Numerical integration methods for the solution of singular integral equations

Abstract

Summary.The evaluation of the stress intensity factors at the tips of a crack in a homogeneous isotropic and elastic medium may be achieved with higher accuracy and much less computation if the Lobatto-Chebyshev method of numerical solution of the corresponding system of singular integral equations is used instead of the method of Gauss-Chebyshev commonly applied to such problems. Comparison of results obtained by the two numerical methods when applied to the problem of a cruciform crack in an infinite medium proves the potentialities of the new approach.

On a method for solving Prandtl's integro-differential equation applied to problems of continuum mechanics using polynomial approximations

“…Using a known procedure [26,27], the SIE (2.17) under condition (2.9), reduces to the following system of linear equations:…”

“…In recent decades, along with the analytical methods of solving this SIE based on boundary value problems of the theory of analytic functions, the effective numerical-analytical method for solving more general SIE, developed by F. Erdogan, G. D. Gupta, and T. S. Cook [26], P. S. Teocaris and N. I. Loakimidis [27] has been widely used. This method is based on Gaussian quadrature formula for the singular integral with Cauchy kernel, obtained by I. Stark [28] and A.…”

“…For this purpose the solution is represented as a product of a function of the Holder function class, the unknown function, and a positive weight function, characterizing singularities of the solution of the original equation. The Holder function is then approximated by Bernstein polynomials or by the Chebyshev polynomials of the first kind or Langrange interpolation polinomial at Chebyshev knots which coincide with the roots of a Chebyshev polynomials of the first and second kinds (Multhopps method), and according to [26][27][28][29] the original Prandtl IDE is reduced to a SLAE. The results obtained by these methods are compared with those calculated by the numerical analysis of the Prandtl IDE in question in particular cases.…”

On the numerical solution of singular integro-differential equations

“…Further, Erdogan and Gupta [3] succeeded in getting rid of the variable transformation, and thus in working along the real integration interval (-1, 1). Theocaris and Ioakimidis [4] proved that the method of Erdogan and Gupta was in reality equivalent to the application of the Gauss-Chebyshev numerical integration rule to Cauchy-type principal value integrals. In the same work, they also developed a new technique for solving singular integral equations, based on the Lobatto-Chebyshev numerical integration rule.…”

“…A considerable number of these contributions were due to Theocaris and Ioakimidis, who in a series of publications (see e.g. [6][7][8][9][10]) have generalized the results of [1][2][3][4] to the most general cases of singular integral equations of the first or the second kind, with constant or variable coefficients, with regular of generalized Fredholm kernels (besides the Cauchy kernels), with weight functions presenting complex singularities at the endpoints of the integration intervals or pairs of such singularities, etc., so that any practical case of singular integral equations could be faced. Although any numerical integration rule can be used for the solution of singular integral equations (Gauss, Radau and Lobatto rules, rules with completely preassigned abscissae, etc.…”