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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider Hammerstein equations of the form

*y*is then obtained by use of the (exact) equation

*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.

**[1]**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****[2]**Christopher T. H. Baker,*The numerical treatment of integral equations*, Clarendon Press, Oxford, 1977. Monographs on Numerical Analysis. MR**0467215****[3]**Richard E. Bellman and Robert E. Kalaba,*Quasilinearization and nonlinear boundary-value problems*, Modern Analytic and Computional Methods in Science and Mathematics, Vol. 3, American Elsevier Publishing Co., Inc., New York, 1965. MR**0178571****[4]**Richard P. Brent,*Some efficient algorithms for solving systems of nonlinear equations*, SIAM J. Numer. Anal.**10**(1973), 327–344. Collection of articles dedicated to the memory of George E. Forsythe. MR**0331764**, https://doi.org/10.1137/0710031**[5]**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**, https://doi.org/10.1017/S0334270000002782**[6]**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****[7]**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**, https://doi.org/10.1093/imanum/5.3.355**[8]**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**, https://doi.org/10.1016/0377-0427(85)90033-0**[9]**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****[10]**Herbert B. Keller,*Geometrically isolated nonisolated solutions and their approximation*, SIAM J. Numer. Anal.**18**(1981), no. 5, 822–838. MR**629667**, https://doi.org/10.1137/0718056**[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]**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**0309333**, https://doi.org/10.1137/0709034**[15]**Jorge J. Moré and Michel Y. Cosnard,*Numerical solution of nonlinear equations*, ACM Trans. Math. Software**5**(1979), no. 1, 64–85. MR**520748**, https://doi.org/10.1145/355815.355820**[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 and L. F. Shampine,*A collocation method for boundary value problems*, Numer. Math.**19**(1972), 1–28. MR**0305607**, https://doi.org/10.1007/BF01395926**[18]**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**0196945****[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. M. Vainikko,*The connection between the mechanical quadrature and finite difference methods*, Ž. Vyčisl. Mat. i Mat. Fiz.**9**(1969), 259–270 (Russian). MR**0260225****[21]**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**, https://doi.org/10.1017/S0334270000002770

Retrieve articles in *Mathematics of Computation*
with MSC:
65R20,
45G10

Retrieve articles in all journals with MSC: 65R20, 45G10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878692-4

Article copyright:
© Copyright 1987
American Mathematical Society