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.

Keywords:

• oscillatory Hilbert transform
• Filon–Clenshaw–Curtis method
• Gauss–Laguerre quadrature
• weak singularity
• Cauchy singularity
• steepest descent method

2010 AMS Subject Classifications:

• 65D32
• 65D30

Disclosure statement

No potential conflict of interest was reported by the author.