A Galerkin solution to a regularized Cauchy singular integro-differential equation

Author:
Jay I. Frankel

Journal:
Quart. Appl. Math. **53** (1995), 245-258

MSC:
Primary 45E05; Secondary 45J05, 65R20

DOI:
https://doi.org/10.1090/qam/1330651

MathSciNet review:
MR1330651

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents a Galerkin approach for solving a regularized version of the Cauchy singular, linear integro-differential equation

**[1]**T. S. Sankar, S. V. Hoa, and V. I. Fabrikant,*Approximate solution of singular integro-differential equations in elastic contact problems*, Internat. J. Numer. Methods Engrg.**18**, 503-519 (1982)**[2]**F. Erdogan,*Approximate solutions of systems of singular integral equations*, SIAM J. Appl. Math.**17**, 1041-1059 (1969)**[3]**F. Erdogan and G. D. Gupta,*On the numerical solution of singular integral equations*, Quart. Appl. Math.**30**, 525-534 (1972)**[4]**A. Gerasoulis and R. P. Srivastav,*A method for the numerical solution of singular integral equations with a principal value integral*, Internat. J. Engrg. Sci.**19**, 1293-1298 (1981)**[5]**S. R. Bland,*The two-dimensional oscillating airfoil in a wind tunnel in subsonic flow*, SIAM J. Appl. Math.**18**, 830-848 (1970)**[6]**J. A. Fromme and M. A. Golberg,*Aerodynamics interface effects on oscillating airfoils with controls in ventilated wind tunnels*, AIAA J.**18**, 417-426 (1980)**[7]**M. A. Golberg,*The convergence of a collocation method for a class of Cauchy singular integral equations*, J. Math. Anal. Appl.**100**, 500-512 (1984)**[8]**M. A. Golberg and J. A. Fromme,*On the convergence of collocation for the generalized airfoil equation*, J. Math. Anal. Appl.**71**, 271-286 (1979)**[9]**A. Plotkin and S. S. Dodbele,*Slender wing in ground effect*, AIAA J.**26**, 493-494 (1988)**[10]**L. Dragos,*Numerical solution of the equation for a thin airfoil in ground effect*, AIAA J.**28**, 2132-2134 (1990)**[11]**R. D. Cess and S. N. Tiwari,*The interaction of thermal conduction and infrared gaseous radiation*, Appl. Sci. Res.**20**, 25-39 (1969)**[12]**E. M. Sparrow and R. D. Cess,*Radiation Heat Transfer*, McGraw-Hill, New York, 1978**[13]**M. A. Golberg, ed.,*Solution Methods for Integral Equations*, Plenum Press, New York, 1979**[14]**M. A. Golberg, ed.,*Numerical Solution of Integral Equations*, Plenum Press, New York, 1990**[15]**P. Linz,*Analytical and Numerical Methods for Volterra Equations*, SIAM, Philadelphia, 1985**[16]**C. T. H. Baker,*The Numerical Treatment of Integral Equations*, Clarendon Press, Oxford, 1978**[17]**L. M. Delves and J. L. Mohamed,*Computational Methods for Integral Equations*, Cambridge Univ. Press, Cambridge, 1988**[18]**K. E. Atkinson,*A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind*, SIAM, Philadelphia, 1976**[19]**F. G. Tricomi,*Integral Equations*, Dover, New York, 1985**[20]**H. Hochstadt,*Integral Equations*, Wiley, New York, 1973**[21]**C. D. Green,*Integral Equation Methods*, Barnes and Noble, New York, 1969**[22]**D. Porter and D. S. G. Stirling,*Integral Equations*, Cambridge Univ. Press, Cambridge, 1990**[23]**J. I. Frankel,*Regularized and preconditioned boundary integral solution to heat transfer in a participating gas flow between parallel plates*, Numer. Heat Transfer**19**, 105-126 (1991)**[24]**J. I. Frankel,*Nonlinear heat transfer: Solution of singular, nonlinear integral equations*, Engrg. Anal.**8**, 231-238 (1991)**[25]**A. S. Peters,*A note on the integral equation of the first kind with a Cauchy kernel*, Comm. Pure Appl. Math.**16**, 57-61 (1963)**[26]**A. S. Peters,*Abel's equation and the Cauchy integral equation of the second kind*, Comm. Pure Appl. Math.**21**, 51-65 (1968)**[27]**A. C. Kaya and F. Erdogan,*On the solution of integral equations with strongly singular kernels*, Quart. Appl. Math.**45**, 105-122 (1987)**[28]**A. C. Kaya and F. Erdogan,*On the solution of integral equations with strong singularities*, in Numerical Solution of Singular Integral Equations (A. Gersoulis and V. Vichnevetsky, eds.), IMACS, 1984, pp. 54-57**[29]**N. I. Ioakimidis,*On the numerical solution of Cauchy type singular integral equations by the collocation method*, Appl. Math. Comp.**12**, 49-60 (1983)**[30]**N. I. Ioakimidis,*Further convergence results for the weighted Galerkin method of numerical solution of Cauchy-type singular integral equations*, Math. Comp.**41**, 79-85 (1983)**[31]**Chien-Ke Lu,*A class of quadrature formulas of Chebyshev type for singular integrals*, J. Math. Anal. Appl.**100**, 416-435 (1984)**[32]**J. I. Frankel, unpublished analysis and computations, 1992**[33]**L. C. Andrews,*Special Functions of Mathematics for Engineers*, 2nd ed., McGraw-Hill, New York, 1992**[34]**T. J. Rivlin,*The Chebyshev Polynomials*, Wiley, New York, 1974**[35]**M. Abramowitz and I. A. Stegun, eds.,*Handbook of Mathematical Functions*, Dover, New York, 1972**[36]**K. E. Atkinson,*An Introduction to Numerical Analysis*, 2nd ed., Wiley, New York, 1989

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DOI:
https://doi.org/10.1090/qam/1330651

Article copyright:
© Copyright 1995
American Mathematical Society