Efficient methods for highly oscillatory integrals with weak and Cauchy singularities

Abstract

In this paper, we present two fast and accurate numerical schemes for the approximation of highly oscillatory integrals with weak and Cauchy singularities. For analytical kernel functions, by using the Cauchy theorem in complex analysis, we transform the integral into two line integrals in complex plane, which can be calculated by some proper Gauss quadrature rules. For general kernel functions, the non-oscillatory and nonsingular part of the integrand is replaced by a polynomial interpolation in Chebyshev points, and the integral is then evaluated by using recurrence relations. Furthermore, several numerical experiments are shown to verify the validity of such methods .

Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels

“…Remark 1. From the convergence rates Corollary 3.1 and Theorem 3.1, compared with that in [19], the new scheme is of much fast convergence rate. It is also illustrated by the numerical results (see Section 4).…”

“…for α = −1, t = 0.3, Table 1 shows the results for relative error compared with results of integral (30) [19] in Table 2.…”

“…For analytic function f (x) the integral was rewritten in the form of sum of line integrals, wherein the integrands do not oscillate and decay exponentially. Moreover, Fang [19] established the Clenshaw-Curtis quadrature for ⨍…”

“…The relative error of Clenshaw-Curtis quadrature rule for integral (30)[19].Tables 3-5 represent results for relative error computed by Clenshaw-Curtis quadrature. As exact value we just have used that returned by the rule when a huge number of points is used.…”

Numerical Steepest Descent Method for Hankel Type of Hypersingular Oscillatory Integrals in Electromagnetic Scattering Problems

“…However, when wðxÞ = ðx + 1Þ α ð1 − xÞ β and α > −1, β > −1, the recursion formula for M j ðkÞ and d j is complicated. In [15], using the numerical steepest descent method, the following Cauchy principal value integral is calculated:…”