Fast Computation of Singular Oscillatory Fourier Transforms

We consider the problem of the numerical evaluation of singular oscillatory Fourier transforms , where . Based on substituting the original interval of integration by the paths of steepest descent, if is analytic in the complex region containing [, ], the computation of integrals can be transformed into the problems of integrating two integrals on [0, ∞) with the integrand that does not oscillate and decays exponentially fast, which can be efficiently computed by using the generalized Gauss Laguerre quadrature rule. The efficiency and the validity of the method are demonstrated by both numerical experiments and theoretical results. More importantly, the presented method in this paper is also a great improvement of a Filon-type method and a Clenshaw-Curtis-Filon-type method shown in Kang and Xiang (2011) and the Chebyshev expansions method proposed in Kang et al. (2013), for computing the above integrals.

[1]  L. Filon.  III.—On a Quadrature Formula for Trigonometric Integrals. , 1930 .

[2]  Yudell L. Luke,et al.  On the computation of oscillatory integrals , 1954, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  A. Erdélyi,et al.  Asymptotic Representations of Fourier Integrals and the Method of Stationary Phase , 1955 .

[4]  E. A. Flinn.  A Modification of Filon's Method of Numerical Integration , 1960, JACM.

[5]  C. W. Clenshaw,et al.  A method for numerical integration on an automatic computer , 1960 .

[6]  By J. N. Lyness.  Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature , 1971 .

[7]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[8]  David Levin,et al.  Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations , 1982 .

[9]  E. Stein,et al.  Singular integrals related to the Radon transform and boundary value problems. , 1983, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Josef Stoer,et al.  Numerische Mathematik 1 , 1989 .

[11]  M. Hamed,et al.  A Numerical Integration Formula for the Solution of the Singular Integral Equation for Classical Crack Problems in Plane and Antiplane Elasticity , 1991 .

[12]  David Levin,et al.  Analysis of a collocation method for integrating rapidly oscillatory functions , 1997 .

[13]  J. R. Webster,et al.  A method to generate generalized quadrature rule for oscillatory integrals , 2000 .

[14]  Roderick Wong,et al.  Asymptotic approximations of integrals , 1989, Classics in applied mathematics.

[15]  A. Iserles,et al.  Efficient quadrature of highly oscillatory integrals using derivatives , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[16]  Sheehan Olver,et al.  Moment-free numerical integration of highly oscillatory functions , 2006 .

[17]  A class of oscillatory singular integrals on triebel-lizorkin spaces , 2006 .

[18]  G. Evans,et al.  Evaluating infinite range oscillatory integrals using generalised quadrature methods , 2007 .

[19]  Ivan G. Graham,et al.  A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering , 2007, Numerische Mathematik.

[20]  S. Xiang.  Efficient Filon-type methods for $$\int_a^bf(x)\,{\rm e}^{{\rm i}\omega g(x)}\,{\rm d}x$$ , 2007 .

[21]  James N. Lyness.  Numerical evaluation of a fixed-amplitude variable-phase integral , 2008, Numerical Algorithms.

[22]  J. N. Lyness,et al.  Asymptotic expansions for oscillatory integrals using inverse functions , 2009 .

[23]  Shuhuang Xiang,et al.  On the evaluation of Cauchy principal value integrals of oscillatory functions , 2010, J. Comput. Appl. Math..

[24]  Xiaojun Chen,et al.  Error bounds for approximation in Chebyshev points , 2010, Numerische Mathematik.

[25]  Shuhuang Xiang,et al.  On the calculation of highly oscillatory integrals with an algebraic singularity , 2010, Appl. Math. Comput..

[26]  Ivan G. Graham,et al.  Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals , 2011 .

[27]  Shuhuang Xiang,et al.  Efficient integration for a class of highly oscillatory integrals , 2011, Appl. Math. Comput..

[28]  Shuhuang Xiang,et al.  Clenshaw–Curtis–Filon-type methods for highly oscillatory Bessel transforms and applications , 2011 .

[29]  Shuhuang Xiang,et al.  Computation of integrals with oscillatory and singular integrands using Chebyshev expansions , 2013, J. Comput. Appl. Math..