Approximate Methods for Calculating Singular and Hypersingular Integrals with Rapidly Oscillating Kernels

Abstract

The article is devoted to the issue of construction of an optimal with respect to order passive algorithms for evaluating Cauchy and Hilbert singular and hypersingular integrals with oscillating kernels. We propose a method for estimating lower bound errors of quadrature formulas for singular and hypersingular integral evaluation. Quadrature formulas were constructed for implementation of the obtained estimates. We constructed quadrature formulas and estimated the errors for hypersingular integrals with oscillating kernels. This method is based on using similar results obtained for singular integrals.

Regular, Singular and Hypersingular Integrals over Fractal Contours

“…Direct calculations are preformed using an eight-point Gauss-Legendre rule, which guarantees eight significant digits in our case. The results of direct calculations of the integral agree well with the calculations performed on the base of Formula (11). Again, the values that we report here are based on the same convention as in the previous example.…”

“…The stability of quadrature and cubature formulas for one-and multi-dimensional singular integrals has been studied in the monograph [11]. Upper bounds of the errors for a number of cubature formulas have been obtained assuming an ε-perturbation of the integrands.…”

“…There are numerous publications devoted to numerical evaluation of SI and HI over smooth curves (see [4,8,[11][12][13] and the literature therein). The numerical approach to the evaluation of SI and HI over fractals is different.…”

“…We perform two versions of calculations for the corresponding contour integrals: the first is based on Formula (7) and its analog for the singular case (11), and the second corresponds to a direct calculation of the integral over the corresponding prefractals with mid-point or trapezoidal rules. The results of calculations for the integral Γ f 1,2 (z)dz for regular functions are given in Table 1.…”

“…Let us estimate the second contribution to the error of cubature (11). This estimate essentially depends on the topology of the fractal Γ.…”