- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 984834, 8 pages
Fast Computation of Singular Oscillatory Fourier Transforms
Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China
Received 1 April 2014; Accepted 3 July 2014; Published 17 July 2014
Academic Editor: Chuanzhi Bai
Copyright © 2014 Hongchao Kang and Xinping Shao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, NY, USA, 2nd edition, 1984.
- L. N. G. Filon, “On a quadrature formula for trigonometric integrals,” Proceedings of the Royal Society of Edinburgh, vol. 49, pp. 38–47, 1928.
- E. A. Flinn, “A modification of Filon's method of numerical integration,” Journal of the Association for Computing Machinery, vol. 7, pp. 181–184, 1960.
- Y. L. Luke, “On the computation of oscillatory integrals,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 50, pp. 269–277, 1954.
- C. W. Clenshaw and A. R. Curtis, “A method for numerical integration on an automatic computer,” Numerische Mathematik, vol. 2, pp. 197–205, 1960.
- S. Xiang, X. Chen, and H. Wang, “Error bounds for approximation in Chebyshev points,” Numerische Mathematik, vol. 116, no. 3, pp. 463–491, 2010.
- A. Iserles and S. P. Nørsett, “Efficient quadrature of highly oscillatory integrals using derivatives,” Proceedings of The Royal Society of London A, vol. 461, no. 2057, pp. 1383–1399, 2005.
- S. Xiang, “Efficient Filon-type methods for ,” Numerische Mathematik, vol. 105, no. 4, pp. 633–658, 2007.
- D. Levin, “Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations,” Mathematics of Computation, vol. 38, no. 158, pp. 531–538, 1982.
- D. Levin, “Analysis of a collocation method for integrating rapidly oscillatory functions,” Journal of Computational and Applied Mathematics, vol. 78, no. 1, pp. 131–138, 1997.
- K. C. Chung, G. A. Evans, and J. R. Webster, “A method to generate generalized quadrature rules for oscillatory integrals,” Applied Numerical Mathematics, vol. 34, no. 1, pp. 85–93, 2000.
- G. A. Evans and K. C. Chung, “Evaluating infinite range oscillatory integrals using generalised quadrature methods,” Applied Numerical Mathematics, vol. 57, no. 1, pp. 73–79, 2007.
- S. Olver, “Moment-free numerical integration of highly oscillatory functions,” IMA Journal of Numerical Analysis, vol. 26, no. 2, pp. 213–227, 2006.
- N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals, Holt, Rinehart and Winston, New York, NY, USA, 1975.
- A. Erdélyi, Asymptotic Expansions, Dover, New York, NY, USA, 1956.
- R. Wong, Asymptotic Approximations of Integrals, SIAM, Philadelphia, PA, USA, 2001.
- H. Wang and S. Xiang, “On the evaluation of Cauchy principal value integrals of oscillatory functions,” Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 95–100, 2010.
- H. Kang and S. Xiang, “On the calculation of highly oscillatory integrals with an algebraic singularity,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 3890–3897, 2010.
- H. Brunner, “On the numerical solution of first-kind Volterra integral equations with highly oscillatory kernels,” in Highly Oscillatory Problems: From Theory to Applications, pp. 13–17, Isaac Newton Institute, 2010.
- H. Brunner, Open Problems in the Computational Solution of Volterra Functional Equations with Highly Oscillatory Kernels, HOP 2007: Effective Computational Methods for Highly Oscillatory Solutions, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, 2007.
- S. Lu, “A class of oscillatory singular integrals,” International Journal of Applied Mathematical Sciences, vol. 2, no. 1, pp. 47–64, 2005.
- D. H. Phong and E. M. Stein, “Singular integrals related to the Radon transform and boundary value problems,” Proceedings of the National Academy of Sciences of the United States of America, vol. 80, no. 24, pp. 7697–7701, 1983.
- M. A. Hamed and B. Cummins, “Numerical integration formula for the solution of the singular integral equation for classical crack problems in plane and antiplane elasticity,” Journal of King Saud University, vol. 3, no. 2, pp. 217–231, 1991.
- V. Domínguez, I. G. Graham, and V. P. Smyshlyaev, “A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering,” Numerische Mathematik, vol. 106, no. 3, pp. 471–510, 2007.
- V. Domínguez, I. G. Graham, and V. P. Smyshlyaev, “Stability and error estimates for Filon-Clenshaw-CURtis rules for highly oscillatory integrals,” IMA Journal of Numerical Analysis, vol. 31, no. 4, pp. 1253–1280, 2011.
- A. Erdélyi, “Asymptotic representations of Fourier integrals and the method of stationary phase,” Journal of the Society for Industrial and Applied Mathematics, vol. 3, pp. 17–27, 1955.
- M. L. Lighthill, Introduction to Fourier Analisis and Generalized Functions, Cambrige University Press, New York, NY, USA, 1958.
- J. N. Lyness, “Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature,” Mathematics of Computation, vol. 25, pp. 87–104, 1971.
- J. N. Lyness, “Numerical evaluation of a fixed-amplitude variable-phase integral,” Numerical Algorithms, vol. 49, no. 1–4, pp. 235–249, 2008.
- J. N. Lyness and J. W. Lottes, “Asymptotic expansions for oscillatory integrals using inverse functions,” BIT: Numerical Mathematics, vol. 49, no. 2, pp. 397–417, 2009.
- H. Kang and S. Xiang, “Efficient integration for a class of highly oscillatory integrals,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3553–3564, 2011.
- H. Kang, S. Xiang, and G. He, “Computation of integrals with oscillatory and singular integrands using Chebyshev expansions,” Journal of Computational and Applied Mathematics, vol. 242, pp. 141–156, 2013.
- S. Xiang, Y. Cho, H. Wang, and H. Brunner, “Clenshaw-curtis-filon-type methods for highly oscillatory bessel transforms and applications,” IMA Journal of Numerical Analysis, vol. 31, no. 4, pp. 1281–1314, 2011.
- P. Henrici, Applied and Computational Complex Analysis, vol. 1, John Wiley & Sons, New York, NY, USA, 1974.
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, USA, 1964.