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Lobatto-Jacobi numerical integration rule; Cauchy type principal value integrals; singular integral equations; Cauchy type kernels
The Lobatto-Jacobi numerical integration rule can be extended so as to apply to the numerical evaluation of Cauchy type principal value integrals and the numerical solution of singular intergral equations with Cauchy type kernels by reduction to systems of linear equations. To this end, the integrals in such a singular integral equation are replaced by sums, as if they were regular integrals, after the singular integral equation is applied at appropriately selected points of the integration interval. An application of this method of numerical solution of singular integral equations is made in the case of a problem of the theory of plane elasticity.
[1] P. S. Theocaris, N. I. Ioakimidis: On the Gauss-Jacobi Numerical Integration Method Applied to the Solution of Singular Integral Equations. To appear in the "Bulletin of the Calcutta Mathematical Society", 1979. MR 0583924 | Zbl 0429.65115
[2] Z. Kopal: Numerical Analysis. Chapman and Hall, London, 1961. Zbl 0101.33701
[3] P. S. Theocaris, N. I. Ioakimidis: Numerical Integration Methods for the Solution of Singular Integral Equations. Quarterly of Applied Mathematics, 1977, Vol.35, 173-183. MR 0445873 | Zbl 0353.45016
[4] P. S. Theocaris, N. I. Ioakimidis: Application of the Gauss, Radau and Lobatto Numerical Integration Rules to the Solution of Singular Integral Equations. To appear in the "Zeitschrift für angewandte Mathemaiik und Mechanik", 1978. MR 0516804 | Zbl 0399.65095
[5] N. I. Ioakimidis, P. S. Theocaris: On the Numerical Solution of a Class of Singular Integral Equations. Journal of Mathematical and Physical Sciences, 1977, Vol. 11, pp. 219-235. MR 0483590
[6] N. I. Ioakimidis, P. S. Theocaris: On the Numerical Evaluation of Cauchy Principal Value Integrals. Revue Roumaine des Sciences Techniques- Série de Mécanique Appliquée, 1977, Vol. 22 pp. 803-818. MR 0483321 | Zbl 0376.65009
[7] F. D. Gakhov: Boundary Value Problems. Pergamon Press, Oxford, 1966 [Translation of: Krayevye Zadachi, Fizmatgiz, Moscow, 1963]. MR 0198152 | Zbl 0141.08001
[8] G. Szegö: Orthogonal Polynomials. (revised edition). American Mathematical Society, New York, 1959. MR 0106295
[9] D. Elliott: Uniform Asymptotic Expansions of the Jacobi Polynomials and an Associated Function. Mathematics of Computation, 1971, Vol. 25, No 114, pp. 309-315. DOI 10.1090/S0025-5718-1971-0294737-5 | MR 0294737 | Zbl 0221.65027
[10] A. Erdélyi(ed.), al.: Higher Transcendental Functions, Vol. II. McGraw-Hill, New York, 1953. Zbl 0052.29502
[11] F. Erdogan G. D. Gupta, T. S. Cook: Numerical Solution of Singular Integral Equations. In: Mechanics of Fracture, Vol. 1: Methods of Analysis and Solutions of Crack Problems (edited by G. C. Sih). Noordhoff, Leyden, the Netherlands, 1973, Chap. 7, pp. 368--425. MR 0471394
[12] M. B. Porter: On the Roots of the Hypergeometric and Bessel's Functions. American Journal of Mathematics, 1898, Vol. 20, pp. 193-214. MR 1505769
[13] T. S. Cook, F. Erdogan: Stresses in Bonded Materials with a Crack Perpendicular to the Interface. International Journal of Engineering Science,, 1972, Vol. 10, pp. 677-697. DOI 10.1016/0020-7225(72)90063-8 | Zbl 0237.73096
[14] F. Erdogan, T. S. Cook: Antiplane Shear Crack Terminating at and Going Through a Bimaterial Interface. International Journal of Fracture, 1974, Vol. 10, pp. 227-240. DOI 10.1007/BF00113928
[15] K. Y. Lin, J. W. Mar: Finite Element Analysis of Stress Intensity Factors for Cracks at a Bi-Material Interface. International Journal of Fracture, 1976, Vol. 12, pp. 521-531.
[16] E. Smith: A Pile-up of Dislocations in a Bi-Metallic Solid. Scripta Metallurgica, 1969, Vol. 3, pp. 415-418.
[17] T. W. Chou: Dislocation Pileups and Elastic Cracks at a Bimaterial Interface. Metallurgical Transactions, 1970, Vol. 1, pp. 1245-1248.
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