Summary
As is well-known, an efficient numerical technique for the solution of Cauchy-type singular integral equations along an open interval consists in approximating the integrals by using appropriate numerical integration rules and appropriately selected collocation points. Without any alterations in this technique, it is proposed that the estimation of the unknown function of the integral equation is further achieved by using the Hermite interpolation formula instead of the Lagrange interpolation formula. Alternatively, the unknown function can be estimated from the error term of the numerical integration rule used for Cauchy-type integrals. Both these techniques permit a significant increase in the accuracy of the numerical results obtained with an insignificant increase in the additional computations required and no change in the system of linear equations solved. Finally, the Gauss-Chebyshev method is considered in its original and modified form and applied to two crack problems in plane isotropic elasticity. The numerical results obtained illustrate the powerfulness of the method.
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References
N. I. Ioakimidis and P. S. Theocaris, The numerical evaluation of a class of generalized stress intensity factors by use of the Lobatto-Jacobi numerical integration rule, International Journal of Fracture 14 (1978) 469–484.
F. Erdogan and G. D. Gupta, On the numerical solution of singular integral equations, Quarterly of Applied Mathematics 29 (1972) 525–534.
F. Erdogan, G. D. Gupta and T. S. Cook, Numerical solution of singular integral equations; contained in: G. C. Sih (ed.), Methods of analysis and solutions of crack problems (Vol. 1 in the series Mechanics of fracture), Noordhoff, Leyden (1973), Chap. 7, pp. 368–425.
P. S. Theocaris and N. I. Ioakimidis, Numerical integration methods for the solution of singular integral equations, Quarterly of Applied Mathematics 35 (1977) 173–183.
N. I. Ioakimidis, General methods for the solution of crack problems in the theory of plane elasticity, Doctoral thesis at the National Technical University of Athens, Athens (1976) [Univ. Micr. order no. 76-21,056].
P. S. Theocaris and N. I. Ioakimidis, Numerical solution of Cauchy-type singular integral equations, Transactions of the Academy of Athens 40 (No. 1) (1977) 1–39.
D. F. Paget and D. Elliott, An algorithm for the numerical evaluation of certain Cauchy principal-value integrals, Numerische Mathematik 19 (1972) 373–385.
S. Krenk, On the use of the interpolation polynomial for solutions of singular integral equations, Quarterly of Applied Mathematics 32 (1975) 479–484.
F. B. Hildebrand, Introduction to numerical analysis, Tata McGraw-Hill Co. Ltd., New Delhi (1956), pp. 314–317.
N. I. Ioakimidis and P. S. Theocaris, On the numerical evaluation of Cauchy principal value integrals, Revue Roumaine des Sciences Techniques — Série de Mécanique Appliquée 22 (1977) 803–818.
L. V. Kantorovich and V. I. Krylov, Approximate methods of higher analysis, Interscience Publishers and P. Noordhoff, Groningen (1958), pp. 98–103.
N. I. Ioakimidis and P. S. Theocaris, On the selection of collocation points for the numerical solution of singular integral equations with generalized kernels appearing in elasticity problems, Computers and Structures (to appear).
N. I. Ioakimidis and P. S. Theocaris, On the numerical solution of singular integrodifferential equations, Quarterly of Applied Mathematics (to appear).
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Theocaris, P.S., Ioakimidis, N.I. A remark on the numerical solution of singular integral equations and the determination of stress-intensity factors. J Eng Math 13, 213–222 (1979). https://doi.org/10.1007/BF00036670
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DOI: https://doi.org/10.1007/BF00036670