Skip to main content
SpringerLink
Log in

Generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations associated to the number operator

  • Original Article
  • Published:
São Paulo Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

By means of the Laplace transform, we give the solution of the generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations in infinite dimensions associated to the number operator. These solutions are given in terms of the Mittag-Leffler function and the convolution product.

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. Barhoumi, A., Ouerdiane, H., Rguigui, H.: Generalized Euler heat equation. Quant.Probab. White Noise Anal. 25, 99–116 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Barhoumi, A., Ouerdiane, H., Rguigui, H.: Stochastic Heat Equation on Algebra of Generalized Functions. Infinite Dimensional Analysis Quantum Probability and Related Topics, 15, No. 4 (2012), 1250026 (18 pages)

  3. Da Silva, J.L., Erraoui, M., Ouerdiane, H.: Generalized fractional evolution equation. Fract. Calcul. Appl. Anal. 10(4), 375–398 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Ettaieb, A., Ouerdiane, H., Rguigui, H.: Powers of quantum white noise derivatives. Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 17, 1450018 (2014) [16 pages]

  5. Ettaieb, A., Khalifa, N. T., Ouerdiane, H., Rguigui, H.: Higher powers of analytical operators and associated -Lie algebras. Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 19, 1650013 (2016) [20 pages]

  6. Gannoun, R., Hachaichi, R., Ouerdiane, H., Rezgi, A.: Un théorème de dualité entre espace de fonction holomorphes à croissance exponentielle. J. Funct. Anal. 171, 1–14 (2000)

    Article  MathSciNet  Google Scholar 

  7. Gel’fand, I.M., Vilenkin, N.Y.: Generalized Functions. Academic Press, New York (1964)

    MATH  Google Scholar 

  8. Kilbas, A.A., Pierantozzi, T., Trujillo, J.J., Vázquez, L.: On the solution of fractional evolution equations. J. Phys. A 37(9), 3271–3283 (2004)

    Article  MathSciNet  Google Scholar 

  9. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland (Elsevier) Series, 204 (2006)

  10. Kuo, H.H.: White Noise Distribution Theory. CRC Press, Boca Raton (1996)

    MATH  Google Scholar 

  11. Obata, N.: White noise calculus and Fock spaces. Lecture notes in Mathematics 1577, Spriger-Verlag (1994)

  12. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, Cambridge (1974)

    MATH  Google Scholar 

  13. Ouerdiane, H., Rguigui, H.: QWN-conservation operator and associated wick differential equation. Commun. Stochast. Anal. 6(3), 437–450 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Piech, M.A.: Parabolic equations associated with the number operator. Trans. Amer. Math. Soc. 194, 213–222 (1974)

    Article  MathSciNet  Google Scholar 

  15. Rguigui, H.: Quantum Ornstein-Uhlenbeck semigroups. Quant. Stud. Math. Found. 2(2), 159–175 (2015)

    Article  MathSciNet  Google Scholar 

  16. Rguigui, H.: Characterization of the QWN-conservation operator. Chaos Solitons Fract. 84, 41–48 (2016)

    Article  MathSciNet  Google Scholar 

  17. Rguigui, H.: Fractional number operator and associated fractional evolution equations. Math. Phys. Anal. Geom. 21, 1–17 (2018)

    Article  Google Scholar 

  18. Rguigui, H.: Characterization theorems for the quantum white noise gross Laplacian and applications. Complex Anal. Oper. Theory 12, 1637–1656 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under project number (RGP-2019-1).

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science of Al-Zulfi, Majmaah University, P.O. Box 66, Al-Majmaah, 11952, Saudi Arabia

    Ziyad A. Alhussain & Anis Riahi

  2. Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraidah, 51452, Saudi Arabia

    Habib Rebei

  3. Department of Mathematics, AL-Qunfudhah University College, Umm Al-Qura University, Al-Qunfudhah, Saudi Arabia

    Hafedh Rguigui

  4. Department of Mathematics, Higher School of Sciences and Technologies of Hammam-Sousse, Sousse, University, Sousse, Tunisia

    Hafedh Rguigui

Authors
  1. Ziyad A. Alhussain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Habib Rebei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hafedh Rguigui
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Anis Riahi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hafedh Rguigui.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by H. M. Srivastava.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alhussain, Z.A., Rebei, H., Rguigui, H. et al. Generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations associated to the number operator. São Paulo J. Math. Sci. 15, 435–449 (2021). https://doi.org/10.1007/s40863-020-00200-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40863-020-00200-5

Keywords

Search

Navigation