## 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, - 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," - 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," - 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

*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.

*ACM Trans. Math. Software*, v. 6, 1980, pp. 240-251.

*U.S.S.R. Comput. Math. and Math. Phys.*, v. 7, no. 4, 1967, pp. 1-41.

## 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