Integral equations on the halfline: a modified finitesection approximation
Authors:
I. H. Sloan and A. Spence
Journal:
Math. Comp. 47 (1986), 589595
MSC:
Primary 65R20; Secondary 45L10
MathSciNet review:
856704
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Abstract: We consider the approximate solution of integral equations of the form , where the conditions on are such that kernels of the WienerHopf form are included in the analysis. The finitesection approximation, in which the infinite integral is replaced by for some , yields an approximate solution that is known, under very general conditions, to converge to as with t fixed. However, the convergence is uniform only on finite intervals, and the approximation is typically very poor for . Under the assumption that f has a limit at infinity, we here introduce a modified finitesection approximation with improved approximation properties, and prove that the new approximate solution converges uniformly to y as . A numerical example illustrates the improvement.
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Ian
H. Sloan, Quadrature methods for integral
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 [1]
 P. M. Anselone & I. H. Sloan, "Integral equations on the halfline," J. Integral Equations. (To appear.) MR 792417 (87f:45036)
 [2]
 K. E. Atkinson, "The numerical solution of integral equations on the halfline," SIAM J. Numer. Anal., v. 6, 1969, pp. 375397. 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. 1941. 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. 16751703.
 [5]
 F. de Hoog & I. H. Sloan, "The finitesection approximation for integral equations on the halfline," 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. 511523. MR 606510 (82c:65091)
 [7]
 I. H. Sloan & A. Spence, "Projection methods for integral equations on the halfline," IMA J. Numer. Anal., v. 6, 1986, pp. 153172. MR 967661 (89h:65225)
 [8]
 E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford Univ. Press, Oxford, 1937.
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DOI:
http://dx.doi.org/10.1090/S00255718198608567040
PII:
S 00255718(1986)08567040
Article copyright:
© Copyright 1986
American Mathematical Society
