Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Integral equations on the half-line: a modified finite-section approximation

Authors: I. H. Sloan and A. Spence
Journal: Math. Comp. 47 (1986), 589-595
MSC: Primary 65R20; Secondary 45L10
MathSciNet review: 856704
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the approximate solution of integral equations of the form $ y(t) - \smallint _0^\infty k(t,s) y(s)\,ds = f(t)$, where the conditions on $ k(t,s)$ are such that kernels of the Wiener-Hopf form $ k(t,s) = \kappa (t - s)$ are included in the analysis. The finite-section approximation, in which the infinite integral is replaced by $ \smallint _0^\beta $ for some $ \beta > 0$, yields an approximate solution $ {y_\beta }(t)$ that is known, under very general conditions, to converge to $ y(t)$ as $ \beta \to \infty $ with t fixed. However, the convergence is uniform only on finite intervals, and the approximation is typically very poor for $ t > \beta $. Under the assumption that f has a limit at infinity, we here introduce a modified finite-section approximation with improved approximation properties, and prove that the new approximate solution converges uniformly to y as $ \beta \to \infty $. A numerical example illustrates the improvement.

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

  • [1] P. M. Anselone & I. H. Sloan, "Integral equations on the half-line," J. Integral Equations. (To appear.) MR 792417 (87f:45036)
  • [2] K. E. Atkinson, "The numerical solution of integral equations on the half-line," SIAM J. Numer. Anal., v. 6, 1969, pp. 375-397. MR 0253579 (40:6793)
  • [3] K. E. Atkinson & F. de Hoog, "The numerical solution of Laplace's equation on a wedge," IMA J. Numer. Anal., v. 4, 1984, pp. 19-41. MR 740782 (86a:65124)
  • [4] G. D. Finn & J. T. Jefferies, "Studies in spectral line formation. I. Formulation and simple applications," J. Quant. Spectrosc. Radiat. Transfer, v. 8, 1968, pp. 1675-1703.
  • [5] F. de Hoog & I. H. Sloan, "The finite-section approximation for integral equations on the half-line," J. Austral. Math. Soc. Ser. B. (Submitted.)
  • [6] I. H. Sloan, "Quadrature methods for integral equations of the second kind over infinite intervals," Math. Comp., v. 36, 1981, pp. 511-523. MR 606510 (82c:65091)
  • [7] I. H. Sloan & A. Spence, "Projection methods for integral equations on the half-line," IMA J. Numer. Anal., v. 6, 1986, pp. 153-172. MR 967661 (89h:65225)
  • [8] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford Univ. Press, Oxford, 1937.

Similar Articles

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

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

Additional Information

Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society