Application of quadrature rules for Cauchytype integrals to the generalized PoincaréBertrand formula
Author:
N. I. Ioakimidis
Journal:
Math. Comp. 44 (1985), 199206
MSC:
Primary 65D32; Secondary 65R20
MathSciNet review:
771041
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Abstract: The classical PoincaréBertrand transposition formula for the inversion of the order of integration in repeated Cauchytype integrals is generalized in accordance with a new interpretation of Cauchytype integrals. Next, the GaussJacobi quadrature rule is applied, in a particular case of the generalized PoincaréBertrand formula, to both members of this formula and it is proved that this formula still remains valid (after the approximation of the integrals by quadrature sums). Two simple applications of this result, one concerning the convergence of a quadrature rule for repeated Cauchytype integrals, and the other the numerical solution of singular integral equations, are made. Further generalizations and applications of the present results follow easily.
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 [1]
 M. M. Chawla & T. R. Ramakrishnan, "Modified GaussJacobi quadrature formulas for the numerical evaluation of Cauchy type singular integrals," BIT, v. 14, 1974, pp. 1421. MR 0331729 (48:10061)
 [2]
 D. Elliott, "On the convergence of Hunter's quadrature rule for Cauchy principal value integrals," BIT, v. 19, 1979, pp. 457462. MR 559954 (81m:65031)
 [3]
 D. Elliott, "The classical collocation method for singular integral equations," SIAM J. Numer. Anal., v. 19, 1982, pp. 816832. MR 664887 (83f:65208)
 [4]
 D. Elliott & D. F. Paget, "Gauss type quadrature rules for Cauchy principal value integrals," Math. Comp., v. 33, 1979, pp. 301309. MR 514825 (81h:65023)
 [5]
 F. D. Gakhov, Boundary Value Problems, Pergamon Press and AddisonWesley, Oxford, 1966, pp. 4953. MR 0198152 (33:6311)
 [6]
 A. Gerasoulis, "On the existence of approximate solutions for singular integral equations of Cauchy type discretized by GaussChebyshev quadrature formulae," BIT, v. 21, 1981, pp. 377380. MR 640939 (82m:65122)
 [7]
 N. I. Ioakimidis, "On the numerical evaluation of singular integrals in interface separation problems," J. Sound Vibration, v. 69, 1980, pp. 167173.
 [8]
 N. I. Ioakimidis, "On the natural interpolation formula for Cauchy type singular integral equations of the first kind," Computing, v. 26, 1981, pp. 7377. MR 620240 (83d:65350)
 [9]
 N. I. Ioakimidis, "On the quadrature methods for the numerical solution of singular integral equations," J. Comput. Appl. Math., v. 8, 1982, pp. 8186. MR 666787 (84b:65130)
 [10]
 N. I. Ioakimidis, "A remark on the application of interpolatory quadrature rules to the numerical solution of singular integral equations," J. Comput. Appl. Math. (To appear.) MR 777102 (86h:65196)
 [11]
 N. I. Ioakimidis, "A new interpretation of Cauchy type singular integrals with an application to singular integral equations," J. Comput. Appl. Math. (To appear.) MR 831073 (88c:30045)
 [12]
 N. I. Ioakimidis, "On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives," Math. Comp., this issue. MR 771040 (86d:65038)
 [13]
 N. I. Ioakimidis & P. S. Theocaris, "On the numerical solution of a class of singular integral equations," J. Math. Phys. Sci., v. 11, 1977, pp. 219235. MR 0483590 (58:3582)
 [14]
 N. I. Ioakimidis & P. S. Theocaris, "A comparison between the direct and the classical numerical methods for the solution of Cauchy type singular integral equations," SIAM J. Numer. Anal., v. 17, 1980, pp. 115118. MR 559466 (81a:65125)
 [15]
 N. I. Ioakimidis & P. S. Theocaris, "A remark on the LobattoChebyshev method for the solution of singular integral equations and the evaluation of stress intensity factors," Serdica, v. 6, 1980, pp. 384390. MR 644291 (83m:45021)
 [16]
 I. K. Lifanov, "Quadrature formulas and the PoincaréBertrand formula for singular integrals," Siberian Math. J., v. 21, 1980 (1981), pp. 787797. MR 601190 (82d:41040)
 [17]
 N. I. Muskhelishvili, Singular Integral Equations, 2nd English ed., WoltersNoordhoff, Groningen, 1958, pp. 5661. MR 0355494 (50:7968)
 [18]
 G. J. Tsamasphyros & P. S. Theocaris, "On the convergence of a Gauss quadrature rule for evaluation of Cauchy type singular integrals," BIT, v. 17, 1977, pp. 458464. MR 0468120 (57:7959)
 [19]
 G. Tsamasphyros & P. S. Theocaris, "Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations," Computing, v. 27, 1981, pp. 7180. MR 623177 (83d:65357)
 [20]
 G. Tsamasphyros & P. S. Theocaris, "On the convergence of some quadrature rules for Cauchy principalvalue and finitepart integrals," Computing, v. 31, 1983, pp. 105114. MR 718908 (85e:65011)
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DOI:
http://dx.doi.org/10.1090/S0025571819850771041X
PII:
S 00255718(1985)0771041X
Article copyright:
© Copyright 1985
American Mathematical Society
