Operational quadrature methods for Wiener-Hopf integral equations
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- by P. P. B. Eggermont and Ch. Lubich PDF
- Math. Comp. 60 (1993), 699-718 Request permission
Abstract:
We study the numerical solution of Wiener-Hopf integral equations by a class of quadrature methods which lead to discrete Wiener-Hopf equations, with quadrature weights constructed from the Fourier transform of the kernel (or rather, from the Laplace transforms of the kernel halves). As the analytical theory of Wiener-Hopf equations is likewise based on the Fourier transform of the kernel, this approach enables us to obtain results on solvability and stability and error estimates for the discretization. The discrete Wiener-Hopf equations are solved by using an approximate Wiener-Hopf factorization obtained with FFT. Numerical experiments with the Milne equation of radiative transfer are included.References
- P. M. Anselone and C. T. H. Baker, Error bounds for integral equations on the half line, J. Integral Equations Appl. 1 (1988), no. 3, 321–342. MR 1003699, DOI 10.1216/JIE-1988-1-3-321
- P. M. Anselone and J. W. Lee, Nonlinear integral equations of the half line, J. Integral Equations Appl. 4 (1992), no. 1, 1–14. MR 1160085, DOI 10.1216/jiea/1181075663
- Philip M. Anselone and Ian H. Sloan, Numerical solutions of integral equations on the half line. II. The Wiener-Hopf case, J. Integral Equations Appl. 1 (1988), no. 2, 203–225. MR 978741, DOI 10.1216/JIE-1988-1-2-203
- P. M. Anselone and I. H. Sloan, Spectral approximations for Wiener-Hopf operators, J. Integral Equations Appl. 2 (1990), no. 2, 237–261. MR 1045771, DOI 10.1216/JIE-1990-2-2-237
- Kendall Atkinson, The numerical solution of integral equations on the half-line, SIAM J. Numer. Anal. 6 (1969), 375–397. MR 253579, DOI 10.1137/0706035
- Raymond H. Chan and Gilbert Strang, Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM J. Sci. Statist. Comput. 10 (1989), no. 1, 104–119. MR 976165, DOI 10.1137/0910009
- G. A. Chandler and I. G. Graham, The convergence of Nyström methods for Wiener-Hopf equations, Numer. Math. 52 (1988), no. 3, 345–364. MR 929577, DOI 10.1007/BF01398884
- P. P. B. Eggermont, On the quadrature error in operational quadrature methods for convolutions, Numer. Math. 62 (1992), no. 1, 35–48. MR 1159044, DOI 10.1007/BF01396219
- P. P. B. Eggermont and Ch. Lubich, Uniform error estimates of operational quadrature methods for nonlinear convolution equations on the half-line, Math. Comp. 56 (1991), no. 193, 149–176. MR 1052091, DOI 10.1090/S0025-5718-1991-1052091-8
- Johannes Elschner, On spline collocation for convolution equations, Integral Equations Operator Theory 12 (1989), no. 4, 486–510. MR 1001654, DOI 10.1007/BF01199456
- Siegfried Gähler and Werner Gähler, Quadrature methods for the solution of integral equations of the second kind on the half line, Math. Nachr. 140 (1989), 321–346. MR 1015403, DOI 10.1002/mana.19891400122
- Gottlob Gienger, On convolution quadratures and their applications to Fredholm integral equations of the second kind, Mitt. Math. Sem. Giessen 186 (1988), ii+121. MR 947877
- I. C. Gohberg and I. A. Fel′dman, Convolution equations and projection methods for their solution, Translations of Mathematical Monographs, Vol. 41, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by F. M. Goldware. MR 0355675
- Gene H. Golub and Charles F. Van Loan, Matrix computations, 2nd ed., Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1989. MR 1002570
- Ivan G. Graham and Wendy R. Mendes, Nystrom-product integration for Wiener-Hopf equations with applications to radiative transfer, IMA J. Numer. Anal. 9 (1989), no. 2, 261–284. MR 1000461, DOI 10.1093/imanum/9.2.261
- E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations. I, Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1987. Nonstiff problems. MR 868663, DOI 10.1007/978-3-662-12607-3
- Peter Henrici, Applied and computational complex analysis. Vol. 3, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1986. Discrete Fourier analysis—Cauchy integrals—construction of conformal maps—univalent functions; A Wiley-Interscience Publication. MR 822470 M. G. Krein, Integral equations on a half-line with kernel depending upon the difference of the arguments, Amer. Math. Soc. Transl. (2) 22 (1962), 163-286.
- C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math. 52 (1988), no. 2, 129–145. MR 923707, DOI 10.1007/BF01398686
- C. Lubich, On convolution quadrature and Hille-Phillips operational calculus, Appl. Numer. Math. 9 (1992), no. 3-5, 187–199. International Conference on the Numerical Solution of Volterra and Delay Equations (Tempe, AZ, 1990). MR 1158482, DOI 10.1016/0168-9274(92)90014-5
- B. Noble, The numerical solution of nonlinear integral equations and related topics, Nonlinear Integral Equations (Proc. Advanced Seminar Conducted by Math. Research Center, U.S. Army, Univ. Wisconsin, Madison, Wis., 1963) Univ. Wisconsin Press, Madison, Wis., 1964, pp. 215–318. MR 0173369
- Siegfried Prössdorf and Bernd Silbermann, Projektionsverfahren und die näherungsweise Lösung singulärer Gleichungen, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1977 (German). Mit einer englischen und einer russischen Zusammenfassung; Teubner-Texte zur Mathematik. MR 0494817 —, Numerical analysis for integral and related operator equations, Birkhäuser, Basel, 1991.
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- 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, DOI 10.1093/imanum/6.2.153
- I. H. Sloan and A. Spence, Integral equations on the half-line: a modified finite-section approximation, Math. Comp. 47 (1986), no. 176, 589–595. MR 856704, DOI 10.1090/S0025-5718-1986-0856704-0
- Frank Stenger, The approximate solution of Wiener-Hopf integral equations, J. Math. Anal. Appl. 37 (1972), 687–724. MR 303783, DOI 10.1016/0022-247X(72)90251-X
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Math. Comp. 60 (1993), 699-718
- MSC: Primary 65R20; Secondary 45E10, 45L05, 47B35, 47G10
- DOI: https://doi.org/10.1090/S0025-5718-1993-1160274-3
- MathSciNet review: 1160274