Skip to main content
SpringerLink
Log in

Imprecisely defined fractional-order Fokker–Planck equation subjected to fuzzy uncertainty

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

Fokker–Planck equation with interval and fuzzy uncertainty has been considered in this paper. Also the derivatives involved with respect to time and space are assumed to be fractional in nature. This problem has been solved using variational iteration method (VIM) along with the double parametric form of fuzzy numbers. For the analysis, both triangular and Gaussian normalised fuzzy sets are taken into consideration. Numerical results for different cases have been obtained and those are depicted in terms of plots and are also compared in special cases for the validation. Moreover, using an important method known as successive approximation method, it has also been verified that the obtained solutions are the same as that of VIM as both methods are equivalent.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. H Risken, The Fokker–Planck equation: Method of solution and applications (Springer, Berlin, 1989)

    Book  MATH  Google Scholar 

  2. F Liu, V Anh and I Turner, J. Comput. Appl. Math. 166, 209 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  3. Z Odibat and S Momani, Phys. Lett. A 369, 349 (2007)

    Article  ADS  Google Scholar 

  4. M Tatari, M Dehghan and M Razzaghi, Math. Comput. Model. 45, 639 (2007)

    Article  Google Scholar 

  5. W Deng, SIAM J. Numer. Anal. 47, 204 (2008)

    Article  MathSciNet  Google Scholar 

  6. S Chen, F Liu, P Zhuang and V Anh, Appl. Math. Model. 33, 256 (2009)

    Article  MathSciNet  Google Scholar 

  7. A Yildirim, J. King Saud Univ. 22, 257 (2010)

    Article  Google Scholar 

  8. L Yan, Abstr. Appl. Anal. 2013, 1 (2013)

    MathSciNet  Google Scholar 

  9. J Gajda and A Wyłomańska, Phys. A Stat. Mech. Appl. 405, 104 (2014)

    Article  Google Scholar 

  10. M S Hashemi, Physica A 417, 141 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  11. D Baleanu, H M Srivastava and X J Yang, Prog. Fract. Differ. Appl. 1, 1 (2015)

    Google Scholar 

  12. M A Malkov, Phys. Rev. D 95, 023007-1 (2017)

  13. M A Firoozjaee, H Jafari, A Lia and D Baleanu, J. Comput. Appl. Math. 339, 367 (2018)

    Article  MathSciNet  Google Scholar 

  14. M A Firoozjaee, S A Yousefia and H Jafari, MATCH Commun. Math. Comput. Chem. 74, 449 (2015)

    MathSciNet  Google Scholar 

  15. H Jafari and H Tajadodi, Therm. Sci. 22, 277 (2018)

    Article  Google Scholar 

  16. S S Roshan, H Jafari and D Baleanu, Math. Methods Appl. Sci. 41, 9134 (2018)

    Article  MathSciNet  Google Scholar 

  17. A Singh, S Das, S H Ong and H Jafari, J. Comput. Nonlinear Dyn. 14(4), 041003-1 (2019)

    Google Scholar 

  18. H Jafari, Appl. Math. Lett. 32, 1 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  19. S L Chang and L A Zadeh, IEEE Trans. Systems Man Cybernet. 2, 30 (1972)

    Article  Google Scholar 

  20. D Dubois and H Prade, Fuzzy Sets Syst. 8, 225 (1982)

    Article  Google Scholar 

  21. N Mikaeilvand and S Khakrangin, Neural Comput. Appl. 21, S307 (2012)

    Article  Google Scholar 

  22. A Khastan, J J Nieto and R R López, Inf. Sci. 222, 544 (2013)

    Article  Google Scholar 

  23. D Qiu, W Zhang and C Lu, Fuzzy Sets Syst. 295, 72 (2016)

    Article  Google Scholar 

  24. S Tapaswini, S Chakraverty and T Allahviranloo, Comput. Math. Model. 28, 278 (2017)

    Article  MathSciNet  Google Scholar 

  25. S Tapaswini, S Chakraverty and J J Nieto, Sadhana 42, 45 (2017)

    Article  Google Scholar 

  26. K Nematollah, S S Roushan and H Jafari, Int. J. Appl. Comput. Math. 4, 33 (2018)

    Article  Google Scholar 

  27. R P Agarwal, V Lakshmikantham and J J Nieto, Nonlinear Anal. Theory Methods Appl. 72, 2859 (2010)

    Article  Google Scholar 

  28. E Khodadadi and E Celik, Fixed Point Theory Appl. 2013, 1 (2013)

    Article  Google Scholar 

  29. S Chakraverty and S Tapaswini, Comput. Model. Eng. Sci. 103, 71 (2014)

    Google Scholar 

  30. A Rivaz, O S Fard and T A Bidgoli, SeMA J. 73, 149 (2016)

    Article  MathSciNet  Google Scholar 

  31. J H He, Int. J. Nonlinear Mech. 34, 699 (1999)

    Article  ADS  Google Scholar 

  32. J H He, Appl. Math. Comput. 114, 115 (2000)

    MathSciNet  Google Scholar 

  33. K Abbaoui and Y Cherruault, Comput. Math. Appl. 29, 103 (1995)

    Article  MathSciNet  Google Scholar 

  34. H Jafari, M Saeidy and D Baleanu, Cent. Eur. J. Phys. 10, 76 (2012)

    Google Scholar 

  35. A F Jameel, Approx. Sci. Comput. 2014, 1 (2014)

    Google Scholar 

  36. S Abbasbandy, T Allahviranloo, P Darabi and O Sedaghatfar, Math. Comput. Appl. 16, 819 (2011)

    MathSciNet  Google Scholar 

  37. M M Hosseini, F Saberirad and B Davvaz, Int. J. Fuzzy Syst. 18, 875 (2016)

    Article  MathSciNet  Google Scholar 

  38. H J Zimmermann, Fuzzy set theory and its application (Kluwer Academic Publishers, Boston\(/\)Dordrecht\(/\)London, 2001)

    Book  Google Scholar 

  39. L Jaulin, M Kieffer, O Didrit and E Walter, Applied interval analysis (Springer, Paris, 2001)

    Book  MATH  Google Scholar 

  40. M Hanss, Applied fuzzy arithmetic: An introduction with engineering applications (Springer, Berlin, 2005)

    MATH  Google Scholar 

  41. S Chakraverty, S Tapaswini and D Behera, Fuzzy arbitrary order systems: Fuzzy fractional differential equations and applications (Wiley, New Jersey, 2016)

    Book  MATH  Google Scholar 

  42. S Chakraverty, S Tapaswini and D Behera, Fuzzy differential equations and applications for engineers and scientists (CRC Press, Florida, 2016)

    Book  MATH  Google Scholar 

  43. I Podlubny, Fractional differential equations (Academic Press, New York, 1999)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewer and editor for their valuable suggestions and comments to improve the quality and clarity of the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Government Autonomous College, Angul, 759 143, India

    Smita Tapaswini

  2. Department of Mathematics, The University of the West Indies, Mona Campus, Kingston 7, Jamaica

    Diptiranjan Behera

Authors
  1. Smita Tapaswini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Diptiranjan Behera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diptiranjan Behera.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tapaswini, S., Behera, D. Imprecisely defined fractional-order Fokker–Planck equation subjected to fuzzy uncertainty. Pramana - J Phys 95, 13 (2021). https://doi.org/10.1007/s12043-020-02033-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-020-02033-5

Keywords

PACS Nos

Search

Navigation