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A universal solution to one-dimensional oscillatory integrals

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Abstract

How to calculate the highly oscillatory integrals is the bottleneck that restraints the research of light wave and electromagnetic wave’s propagation and scattering. Levin method is a classical quadrature method for this type of integrals. Unfortunately it is susceptible to the system of linear equations’ ill-conditioned behavior. We bring forward a universal quadrature method in this paper, which adopts Chebyshev differential matrix to solve the ordinary differential equation (ODE). This method can not only obtain the indefinite integral’ function values directly, but also make the system of linear equations well-conditioned for general oscillatory integrals. Furthermore, even if the system of linear equations in our method is ill-conditioned, TSVD method can be adopted to solve them properly and eventually obtain accurate integral results, thus making a breakthrough in Levin method’s susceptivity to the system of linear equations’ ill-conditioned behavior.

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References

  1. Zhang Y X, Chi Z Y. Propagation and Imaging of Light in Atmosphere (in Chinese). Beijing: National Defence Industry Press, 1997

    Google Scholar 

  2. Ishimaru A. Wave Propagation and Scattering in Random Media. New York: Academic, 1978

    Google Scholar 

  3. Kong J A. Electromagnetic Wave Theory. New York: Wiley, 1986

    Google Scholar 

  4. Shariff K, Wray A. Analysis of the radar reflectivity of aircraft vortex wakes. J Fluid Mech, 2002, 463: 121–161

    Article  MATH  Google Scholar 

  5. Kincaid D, Cheney W. Numerical Analysis: Mathematics of Scientific Computing. Pacific Grove: Brooks/Cole Publishing Company, 2002

    Google Scholar 

  6. Lu W Z. Theory and Technology of Antenna (in Chinese). Xi’an: Press of Xidian University, 2004

    Google Scholar 

  7. Li S X. High Frequency Approximation of Wave Equation and Symplectic Geometry (in Chinese). Beijing: Science Press, 2001

    Google Scholar 

  8. Zhang C B. The Theory Analysis and Application of Synthetic Aperture Radar (in Chinese). Beijing: Science Press, 1989

    Google Scholar 

  9. Chew W C. Waves and Fields in Inhomogeneous Media. New York: Van Nostrand Reinhold, 1990

    Google Scholar 

  10. Li J C, Zhou X C. Approximate Method in Mathematics and Physics (in Chinese). Beijing: Science Press, 1998

    Google Scholar 

  11. Tatarskii V I. Theory of single scattering by random distributed scatterers. IEEE Trans Ant Prop, 2003, 51: 2806–2813

    Article  Google Scholar 

  12. Filon L N G. On a quadrature formula for trigonometric integrals. Proc Roy Soc, 1928. 49: 38–47

    Google Scholar 

  13. Levin D. Procedures for computing one-and two-dimensional integrals of functions with rapid irregular oscillations. Math Comp, 1982, 38(158): 531–538

    Article  MATH  MathSciNet  Google Scholar 

  14. Evans G A. An expansion method for irregular oscillatory integrals. J Comp Math, 1997, 63: 137–148

    Article  MATH  Google Scholar 

  15. Evans G A. An alternative method for irregular oscillatory integrals over a finite range. J Comp Math, 1994, 53: 185–193

    Article  Google Scholar 

  16. Evans G A, Webster J R. A comparison of some methods for the evaluation of highly oscillatory integrals. J Comp Appl Math, 1999, 112: 55–69

    Article  MATH  MathSciNet  Google Scholar 

  17. Evans G A, Webster J R. A high order, progressive method for the evaluation of irregular oscillatory integrals. Appl Num Math, 1997, 23: 205–218

    Article  MATH  MathSciNet  Google Scholar 

  18. Iserles A, Norsett S P. Efficient quadrature of highly oscillatory integrals using derivatives. Royal Society of London Proceedings Series A, 2005, 461(2057): 1383–1399

    Article  MATH  MathSciNet  Google Scholar 

  19. Iserles A, Norsett S P. Quadrature methods for multivariate highly oscillatory integrals using derivatives. Tech Rep, Cambridge: University of Cambridge, 2005.

    Google Scholar 

  20. Olver S. Moment-free numerical integration of highly oscillatory functions. IMA J Num Anal, 2006, 26(2): 213–227

    Article  MATH  MathSciNet  Google Scholar 

  21. Boyd J P. Chebyshev and Fourier Spectral Methods. New York: DOVER Publications, 2000

    Google Scholar 

  22. Gourgoulhon E. Introduction to spectral methods. Tech Rep, LUTH, 2002

  23. Fuming M, Yutang C. Numerical Approximation (in Chinese). Changchun: Publishing House of Jilin University, 2000

    Google Scholar 

  24. Canuto C, Hussaini M Y, Quarteroni A, et al. Spectral Methods in Fluid Dynamics. New York: Springer, 1988

    MATH  Google Scholar 

  25. Baltensperger R. Improving the accuracy of the matrix differentiation method for arbitrary collocation points. App Num Math, 2000, 33: 143–149

    Article  MATH  MathSciNet  Google Scholar 

  26. Baltensperger R, Berrut J P. Errata to “the erros in calculating the pseudospectral differentiation matrices for chebyshevgauss-lobatto points”. Comp Math Appl, 1999, 38: 119

    MathSciNet  Google Scholar 

  27. Baltensperger R, Berrut J P. The erros in calculating the pseudospectral differentiation matrices for chebyshev-gauss-lobatto points. Comp Math Appl, 1999, 37: 41–48

    Article  MATH  MathSciNet  Google Scholar 

  28. Elbarbary M, ElSayed M. Higher order pseudospectral differentiation matrices. Appl Num Math, 2005, 55: 425–438

    Article  MATH  MathSciNet  Google Scholar 

  29. Costa B, Don W S. On the computation of high order pseudospectral derivatives. Appl Num Math, 2000, 33: 151–159

    Article  MATH  MathSciNet  Google Scholar 

  30. Jiannbing L, Xuesong W, Tao W. Performance analysis of a unified quadrature method for one-dimensional oscillatory integrals. Appl Math Comp, Submitted for publication.

  31. Bakhvalov N S, Vasil’eva L G. Evaluation of the integrals of oscillating functions by interpolation at nodes of gaussian quadratures. Comp Math Phys, 1968, 8(1): 241–249

    Article  Google Scholar 

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Authors and Affiliations

  1. College of Electronic Science and Engineering, National University of Defense Technology, Changsha, 410073, China

    JianBing Li, XueSong Wang & Tao Wang

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  1. JianBing Li
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  2. XueSong Wang
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  3. Tao Wang
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Correspondence to JianBing Li.

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Li, J., Wang, X. & Wang, T. A universal solution to one-dimensional oscillatory integrals. Sci. China Ser. F-Inf. Sci. 51, 1614–1622 (2008). https://doi.org/10.1007/s11432-008-0121-2

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  • DOI: https://doi.org/10.1007/s11432-008-0121-2

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