Fast integration of highly oscillatory integrals with exotic oscillators

In this paper, we present an efficient Filon-type method for the integration of systems containing Bessel functions with exotic oscillators based on a diffeomorphism transformation and give applications to Airy transforms. Preliminary numerical results show the effectiveness and accuracy of the quadrature for large arguments of integral systems.

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

[2]  Sheehan Olver,et al.  Moment-free numerical approximation of highly oscillatory integrals with stationary points , 2007, European Journal of Applied Mathematics.

[3]  Sheehan Olver,et al.  Numerical approximation of vector-valued highly oscillatory integrals , 2007 .

[4]  Shuhuang Xiang,et al.  Numerical quadrature for Bessel transformations , 2008 .

[5]  Robert Piessens,et al.  Automatic computation of Bessel function integrals , 1984 .

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

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

[8]  Shuhuang Xiang,et al.  Efficient Filon-type methods for (∫abf(x), eiωg(x), dx) , 2007, Numerische Mathematik.

[9]  Arieh Iserles,et al.  Highly Oscillatory Quadrature and Its Applications , 2005 .

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

[11]  Jean-François Richard,et al.  Methods of Numerical Integration , 2000 .

[12]  Daan Huybrechs,et al.  A Sparse Discretization for Integral Equation Formulations of High Frequency Scattering Problems , 2007, SIAM J. Sci. Comput..

[13]  A. Iserles,et al.  On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation , 2004 .

[14]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

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

[16]  Lloyd N. Trefethen,et al.  Is Gauss Quadrature Better than Clenshaw-Curtis? , 2008, SIAM Rev..

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

[18]  R. Piessens,et al.  Modified clenshaw-curtis method for the computation of Bessel function integrals , 1983 .

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

[20]  Daan Huybrechs,et al.  On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation , 2006, SIAM J. Numer. Anal..

[21]  Shuhuang Xiang,et al.  Numerical analysis of a fast integration method for highly oscillatory functions , 2007 .

[22]  Shuhuang Xiang,et al.  On generalized quadrature rules for fast oscillatory integrals , 2008, Appl. Math. Comput..

[23]  A. Iserles,et al.  Highly Oscillatory Quadrature: The Story soFar , 2006 .

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

[25]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[26]  David Levin,et al.  Fast integration of rapidly oscillatory functions , 1996 .

[27]  G. A. Evans,et al.  Some theoretical aspects of generalised quadrature methods , 2003, J. Complex..

[28]  Timothy S. Murphy,et al.  Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals , 1993 .

[29]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .