On solving singular integral equations via a hyperbolic tangent quadrature rule
Author:
Ezio Venturino
Journal:
Math. Comp. 47 (1986), 159167
MSC:
Primary 65R20; Secondary 45E05
MathSciNet review:
842128
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Abstract: We propose a scheme for solving singular integral equations based on a "hyperbolic tangent" quadrature rule. The integral equation is reduced to a system of linear equations, after quadrature and collocation. The matrix of the system is shown to be nonsingular for every choice of the number of quadrature nodes by producing a lower bound for its determinant.
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 F. R. Gantmacher, The Theory of Matrices, Chelsea, New York, 1959. MR 0107649 (21:6372c)
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 J. McNamee, F. Stenger & E. L. Whitney, "Whittaker's cardinal function in retrospect," Math. Comp., v. 25, 1971, pp. 141154. MR 0301428 (46:586)
 [3]
 R. A. Sack, "Comments on some quadrature formulas by F. Stenger," J. Inst. Math. Appl., v. 21, 1978, pp. 359361. MR 0494861 (58:13644)
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 F. Stenger, "Integration formulae based on the trapezoidal formula," J. Inst. Math. Appl., v. 12, 1973, pp. 103114. MR 0381261 (52:2158)
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 F. Stenger, "Approximations via Whittaker's cardinal function," J. Approx. Theory, v. 17, 1976, pp. 222240. MR 0481786 (58:1885)
 [6]
 F. Stenger, "Remarks on "Integration formulas based on the trapezoidal formula"," J. Inst. Math. Appl., v. 19, 1977, pp. 145147. MR 0440879 (55:13747)
 [7]
 F. Stenger & D. Elliott, "SINC method of solution for singular integral equations," in Numerical Solution of Singular Integral Equations (A. Gerasoulis, R. Vichnevetsky, eds.), Proceedings of an IMACS International Symposium held at Lehigh University, Bethlehem, PA, USA, June 2122, 1984, pp. 2735.
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 J. M. Whittaker, "On the cardinal function of interpolation theory," Proc. Edinburgh Math. Soc. (1), v. 2, 1927, pp. 4146.
 [9]
 E. Venturino, An Analysis of Some Direct Methods for the Numerical Solution of Singular Integral Equations, Ph. D. thesis, SUNY at Stony Brook, 1984.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608421289
PII:
S 00255718(1986)08421289
Article copyright:
© Copyright 1986
American Mathematical Society
