Skip to main content
SpringerLink
Log in
Book cover

Theoretical Developments and Applications of Non-Integer Order Systems pp 3–16Cite as

Fractional Sturm-Liouville Problem in Terms of Riesz Derivatives

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 357))

Abstract

In the paper, a regular fractional Sturm-Liouville problem on a bounded domain is formulated in terms of Riesz derivatives. The considered case includes vanishing Dirichlet boundary conditions and we prove that its eigenvalues are real, eigenfunctions are continuous and form orthogonal sets of functions in the respective Hilbert spaces. In addition, a boundedness results for eigenvalues are derived and a connection between the discussed fractional Sturm-Liouville equations and Euler-Lagrange equations for the corresponding action functionals is established.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
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

Learn about institutional subscriptions

References

  1. Al-Mdallal, Q.M.: An efficient method for solving fractional Sturm-Liouville problems. Chaos Solitons Fractals 40, 183–189 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Al-Mdallal, Q.M.: On the numerical solution of fractional Sturm-Liouville problem. Int. J. Comput. Math. 87(12), 2837–2845 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Erturk, V.S.: Computing eigenelements of Sturm-Liouville problems of fractional order via fractional differential transform method. Math. Comput. Appl. 16, 712–720 (2011)

    MathSciNet  Google Scholar 

  4. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  5. Klimek, M.: On Solutions of Linear Fractional Differential Equations of a Variational Type. The Publishing Office of Czestochowa University of Technology, Czestochowa (2009)

    Google Scholar 

  6. Klimek, M., Agrawal, O.P.: On a regular fractional Sturm-Liouville problem with derivatives of order in \((0,1)\). Proceedings of the 13th International Carpathian Control Conference, Vysoke Tatry (Podbanske), Slovakia, 28–31 May 2012. doi:10.1109/CarpathianCC.2012.6228655

  7. Klimek, M., Agrawal, O.P.: Fractional Sturm-Liouville problem. Comput. Math. Appl. 66, 795–812 (2013)

    Article  MathSciNet  Google Scholar 

  8. Klimek, M., Agrawal, O.P.: Space- and time-fractional Legendre-Pearson diffusion equation. Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (DETC2013), Portland, Oregon USA, 4–7 August 2013. doi:10.1115/DETC2013-12604

  9. Klimek, M., Błasik, M.: Regular fractional sturm-liouville problem with discrete spectrum: solutions and applications. Proceedings of the 2014 International Conference on Fractional Differentiation and Its Applications, ICFDA 2014, Catania, Italy, 23–25 June 2014. doi:10.1109/ICFDA.2014.6967383

  10. Klimek, M., Błasik, M.: Regular Sturm-Liouville problem with Riemann-Liouville derivatives of order in (1,2): discrete spectrum, solutions and applications. In: Latawiec, K.J., Łukaniszyn, M., Stanisławski, R. (eds.) Advances in Modelling and Control of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol. 320, pp. 25–36. Springer, Heidelberg (2015)

    Google Scholar 

  11. Klimek, M.: Fractional Sturm-Liouville problem and 1D space-time fractional diffusion with mixed boundary conditions. To appear. In: Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (DETC2015), Boston, Massachussets USA, 2–5 August 2015. Paper DETC2015/MESA-46808

    Google Scholar 

  12. Lin, Y., He, T., Shi, H.: Existence of positive solutions for Sturm-Liouville BVPs of singular fractional differential equations, U. P. B. Sci. Bull. 74, 1 (Series A) (2012)

    Google Scholar 

  13. Malinowska, A.B., Torres, D.F.M.: Introduction to the Fractional Calculus of Variations. Imperial College Press, London (2012)

    Google Scholar 

  14. Neamaty, A., Darzi, R., Dabbaghian, A., Golipoor, J.: Introducing an iterative method for solving a special FDE. Int. Math. Forum 4, 1449–1456 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Qi, J., Chen, S.: Eigenvalue problems of the model from nonlocal continuum mechanics. J. Math. Phys. 52, 073516, 14pp (2011)

    Google Scholar 

  16. Rivero, M., Trujillo, J.J., Velasco, M.P.: A fractional approach to the Sturm-Liouville problem. Cent. Eur. J. Phys. (2013). doi:10.2478/s11534-013-0216-2

  17. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  18. Zayernouri, M., Karniadakis, G.E.: Fractional Sturm-Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013)

    Article  MathSciNet  Google Scholar 

  19. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation methods for linear and nonlinear variable order FPDEs. J. Comput. Phys. doi:10.1016/j.jcp.2014.12.001

    Google Scholar 

  20. Zayernouri, M., Ainsworth, M., Karniadakis, G.E.: A unified Petrov-Galerkin spectral method for fractional PDEs. Comput. Meth. Appl. Mech. Eng. 283, 1545–1569 (2015)

    Google Scholar 

  21. Zettl, A.: Sturm-Liouville theory. Mathematical Surveys and Monographs, vol. 121. American Mathematical Society (2005)

    Google Scholar 

Download references

Acknowledgments

This research was supported by CUT (Czestochowa University of Technology) grant No BS/PB-1-105-3010/2011/S.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Czestochowa University of Technology, Armii Krajowej 21, 42-200, Czestochowa, Poland

    Malgorzata Klimek

Authors
  1. Malgorzata Klimek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Malgorzata Klimek .

Editor information

Editors and Affiliations

  1. Faculty of Electrical Engineering, West Pomeranian University of Technology, Szczecin, Poland

    Stefan Domek

  2. Faculty of Electrical Engineering, West Pomeranian University of Technology, Szczecin, Poland

    Paweł Dworak

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Klimek, M. (2016). Fractional Sturm-Liouville Problem in Terms of Riesz Derivatives. In: Domek, S., Dworak, P. (eds) Theoretical Developments and Applications of Non-Integer Order Systems. Lecture Notes in Electrical Engineering, vol 357. Springer, Cham. https://doi.org/10.1007/978-3-319-23039-9_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-23039-9_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23038-2

  • Online ISBN: 978-3-319-23039-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation