Chebyshev iteration methods for integral equations of the second kind.

Author:
T. W. Sag

Journal:
Math. Comp. **24** (1970), 341-355

MSC:
Primary 65.75

MathSciNet review:
0278564

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Abstract: In this paper the numerical solution of Fredholm integral equations of the second kind using an iterative method in which the solution is represented by a Chebyshev series is discussed. A description of a technique of Chebyshev reduction of the norm of the kernel for use in cases when the iterations converge slowly or not at all is also given. Finally, the application of the methods to other types of second-kind equations is considered.

**[1]**F. G. Tricomi,*Integral equations*, Pure and Applied Mathematics. Vol. V, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR**0094665****[2]**L. Fox and E. T. Goodwin,*The numerical solution of non-singular linear integral equations*, Philos. Trans. Roy. Soc. London. Ser. A.**245**(1953), 501–534. MR**0054355****[3]**David Elliott,*Chebyshev series method for the numerical solution of Fredholm integral equations*, Comput. J.**6**(1963/1964), 102–111. MR**0155452****[4]**C. W. Clenshaw,*Chebyshev Series for Mathematical Functions*, National Physical Laboratory Mathematical Tables, vol. 5, H.M.S.O., London, 1962. MR**26**#362.**[5]**T. W. Sag,*Numerical Methods for the Solution of Integral Equations*, Ph.D. Thesis, University of Manchester, 1966.**[6]**C. Lanczos.*Tables of Chebyshev Polynomials (Introduction)*, Nat. Bur. Standards Appl. Math. Series 9, U. S. Government Printing Office, Washington, D. C., 1952.**[7]**E. R. Love,*The electrostatic field of two equal circular co-axial conducting disks*, Quart. J. Mech. Appl. Math.**2**(1949), 428–451. MR**0034700****[8]**John Todd and S. E. Warschawski,*On the solution of the Lichtenstein-Gershgorin integral equation in conformal mapping. II. Computational experiments*, Experiments in the computation of conformal maps, National Bureau of Standards Applied Mathematics Series, No. 42, U. S. Government Printing Office, Washington, D. C., 1955, pp. 31–44. MR**0074122****[9]**Donald G. Anderson,*Iterative procedures for nonlinear integral equations*, J. Assoc. Comput. Mach.**12**(1965), 547–560. MR**0184447****[10]**C. B. Haselgrove,*The solution of non-linear equations and of differential equations with two-point boundary conditions*, Comput. J.**4**(1961/1962), 255–259. MR**0130114**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0278564-X

Keywords:
Fredholm integral equations,
iteration,
Chebyshev series approximation,
numerical quadrature,
Chebyshev reduction of kernel,
nonlinear integral equations

Article copyright:
© Copyright 1970
American Mathematical Society