Application of quadrature rules for Cauchy-type integrals to the generalized Poincaré-Bertrand formula

Author:
N. I. Ioakimidis

Journal:
Math. Comp. **44** (1985), 199-206

MSC:
Primary 65D32; Secondary 65R20

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771041-X

MathSciNet review:
771041

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Abstract | References | Similar Articles | Additional Information

Abstract: The classical Poincaré-Bertrand transposition formula for the inversion of the order of integration in repeated Cauchy-type integrals is generalized in accordance with a new interpretation of Cauchy-type integrals. Next, the Gauss-Jacobi quadrature rule is applied, in a particular case of the generalized Poincaré-Bertrand 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 Cauchy-type integrals, and the other the numerical solution of singular integral equations, are made. Further generalizations and applications of the present results follow easily.

**[1]**M. M. Chawla and T. R. Ramakrishnan,*Modified Gauss-Jacobi quadrature formulas for the numerical evaluation of Cauchy type singular integrals*, Nordisk Tidskr. Informationsbehandling (BIT)**14**(1974), 14–21. MR**0331729****[2]**David Elliott,*On the convergence of Hunter’s quadrature rule for Cauchy principal value integrals*, BIT**19**(1979), no. 4, 457–462. MR**559954**, https://doi.org/10.1007/BF01931261**[3]**David Elliott,*The classical collocation method for singular integral equations*, SIAM J. Numer. Anal.**19**(1982), no. 4, 816–832. MR**664887**, https://doi.org/10.1137/0719057**[4]**David Elliott and D. F. Paget,*Gauss type quadrature rules for Cauchy principal value integrals*, Math. Comp.**33**(1979), no. 145, 301–309. MR**514825**, https://doi.org/10.1090/S0025-5718-1979-0514825-2**[5]**F. D. Gakhov,*Boundary value problems*, Translation edited by I. N. Sneddon, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. MR**0198152****[6]**Apostolos Gerasoulis,*On the existence of approximate solutions for singular integral equations of Cauchy type discretized by Gauss-Chebyshev quadrature formulae*, BIT**21**(1981), no. 3, 377–380. MR**640939**, https://doi.org/10.1007/BF01941474**[7]**N. I. Ioakimidis, "On the numerical evaluation of singular integrals in interface separation problems,"*J. Sound Vibration*, v. 69, 1980, pp. 167-173.**[8]**N. I. Ioakimidis,*On the natural interpolation formula for Cauchy-type singular integral equations of the first kind*, Computing**26**(1981), no. 1, 73–77 (English, with German summary). MR**620240**, https://doi.org/10.1007/BF02243425**[9]**N. I. Ioakimidis,*On the quadrature methods for the numerical solution of singular integral equations*, J. Comput. Appl. Math.**8**(1982), no. 2, 81–86. MR**666787**, https://doi.org/10.1016/0771-050X(82)90059-6**[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.**11**(1984), no. 3, 267–276. MR**777102**, https://doi.org/10.1016/0377-0427(84)90001-3**[11]**N. I. Ioakimidis,*A new interpretation of Cauchy type singular integrals with an application to singular integral equations*, J. Comput. Appl. Math.**14**(1986), no. 3, 271–278. MR**831073**, https://doi.org/10.1016/0377-0427(86)90065-8**[12]**N. I. Ioakimidis,*On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives*, Math. Comp.**44**(1985), no. 169, 191–198. MR**771040**, https://doi.org/10.1090/S0025-5718-1985-0771040-8**[13]**N. I. Ioakimidis and P. S. Theocaris,*On the numerical solution of a class of singular integral equations*, J. Mathematical and Physical Sci.**11**(1977), no. 3, 219–235. MR**0483590****[14]**N. I. Ioakimidis and 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.**17**(1980), no. 1, 115–118. MR**559466**, https://doi.org/10.1137/0717012**[15]**N. I. Ioakimidis and P. S. Theocaris,*A remark on the Lobatto-Chebyshev method for the solution of singular integral equations and the evaluation of stress intensity factors*, Serdica**6**(1980), no. 4, 384–390 (1981). MR**644291****[16]**I. K. Lifanov,*Quadrature formulas and the Poincaré-Bertrand formula for singular integrals*, Sibirsk. Mat. Zh.**21**(1980), no. 6, 46–60, 219 (Russian). MR**601190****[17]**N. I. Muskhelishvili,*Singular integral equations*, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR**0355494****[18]**G. J. Tsamasphyros and P. S. Theocaris,*On the convergence of a Gauss quadrature rule for evaluation of Cauchy type singular integrals*, Nordisk Tidskr. Informationsbehandling (BIT)**17**(1977), no. 4, 458–464. MR**0468120****[19]**G. Tsamasphyros and P. S. Theocaris,*Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations*, Computing**27**(1981), no. 1, 71–80 (English, with German summary). MR**623177**, https://doi.org/10.1007/BF02243439**[20]**G. Tsamasphyros and P. S. Theocaris,*On the convergence of some quadrature rules for Cauchy principal-value and finite-part integrals*, Computing**31**(1983), no. 2, 105–114 (English, with German summary). MR**718908**, https://doi.org/10.1007/BF02259907

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DOI:
https://doi.org/10.1090/S0025-5718-1985-0771041-X

Article copyright:
© Copyright 1985
American Mathematical Society