Skip to main content
SpringerLink
Log in

Exact solution for the fractional cable equation with nonlocal boundary conditions

  • Research Article
  • Published:
Central European Journal of Physics

Abstract

The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied.

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. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. A. V. Chechkin, R. Gorenflo, I. M. Sokolov, J. Phys. A 38, 679 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  3. B. I. Henry, T. A. M. Langlands, S. L. Wearne, Phys. Rev. E 74, 031116 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  4. B. I. Henry, T. A. M. Langlands, S. L. Wearne, Phys. Rev. Lett. 100, 128103 (2008)

    Article  ADS  Google Scholar 

  5. T. A. M. Langlands, B. I. Henry, S. L. Wearne, J. Math. Biol. 59, 761 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. T. A. M. Langlands, B. I. Henry, S. L. Wearne, SIAM J. Appl. Math. 71, 1168 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. F. Liu, Q. Yang, I. Turner, J. Comput. Nonlinear Dyn. 6, 0110091 (2011)

    Google Scholar 

  8. X. Hu, L. Zhang, Appl. Math. Model. 36, 4027 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Y. M. Lin, X. J. Li, C. J. Xu, Math. Comput. 80, 1369 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Bajlekova, Ph.D. thesis, Eindhoven University of Technology (Eindhoven, The Netherlands, 2001) 10 http://alexandria.tue.nl/extra2/200113270.pdf

  11. A. Schot et al., Phys. Lett. A 366, 346 (2007)

    Article  ADS  Google Scholar 

  12. V. Daftardar-Gejji, S. Bhalekar, J. Math. Anal. Appl. 345, 754 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Yu. Luchko, J. Math. Anal. Appl. 374, 538 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. H. Jiang, F. Liu, I. Turner, K. Burrage, Comput. Math. Appl. 64, 3377 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Dehghan, Chaos Soliton. Fract. 32, 661 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations (Elsevier, Amsterdam, 2006)

    MATH  Google Scholar 

  17. I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999)

    MATH  Google Scholar 

  18. R. K. Saxena, A. M. Mathai, H. J. Haubold, Astrophys. Space Sci. 209, 299 (2004)

    Article  ADS  Google Scholar 

  19. N. S. Bozhinov, Differ. Equ. 26, 741 (1990) (in Russian)

    MATH  MathSciNet  Google Scholar 

  20. I. H. Dimovski, Convolutional Calculus (Kluwer, Dordrecht, 1990)

    Book  MATH  Google Scholar 

  21. I. H. Dimovski, R. I. Petrova, In: Generalized functions and convergence, Katowice 1988 (World Sci. Publ., Teaneck, NJ, 1990) 89

    Google Scholar 

  22. Y. Tsankov, C. R. Acad. Bulg. Sci. (2013) (to appear)

    Google Scholar 

  23. K. Diethelm, The analysis of fractional differential equations (Springer-Verlag Berlin Heidelberg, 2010)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, Sofia, 1113, Bulgaria

    Emilia G. Bazhlekova & Ivan H. Dimovski

Authors
  1. Emilia G. Bazhlekova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ivan H. Dimovski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emilia G. Bazhlekova.

About this article

Cite this article

Bazhlekova, E.G., Dimovski, I.H. Exact solution for the fractional cable equation with nonlocal boundary conditions. centr.eur.j.phys. 11, 1304–1313 (2013). https://doi.org/10.2478/s11534-013-0213-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11534-013-0213-5

Keywords

Search

Navigation