Gauss-Chebyshev quadrature formulae for strongly singular integrals
Author:
Alexander M. Korsunsky
Journal:
Quart. Appl. Math. 56 (1998), 461-472
MSC:
Primary 65D32
DOI:
https://doi.org/10.1090/qam/1637040
MathSciNet review:
MR1637040
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Abstract: This paper presents some explicit results concerning an extension of the mechanical quadrature technique, namely, the Gauss-Jacobi numerical integration scheme, to the class of integrals whose kernels exhibit second order of singularity (i.e., one degree more singular than Cauchy). In order to ascribe numerical values to these integrals they must be understood in Hadamard’s finite-part sense. The quadrature formulae are derived from those for Cauchy singular integrals.
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G. Eason, B. Noble, and I. N. Sneddon, On certain integrals of Lipschitz-Hankel type involving products of Bessel functions, Trans. Roy. Soc. 247, 529–551 (1955)
A. Erdelyi, Higher Transcendental Functions, vol. 2, McGraw-Hill, New York, 1953
F. Erdogan, G. D. Gupta, and T. S. Cook, Numerical Solution for Singular Integral Equations, Mechanics of Fracture, vol. 1, Noordhoff, Leiden, 1973, pp. 368–425
I. Gel’fand and V. Shilov, Generalized Functions, Academic Press, 1967
J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1952
D. A. Hills, D. N. Dai, P. A. Kelly, and A. M. Korsunsky, Singular Integral Equations in the Mechanics of Fracture, Kluwer, Dordrecht, 1995
N. I. Ioakimidis and P. S. Theocaris, On the numerical solution of singular integrodifferential equations, Quart. Appl. Math. 37, 325–331 (1979)
A. C. Kaya, Applications of Integral Equations with Strong Singularities in Fracture Mechanics, Ph. D. Thesis, Lehigh University, 1984
A. C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. Appl. Math. 45, 105–122 (1987)
A. M. Korsunsky, The Solution of Axisymmetric Crack Problems in Inhomogeneous Media, D. Phil. Thesis, University of Oxford, 1994
A. M. Korsunsky and D. A. Hills, The solution of plane crack problems by dislocation dipole procedures, J. Strain Anal. 30, 21–27 (1995)
A. M. Korsunsky, Fundamental eigenstrain solutions for axisymmetric crack problems, J. Mech. Phys. Solids 43, 1221–1241 (1995)
H. R. Kutt, Quadrature Formulae for Finite-Part Integrals, Spec. Report WISK 178, Nat. Res. Inst. for Math. Sci., Pretoria, 1975
H. R. Kutt, The numerical evaluation of principal value integrals by finite-part integration, Numer. Math. 24, 205–210 (1975)
M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions, Cambridge University Press, Cambridge, 1959
G. Monegato, Numerical evaluation of hypersingular integrals, J. Comp. Appl. Math. 50, 9–31 (1994)
G. Monegato, On the weights of certain quadratures for the numerical evaluation of Cauchy principal value integrals and their derivatives, Numer. Math. 50, 273–281 (1987)
Y. Murakami, ed., Stress Intensity Factors Handbook, Pergamon Press, 1987
D. F. Paget, The numerical evaluation of Hadamard finite-part integrals, Numer. Math. 36, 447–453 (1981)
N. J. Salamon and G. G. Walter, Limits of Lipschitz-Hankel integrals, J. Inst. Math. Appl. 24, 237–254 (1979)
G. Szegö, Orthogonal Polynomials, Colloquium Publications, vol. XXIII, Amer. Math. Soc., Providence, RI, 1939
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