Skip to main content
SpringerLink
Log in

Precise integration method for solving singular perturbation problems

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Su, Y. C. The Boundary Layer Correction Method in the Singular Perturbations (in Chinese), Science and Technology Press, Shanghai (1983)

    Google Scholar 

  2. Kadalbajoo, M. K. and Reddy, Y. N. Asymptotic and numerical analysis of singular perturbation problems: a survey. Applied Mathematics and Computation 30, 223–259 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chawla, M. M. A fourth-order tridiagonal finite difference method for general non-linear two-point boundary value problems with mixed boundary conditions. Journal of the Institute of Mathematics and Its Applications 21, 83–93 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Su, Y. C. and Wu, Q. G. An Introduction to Numerical Methods for the Singular Perturbation Problems (in Chinese), Chongqing Publishing House, Chongqing (1991)

    Google Scholar 

  5. Andargie, A. and Reddy, Y. N. Fitted fourth-order tridiagonal finite difference method for singular perturbation problems. Applied Mathematics and Computation 192, 90–100 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Stynes, M. and O’Riordan, E. A uniformly accurate finite-element method for a singular perturbation problem in conservative form. SIAM Journal on Numerical Analysis 23, 369–375 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  7. Vigo-Aguiar, J. and Natesan, S. A parallel boundary value technique for singularly perturbed two-point boundary value problems. The Journal of Supercomputing 27, 195–206 (2004)

    Article  MATH  Google Scholar 

  8. Reddy, Y. N. and Chakravarthy, P. P. An initial-value approach for singularly perturbed two-point boundary value problems. Applied Mathematics and Computation 155, 95–110 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kadalbajoo, M. K. and Kumar, D. Initial value technique for singularly perturbed two-point boundary value problems using an exponentially fitted finite difference scheme. Computers and Mathematics with Applications 57, 1147–1156 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Aziz, T. and Khan, A. A spline method for second-order singularly perturbed boundary-value problems. Journal of Computational and Applied Mathematics 147, 445–452 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tirmizi, I. A., Fazal-i-Haq, and Siraj-ul-Islam. Non-polynomial spline solution of singularly perturbed boundary-value problems. Applied Mathematics and Computation 196, 6–16 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Zhong, W. X. and Williams, F. W. A precise time step integration method. Proceedings of the Institute of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 208(C6), 427–430 (1994)

    Article  Google Scholar 

  13. Lin, J. H., Sun, D. K., Zhong, W. X., and Zheng, W. S. High efficiency computation of the variances of structural evolutionary random responses. Shock and Vibration 7(4), 209–216 (2000)

    Google Scholar 

  14. Gu, Y. X., Chen, B. S., Zhang, H. W., and Grandhi, R. V. A sensitivity analysis method for linear and nonlinear transient heat conduction with precise time integration. Structural and Multidisciplinary Optimization 24, 23–37 (2002)

    Article  Google Scholar 

  15. Zhang, H. W., Zhang, X. W., and Chen, J. S. A new algorithm for numerical solution of dynamic elastic-plastic hardening and softening problems. Computers and Structures 81(17), 1739–1749 (2003)

    Article  MathSciNet  Google Scholar 

  16. Zhong, W. X. Combined method for the solution of asymmetric Riccati differential equations. Computer Methods in Applied Mechanics and Engineering 191, 93–102 (2001)

    Article  MathSciNet  Google Scholar 

  17. Chen, B. S., Tong, L. Y., and Gu, Y. X. Precise time integration for linear two-point boundary value problems. Applied Mathematics and Computation 175, 182–211 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Fu, M. H. and Lin, J. H. An exponential matrix function and its application (in Chinese). Acta Mechanica Sinica 41(5), 808–814 (2009)

    MathSciNet  Google Scholar 

  19. Tan, S. J. and Zhong, W. X. Precise integration method for Duhamel terms arising from nonhomogenous systems (in Chinese). Acta Mechanica Sinica 39(3), 374–381 (2007)

    Google Scholar 

  20. Wang, M. F. and Zhou, X. Y. Renewal precise time step integration method of structural dynamic analysis. Acta Mechanica Sinica 36(2), 191–195 (2004)

    Google Scholar 

  21. Ren, C. B., He, G. Z., and Li, Z. F. A high-precision and general computational scheme in precise integration of structural dynamics (in Chinese). Mechanical Science and Technology 24(2), 1507–1509 (2005)

    Google Scholar 

  22. Fu, M. H. and Liang, H. L. An improved precise Runge-Kutta integration (in Chinese). Acta Scientiarum Naturalium Universitatis Sunyatseni 48(5), 1–5 (2009)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou, 510275, P. R. China

    Ming-hui Fu  (富明慧) & Man-chi Cheung  (张文志)

  2. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119992, Russia

    S. V. Sheshenin

Authors
  1. Ming-hui Fu  (富明慧)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Man-chi Cheung  (张文志)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. V. Sheshenin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ming-hui Fu  (富明慧).

Additional information

Communicated by Xing-ming GUO

Project supported by the National Natural Science Foundation of China (No. 10672194) and the China-Russia Cooperative Project (the National Natural Science Foundation of China and the Russian Foundation for Basic Research) (No. 10811120012)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fu, Mh., Cheung, Mc. & Sheshenin, S.V. Precise integration method for solving singular perturbation problems. Appl. Math. Mech.-Engl. Ed. 31, 1463–1472 (2010). https://doi.org/10.1007/s10483-010-1376-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-010-1376-x

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation