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Mathematics of Computation

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A new collocation-type method for Hammerstein integral equations
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by Sunil Kumar and Ian H. Sloan PDF
Math. Comp. 48 (1987), 585-593 Request permission

Abstract:

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.
References
  • 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
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • 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