Skip to main content
SpringerLink
Log in

Numerical Solution of Highly Oscillatory Nonlinear Integrals Using Quasi-Monte Carlo Methods

  • Conference paper
  • First Online:
Mathematical Analysis and its Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 143))

  • 1695 Accesses

Abstract

Highly oscillating integrals occur in many engineering applications. This paper discusses the quasi-Monte Carlo methods for calculation of the highly oscillating integrals using a low discrepancy sequence. We evaluated the highly oscillating integrals using a low discrepancy sequence known as Vander Corput sequence. The theoretical error bounds are calculated and are compared with analytical results. The reliability of the quasi-Monte Carlo methods is compared with He’s homotopy perturbation method.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Asheim, S.: A combined Filon/Asymptotic quadrature method for highly oscillatory problems. BIT Numer. Math. 48, 425–448 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  3. Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer-verlag, Berlin (2008)

    Book  MATH  Google Scholar 

  4. Iserles, A., Norsett, S.P., Efficient quadrature of highly oscillatory integrals using derivatives, Proc. Roy. Soc. Edinburgh, A. 461,1383–1399 (2005)

    Google Scholar 

  5. Iserles, A., Norsett, S.P., Olver, S., Highly oscillating quadrature: The story so far

    Google Scholar 

  6. Jain, M.K., Iyengar, S.R.K., Jain, R.K.: Introduction to Numerical Analysis. Narosa Publishers, New Delhi (1993)

    Google Scholar 

  7. Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. John-Wiley & Sons, Newyork (1974)

    MATH  Google Scholar 

  8. Kunik, M., Skrzpacz, P.: Diffraction of light revisited. Math. Methods. Appl. Sci. 31(7), 793–820 (2007)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Molabahrami, A., Khani. F.: Numerical solutions of highly oscillatory integrals. Appl. Math. Comput. 198, 657–664 (2008)

    Google Scholar 

  11. Oseledets, I.V., Stavtsev, S.L., Tyrtyshnikov, E.E.: Integration of oscillating functions in a quasi-three-dimensional electrodynamic problem. Comput. Math. Math. Phys. 49(2), 292–303 (2009)

    Article  MathSciNet  Google Scholar 

  12. Walter R.: Real and complex analysis, 3rd edition, McGraw-Hill Company, New Delhi, 1987

    Google Scholar 

  13. Wu, X.Y., Xia, J.L.: Two low accuracy methods for stiff systems. Appl. Math. Comput. 123, 141–153 (2001)

    Google Scholar 

  14. Xiang, S.: Efficient quadrature for highly oscillatory integrals involving critical points. J. Comput. Appl. Math. 206, 688–698 (2006). doi:10.1016/j.cam.2006.08.018

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rajiv Gandhi University of Knowledge Technologies, RK Valley, YSR Dist., AP, 516329, India

    Nageswara Rao Narni

Authors
  1. Nageswara Rao Narni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nageswara Rao Narni .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    P. N. Agrawal

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    R. N. Mohapatra

  3. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    Uaday Singh

  4. Dept. of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    H. M. Srivastava

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer India

About this paper

Cite this paper

Narni, N.R. (2015). Numerical Solution of Highly Oscillatory Nonlinear Integrals Using Quasi-Monte Carlo Methods. In: Agrawal, P., Mohapatra, R., Singh, U., Srivastava, H. (eds) Mathematical Analysis and its Applications. Springer Proceedings in Mathematics & Statistics, vol 143. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2485-3_27

Download citation

Publish with us

Policies and ethics

Search

Navigation