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

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856704-0

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 Wiener-Hopf form are included in the analysis. The finite-section 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 finite-section approximation with improved approximation properties, and prove that the new approximate solution converges uniformly to *y* as . A numerical example illustrates the improvement.

**[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.

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0856704-0

Article copyright:
© Copyright 1986
American Mathematical Society