A new collocationtype method for Hammerstein integral equations
Authors:
Sunil Kumar and Ian H. Sloan
Journal:
Math. Comp. 48 (1987), 585593
MSC:
Primary 65R20; Secondary 45G10
MathSciNet review:
878692
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Abstract: We consider Hammerstein equations of the form and present a new method for solving them numerically. The method is a collocation method applied not to the equation in its original form, but rather to an equivalent equation for . The desired approximation to y is then obtained by use of the (exact) equation Advantages of this method, compared with the direct collocation approximation for y, are discussed. The main result in the paper is that, under suitable conditions, the resulting approximation to y converges to the exact solution at a rate at least equal to that of the best approximation to z from the space in which the collocation solution is sought.
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 [1]
 K. E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, Pa., 1976. MR 0483585 (58:3577)
 [2]
 C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press, Oxford, 1977. MR 0467215 (57:7079)
 [3]
 R. E. Bellman & R. E. Kalaba, Quasilinearization and Nonlinear BoundaryValue Problems, Elsevier, New York, 1965. MR 0178571 (31:2828)
 [4]
 R. P. Brent, "Some efficient algorithms for solving systems of nonlinear equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 327344. MR 0331764 (48:10096)
 [5]
 F. Chatelin & R. Lebbar, "The iterated projection solution for Fredholm integral equations of second kind," J. Austral. Math. Soc. Ser. B, v. 22, 1981, pp. 439451. MR 626935 (82h:65096)
 [6]
 F. Chatelin & R. Lebbar, "Superconvergence results for the iterated projection method applied to a Fredholm integral equation of the second kind and the corresponding eigenvalue problem," J. Integral Equations, v. 6, 1984, pp. 7191. MR 727937 (85i:65167)
 [7]
 I. G. Graham, S. Joe & I. H. Sloan, "Iterated Galerkin versus iterated collocation for integral equations of the second kind," IMA J. Numer. Anal., v. 5, 1985, pp. 355369. MR 800020 (86j:65178)
 [8]
 S. Joe, "Collocation methods using piecewise polynomials for second kind integral equations," J. Comput. Appl. Math., v. 12 & 13, 1985, pp. 391400. MR 793970
 [9]
 L. V. Kantorovich & G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. MR 664597 (83h:46002)
 [10]
 H. B. Keller, "Geometrically isolated nonisolated solutions and their approximation," SIAM J. Numer. Anal., v. 18, 1981, pp. 822838. MR 629667 (82j:58013)
 [11]
 M. A. Krasnosel'skiĭ, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford, 1964.
 [12]
 M. A. Krasnosel'skiĭ, G. M. Vaĭnikko, P. P. Zabreĭko, Ya. B. Rutitskiĭ & V. Ya. Stetsenko, Approximate Solution of Operator Equations, WoltersNoordhoff, Groningen, 1972.
 [13]
 M. A. Krasnosel'skiĭ & P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis, SpringerVerlag, Berlin, 1984.
 [14]
 T. R. Lucas & G. W. Reddien, "Some collocation methods for nonlinear boundary value problems," SIAM J. Numer. Anal., v. 9, 1972, pp. 341356. MR 0309333 (46:8443)
 [15]
 J. J. Moré & M. Y. Cosnard, "Numerical solution of nonlinear equations," ACM Trans. Math. Software, v. 5, 1979, pp. 6485. MR 520748 (80c:65110)
 [16]
 J. J. Moré & M. Y. Cosnard, "ALGORITHM 554: BRENTM, A Fortran subroutine for the numerical solution of systems of nonlinear equations," ACM Trans. Math. Software, v. 6, 1980, pp. 240251.
 [17]
 R. D. Russell & L. F. Shampine, "A collocation method for boundary value problems," Numer. Math., v. 19, 1972, pp. 128. MR 0305607 (46:4737)
 [18]
 G. Vaĭnikko, "The convergence of the collocation method for nonlinear differential equations," U.S.S.R. Comput. Math. and Math. Phys., v. 6, no. 1, 1966, pp. 4758. MR 0196945 (33:5129)
 [19]
 G. Vaĭnikko, "Galerkin's perturbation method and the general theory of approximate methods for nonlinear equations," U.S.S.R. Comput. Math. and Math. Phys., v. 7, no. 4, 1967, pp. 141.
 [20]
 G. Vaĭnikko, "The connection between mechanical quadrature and finite difference methods," U.S.S.R. Comput. Math. and Math. Phys., v. 9, no. 2, 1969, pp. 116. MR 0260225 (41:4853)
 [21]
 G. Vaĭnikko & P. Uba, "A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel," J. Austral. Math. Soc. Ser. B, v. 22, 1981, pp. 431438. MR 626934 (82h:65100)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198708786924
PII:
S 00255718(1987)08786924
Article copyright:
© Copyright 1987
American Mathematical Society
