Galerkin methods for singular integral equations

Author:
K. S. Thomas

Journal:
Math. Comp. **36** (1981), 193-205

MSC:
Primary 65R20; Secondary 45F99

MathSciNet review:
595052

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Abstract | References | Similar Articles | Additional Information

Abstract: The approximate solution of a singular integral equation by Galerkin's method is studied. We discuss the theoretical aspects of such problems and give error bounds for the approximate solution.

**[1]**Carl de Boor,*Good approximation by splines with variable knots*, Spline functions and approximation theory (Proc. Sympos., Univ. Alberta, Edmonton, Alta., 1972) Birkhäuser, Basel, 1973, pp. 57–72. Internat. Ser. Numer. Math., Vol. 21. MR**0403169****[2]**Hermann G. Burchard,*Splines (with optimal knots) are better*, Applicable Anal.**3**(1973/74), 309–319. MR**0399708****[3]**T. Carleman, "Sur la résolution de certaines équations intégrales,"*Ark. Mat. Astronom. Fys.*, v. 16, 1921, pp. 1-19.**[4]**Y. Cherruault,*Approximation d’opérateurs linéaires et applications*, Monographies d’informatique, No. 4, Dunod, Paris, 1968 (French). MR**0236584****[5]**D. S. Dodson,*Optimal Order Approximation by Polynomial Spline Functions*, Ph. D. Thesis, Comp. Sci. Dept., Purdue Univ., Lafayette, Ind., 1972.**[6]**M. L. Dow and David Elliott,*The numerical solution of singular integral equations over (-1,1)*, SIAM J. Numer. Anal.**16**(1979), no. 1, 115–134. MR**518688**, 10.1137/0716009**[7]**F. Erdogan, G. D. Gupta, and T. S. Cook,*Numerical solution of singular integral equations*, Mechanics of fracture, Vol. 1, Noordhoff, Leiden, 1973, pp. 368–425. MR**0471394****[8]**V. V. Ivanov,*The theory of approximate methods and their application to the numerical solution of singular integral equations*, Noordhoff International Publishing, Leyden, 1976. Translated from the Russian by A. Ideh; Edited by R. S. Anderssen and D. Elliott; Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, No. 2. MR**0405045****[9]**L. N. Karpenko,*Approximation solution of a singular integral equation by means of Jacobi polynomials*, J. Appl. Math. Mech.**30**(1966), 668–675. MR**0231156****[10]**Steen Krenk,*On quadrature formulas for singular integral equations of the first and the second kind*, Quart. Appl. Math.**33**(1975/76), no. 3, 225–232. MR**0448967****[11]**Peter Linz,*A general theory for the approximate solution of operator equations of the second kind*, SIAM J. Numer. Anal.**14**(1977), no. 3, 543–554. MR**0431665****[12]**N. I. Mushkhelishvili,*Singular Integral Equations*, Noordhoff, Groningen, 1953.**[13]**John R. Rice,*On the degree of convergence of nonlinear spline approximation*, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ. of Wisconsin, Madison, Wis., 1969) Academic Press, New York, 1969, pp. 349–365. MR**0267324****[14]**K. S. Thomas,*On the approximate solution of operator equations*, Numer. Math.**23**(1974/75), 231–239. MR**0373275****[15]**F. G. Tricomi,*Integral equations*, Pure and Applied Mathematics. Vol. V, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR**0094665**

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1981-0595052-9

Article copyright:
© Copyright 1981
American Mathematical Society