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The numerical solution of Hammerstein equations by a method based on polynomial collocation

Published online by Cambridge University Press:  17 February 2009

Sunil Kumar
Sunil Kumar
Affiliation:
School of Mathematics, The University of New South Wales, Sydney, NSW 2033, Australia.
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Abstract

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In recent papers we have considered the numerical solution of the Hammerstein equation

by a method which first applies the standard collocation procedure to an equivalent equation for z(t):= g(t, y(t)), and then obtains an approximation to y by use of the equation

In this paper we approximate z by a polynomial zn of degree ≤ n − 1, with coefficients determined by collocation at the zeros of the nth degree Chebyshev polynomial of the first kind. We then define the approximation to y to be

and establish that, under suitable conditions, , uniformly in t.

Type
Research Article
Information
The ANZIAM Journal , Volume 31 , Issue 3 , January 1990 , pp. 319 - 329
Copyright
Copyright © Australian Mathematical Society 1990

References

[1] Askey, R., “Mean convergence of orthogonal series and Lagrange interpolation,” Acta Math. Acad. Sci. Hungar. 23 (1972) 71–85.CrossRefGoogle Scholar
[2] Askey, R., “Summability of Jacobi series”, Trans. Amer. Math. Soc. 179 (1973) 71–84.CrossRefGoogle Scholar
[3] Bellman, R. E. and Kalaba, R. E., Quasilinearization and nonlinear boundary-value problems (Elsevier, New York, 1965).Google Scholar
[4] Cheney, E. W., Introduction to approximation theory (McGraw-Hill, New York, 1966).Google Scholar
[5] Davis, P. J., Interpolation and approximation (Blaisdell, New York, 1963).Google Scholar
[6] Erdös, P. and Feldheim, E., “Sur le mode de convergence pour l'interpolation de Lagrange”, C. R. Acad. Sci. Paris Sér. A-B 203 (1936) 913–915.Google Scholar
[7] Graham, I. G. and Sloan, I. H., “On the compactness of certain integral operators”, J. Math. Anal. Appl. 68 (1979) 580–594.CrossRefGoogle Scholar
[8] Kantorovich, L. V. and Akilov, G. P., Functional analysis (Pergamon Press, Oxford, 1982).Google Scholar
[9] Kumar, S., “Superconvergence of a collocation-type method for Hammerstein equations”, IMA J. Numer. Anal. 7 (1987) 313–325.CrossRefGoogle Scholar
[10] Kumar, S. and Sloan, I. H., “A new collocation-type method for Hammerstein integral equations”, Math. Comp. 48 (1987) 585–593.CrossRefGoogle Scholar
[11] Marcinkiewicz, J., “Sur I’interpolation (1)”, Studia Math. 6 (1936) 1–17.CrossRefGoogle Scholar
[12] Nevai, G. P., “Lagrange interpolation at zeros of orthogonal polynomials”, in Approximation theory II (eds. Lorentz, G. G., Chui, C. K. and Schumaker, L. L.), (Academic Press, New York, 1976), 163–201.Google Scholar
[13] Piessens, R. and Branders, M., “Numerical solution of integral equations of mathematical physics, using Chebyshev polynomials”, J. Comput. Phys. 21 (1976) 178–196.CrossRefGoogle Scholar
[14] Rivlin, T. J., The Chebyshev polynomials (Wiley, New York, 1974).Google Scholar
[15] Sloan, I. H. and Bum, B. J., “Collocation with polynomials for integral equations of the second kind: a new approach to the theory”, J. Integral Equations 1 (1979) 77–94.Google Scholar