Skip to main content
SpringerLink
Log in

Finite Element Approximation for the Sturm-Liouville Problem with Quadratic Eigenvalue Parameter

  • Conference paper
  • First Online:
Advances in High Performance Computing (HPC 2019)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 902))

  • 459 Accesses

Abstract

The eigenvalue problem considered in this paper contains both spectral parameter \(\lambda \) as well as \(\lambda ^2\). A method for linearization is presented and finite element approximation is used. Error estimate of eigenpairs is proved. Computational aspects for more general problems in this type are discussed. Finally, for purpose of illustration numerical implementation is given.

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. Al-Gwaiz, M.A.: Sturm-Liouville Theory and Its Applications, vol. 7. Springer, Berlin (2008)

    MATH  Google Scholar 

  2. Andreev, A.B., Racheva, M.R.: A method for linearization of a beam problem. In: International Conference on Numerical Analysis and Its Applications, pp. 180–186. Springer, Cham (2016)

    Google Scholar 

  3. Babuška, I., Osborn, J.: Eigenvalue problems. In: Lions, P.G., Ciarlet P.G. (eds.) Handbook of Numerical Analysis, vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam, pp. 641–787 (1991)

    Google Scholar 

  4. Barkwell, L., Lancaster, P., Markus, A.S.: Gyroscopically stabilized systems: a class of quadratic eigenvalue problems with real spectrum. Can. J. Math. 44(1), 42–53 (1991)

    Article  MathSciNet  Google Scholar 

  5. Brenner, S., Scott, R.: The Mathematical Theory of Finite Element Methods, vol. 15. Springer Science & Business Media (2007)

    Google Scholar 

  6. Solov’ev, S.I.: Approximation of differential eigenvalue problems with a nonlinear dependence on the parameter. Differ. Equ. 50(7), 947–954 (2014)

    Article  MathSciNet  Google Scholar 

  7. Solov’ev, S.I.: Finite element approximation with numerical integration for differential eigenvalue problems. Appl. Numer. Math. 93, 206–214 (2015)

    Article  MathSciNet  Google Scholar 

  8. Turner, R.E.: Some variational principles for a nonlinear eigenvalue problem. J. Math. Anal. Appl. 17(1), 151–160 (1967)

    Article  MathSciNet  Google Scholar 

  9. Weinberger, H.F.: On a nonlinear eigenvalue problem. J. Math. Anal. Appl. 21, 506–509 (1968)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Information and Communication Technologies, BAS, Sofia, Bulgaria

    A. B. Andreev

  2. Technical University of Gabrovo, Gabrovo, Bulgaria

    A. B. Andreev & M. R. Racheva

Authors
  1. A. B. Andreev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. R. Racheva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. B. Andreev .

Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Institute of Information and Communication Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Stefka Fidanova

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Andreev, A.B., Racheva, M.R. (2021). Finite Element Approximation for the Sturm-Liouville Problem with Quadratic Eigenvalue Parameter. In: Dimov, I., Fidanova, S. (eds) Advances in High Performance Computing. HPC 2019. Studies in Computational Intelligence, vol 902. Springer, Cham. https://doi.org/10.1007/978-3-030-55347-0_31

Download citation

Publish with us

Policies and ethics

Search

Navigation