [9] D. S. Lubinsky and P. Rabinowitz, Rates of convergence of Gaussian quadrature for singular integrands, Math. Comp. 47 (1984), 219-242. MR 744932 (86b:65018)
  • [10] R. Piessens and M. Branders, Numerical solution of integral equations of mathematical physics using Chebyshev polynomials, J. Comp. Phys. 21 (1976), 178-196. MR 0458972 (56:17171)
  • [11] P. Rabinowitz and I. H. Sloan, Product integration in the presence of a singularity, SIAM J. Numer. Anal. 21 (1984), 149-166. MR 731219 (85c:65023)
  • [12] J. R. Rice, On the degree of convergence of nonlinear spline approximation, Approx. with Emphasis on Spline Functions (I. J. Schoenberg, ed.), Academic Press, New York, 1969, pp. 349-365. MR 0267324 (42:2226)
  • [13] G. Richter, On weakly singular Fredholm integral equations with displacement kernels, J. Math. Anal. Appl 55 (1976), 32-42. MR 0407549 (53:11322)
  • [14] C. Schneider, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Equations Operator Theory 2 (1979), 62-68. MR 532739 (80f:45002)
  • [15] -, Product integration for weakly singular integral equations, Math. Comp. 36 (1981), 207-213. MR 595053 (82c:65090)
  • [16] I. H. Sloan, On the numerical evaluation of singular integrals, BIT 18 (1978), 91-102. MR 0501799 (58:19054)
  • [17] I. H. Sloan and W. E. Smith, Product-integration with the Clenshaw-Curtis and related points, Numer. Math. 30 (1978), 415-428. MR 0494863 (58:13646)
  • [18] F. Stenger, Numerical methods based on Whittaker cardinal, or sinc quadrature, SIAM Rev. 23 (1981), 165-224. MR 618638 (83g:65027)
  • [19] M. H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space. I, J. Math. Mech. 13 (1964), 407-479; II, J. Math. Mech. 14 (1965), 813-839. MR 0163159 (29:462)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D32, 45L10, 65R20

Retrieve articles in all journals with MSC: 65D32, 45L10, 65R20

Additional Information

Keywords: Gaussian quadrature, singular integrals, weakly singular Fredholm integral equations, product-integration scheme
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society