Quadrature methods for highly oscillatory singular integrals

نویسندگان

  • Jing GAO
  • Marissa CONDON
چکیده

We study asymptotic expansions, Filon-type methods and complex-valued Gaussian quadrature for highly oscillatory integrals with power-law and logarithmic singularities. We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand, the stationary points and the endpoints of the integral. A truncated asymptotic expansion achieves an error that decays faster for increasing frequency. Based on the asymptotic analysis, a Filon-type method is constructed to approximate the integral. Unlike the asymptotic method, the Filon method achieves high accuracy for both small ω and large ω. The complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function. Numerical results indicate that the complexvalued Gaussian quadrature achieves the highest accuracy when the three methods are compared. 2010 Mathematics Subject Classification: 65D32, 41A55.

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تاریخ انتشار 2016