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 , 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 and Ian H. Sloan,*Integral equations on the half line*, J. Integral Equations**9**(1985), no. 1, suppl., 3–23. MR**792417****[2]**Kendall Atkinson,*The numerical solution of integral equations on the half-line*, SIAM J. Numer. Anal.**6**(1969), 375–397. MR**0253579****[3]**K. Atkinson and F. de Hoog,*The numerical solution of Laplace’s equation on a wedge*, IMA J. Numer. Anal.**4**(1984), no. 1, 19–41. MR**740782**, 10.1093/imanum/4.1.19**[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]**Ian H. Sloan,*Quadrature methods for integral equations of the second kind over infinite intervals*, Math. Comp.**36**(1981), no. 154, 511–523. MR**606510**, 10.1090/S0025-5718-1981-0606510-2**[7]**I. H. Sloan and A. Spence,*Projection methods for integral equations on the half-line*, IMA J. Numer. Anal.**6**(1986), no. 2, 153–172. MR**967661**, 10.1093/imanum/6.2.153**[8]**E. C. Titchmarsh,*Introduction to the Theory of Fourier Integrals*, Oxford Univ. Press, Oxford, 1937.

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

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

Additional Information

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

Article copyright:
© Copyright 1986
American Mathematical Society