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A new collocation-type method for Hammerstein integral equations


Authors: Sunil Kumar and Ian H. Sloan
Journal: Math. Comp. 48 (1987), 585-593
MSC: Primary 65R20; Secondary 45G10
DOI: https://doi.org/10.1090/S0025-5718-1987-0878692-4
MathSciNet review: 878692
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Abstract: We consider Hammerstein equations of the form

$\displaystyle y(t) = f(t) + \int_a^b {k(t,s)g(s,y(s))\,ds,\quad t \in [a,b],} $

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 $ z(t): = g(t,y(t))$. The desired approximation to y is then obtained by use of the (exact) equation

$\displaystyle y(t) = f(t) + \int_a^b {k(t,s)z(s)\,ds,\quad t \in [a,b].} $

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 Boundary-Value 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. 327-344. 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. 439-451. 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. 71-91. 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. 355-369. 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. 391-400. 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. 822-838. 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, Wolters-Noordhoff, Groningen, 1972.
  • [13] M. A. Krasnosel'skiĭ & P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, 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. 341-356. MR 0309333 (46:8443)
  • [15] J. J. Moré & M. Y. Cosnard, "Numerical solution of nonlinear equations," ACM Trans. Math. Software, v. 5, 1979, pp. 64-85. 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. 240-251.
  • [17] R. D. Russell & L. F. Shampine, "A collocation method for boundary value problems," Numer. Math., v. 19, 1972, pp. 1-28. 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. 47-58. 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. 1-41.
  • [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. 1-16. 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. 431-438. MR 626934 (82h:65100)

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DOI: https://doi.org/10.1090/S0025-5718-1987-0878692-4
Article copyright: © Copyright 1987 American Mathematical Society

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