Skip to main content

An analogue of the four-point Newton-Cotes formula for a function with a boundary-layer component

Numerical Analysis and Applications volume 6, pages 268–278 (2013)Cite this article

Abstract

The construction of the Newton-Cotes formulas is based on approximating the integrand by a Lagrange polynomial. The error of such quadrature formulas can be great for a function with a boundary-layer component. In this paper, an analog of the four-point Newton-Cotes rule is constructed. The construction is based on using a nonpolynomial interpolation that is exact for the boundary layer component. Error estimates of the quadrature rule independent of the boundary layer component gradients are obtained. Numerical experiments are performed.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Berezin, I.S. and Zhidkov, N.P., Metody vychislenii (Calculation Methods), Moscow: Nauka, 1966.

    Google Scholar 

  2. Bakhvalov, N.S., Chislennye metody (Numerical Methods), Moscow: Nauka, 1975.

    Google Scholar 

  3. Zadorin, A.I., An Interpolation Method for a Problem with a Boundary Layer, Sib. Zh. Vych. Mat., 2007, vol. 10, no. 3, pp. 267–275.

    MathSciNet  Google Scholar 

  4. Zadorin, A.I. and Zadorin, N.A., Spline Interpolation on a Uniform Grid of a Function with a Boundary-Layer Component, Zh. Vych. Mat. Mat. Fiz., 2010, vol. 50, no. 2, pp. 221–233.

    MathSciNet  MATH  Google Scholar 

  5. Zadorin, A.I., Spline Interpolation of Functions with a Boundary Layer Component, Int. J. Num. Anal. Model., Ser. B., 2011, vol. 2, nos. 2/3, pp. 562–579.

    MathSciNet  Google Scholar 

  6. Zadorin, A.I. and Zadorin, N.A., Quadrature Formulas for Functions with a Boundary-Layer Component, Zh. Vych. Mat. Mat. Fiz., 2011, vol. 51, no. 11, pp. 1952–1962.

    MathSciNet  MATH  Google Scholar 

  7. Shishkin, G.I., Setochnye approksimatsii singulyarno vozmushchennykh ellipticheskikh i parabolicheskikh uravnenii (Grid Approximations of Singular Perturbation Elliptic and Parabolic Equations), Yekaterinburg: UrO AN SSSR, 1992.

    Google Scholar 

  8. Miller, J.J., O’Riordan, E., and Shishkin, G.I., Fitted Numerical Methods for Singular Perturbation Problems, Singapore: World Scientific, 1996.

    MATH  Google Scholar 

  9. Kellogg, R.B. and Tsan, A., Analysis of SomeDifference Approximations for a Singular Perturbation Problem without Turning Points, Math. Comp., 1978, vol. 32, pp. 1025–1039.

    MathSciNet  Article  MATH  Google Scholar 

  10. Miln, V.E., Chislennyi analiz (Numerical Analysis), Moscow: IL, 1951.

    Google Scholar 

  11. Nikolskii, S.M., Kvadraturnye formuly (Quadrature Formulas), Moscow: Nauka, 1974.

    Google Scholar 

  12. Doolan, E., Miller, J., and Schilders, W., Ravnomernye chislennyemetody resheniya zadach s pogranichnym sloem (Uniform Numerical Methods for Problems with Initial and Boundary Layers), Moscow: Mir, 1983.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, ul. Pevtsova 13, Omsk, 644099, Russia

    A. I. Zadorin & N. A. Zadorin

Authors
  1. A. I. Zadorin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. A. Zadorin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. I. Zadorin.

Additional information

Original Russian Text © A.I. Zadorin, N.A. Zadorin, 2013, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2013, Vol. 16, No. 4, pp. 313–323.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Zadorin, A.I., Zadorin, N.A. An analogue of the four-point Newton-Cotes formula for a function with a boundary-layer component. Numer. Analys. Appl. 6, 268–278 (2013). https://doi.org/10.1134/S1995423913040022

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995423913040022

Keywords

  • one-variable function
  • boundary-layer component
  • high gradients
  • definite integral
  • nonpolynomial interpolation
  • quadrature rule
  • error estimate