Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


A new collocation-type method for Hammerstein integral equations
HTML articles powered by AMS MathViewer

by Sunil Kumar and Ian H. Sloan PDF
Math. Comp. 48 (1987), 585-593 Request permission


We consider Hammerstein equations of the form \[ y(t) = f(t) + \int _a^b {k(t,s)g(s,y(s)) ds,\quad t \in [a,b],} \] 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 \[ y(t) = f(t) + \int _a^b {k(t,s)z(s) ds,\quad t \in [a,b].} \] 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.
  • Kendall E. Atkinson, A survey of numerical methods for the solution of Fredholm integral equations of the second kind, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. MR 0483585
  • Christopher T. H. Baker, The numerical treatment of integral equations, Monographs on Numerical Analysis, Clarendon Press, Oxford, 1977. MR 0467215
  • Richard E. Bellman and Robert E. Kalaba, Quasilinearization and nonlinear boundary-value problems, Modern Analytic and Computational Methods in Science and Mathematics, Vol. 3, American Elsevier Publishing Co., Inc., New York, 1965. MR 0178571
  • Richard P. Brent, Some efficient algorithms for solving systems of nonlinear equations, SIAM J. Numer. Anal. 10 (1973), 327–344. MR 331764, DOI 10.1137/0710031
  • Françoise Chatelin and Rachid Lebbar, The iterated projection solution for the Fredholm integral equation of second kind, J. Austral. Math. Soc. Ser. B 22 (1980/81), no. 4, 439–451. MR 626935, DOI 10.1017/S0334270000002782
  • Françoise Chatelin and Rachid 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 6 (1984), no. 1, 71–91. MR 727937
  • Ivan G. Graham, Stephen Joe, and Ian H. Sloan, Iterated Galerkin versus iterated collocation for integral equations of the second kind, IMA J. Numer. Anal. 5 (1985), no. 3, 355–369. MR 800020, DOI 10.1093/imanum/5.3.355
  • S. Joe, Collocation methods using piecewise polynomials for second kind integral equations, Proceedings of the international conference on computational and applied mathematics (Leuven, 1984), 1985, pp. 391–400. MR 793970, DOI 10.1016/0377-0427(85)90033-0
  • L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597
  • Herbert B. Keller, Geometrically isolated nonisolated solutions and their approximation, SIAM J. Numer. Anal. 18 (1981), no. 5, 822–838. MR 629667, DOI 10.1137/0718056
  • M. A. Krasnosel’skiĭ, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford, 1964. 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. M. A. Krasnosel’skiĭ & P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin, 1984.
  • Thomas R. Lucas and George W. Reddien Jr., Some collocation methods for nonlinear boundary value problems, SIAM J. Numer. Anal. 9 (1972), 341–356. MR 309333, DOI 10.1137/0709034
  • Jorge J. Moré and Michel Y. Cosnard, Numerical solution of nonlinear equations, ACM Trans. Math. Software 5 (1979), no. 1, 64–85. MR 520748, DOI 10.1145/355815.355820
  • 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.
  • R. D. Russell and L. F. Shampine, A collocation method for boundary value problems, Numer. Math. 19 (1972), 1–28. MR 305607, DOI 10.1007/BF01395926
  • G. Vainikko, On the convergence of the collocation method for nonlinear differential equations, Ž. Vyčisl. Mat i Mat. Fiz. 6 (1966), no. 1, 35–42 (Russian). MR 196945
  • 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.
  • G. M. Vainikko, The connection between the mechanical quadrature and finite difference methods, Ž. Vyčisl. Mat i Mat. Fiz. 9 (1969), 259–270 (Russian). MR 260225
  • G. Vainikko and P. Uba, A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, J. Austral. Math. Soc. Ser. B 22 (1980/81), no. 4, 431–438. MR 626934, DOI 10.1017/S0334270000002770
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 65R20, 45G10
  • Retrieve articles in all journals with MSC: 65R20, 45G10
Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Math. Comp. 48 (1987), 585-593
  • MSC: Primary 65R20; Secondary 45G10
  • DOI:
  • MathSciNet review: 878692