Skip to main content
SpringerLink
Log in

A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler-type kernel

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract.

The mathematical model of breaking of non-linear dispersive water waves with memory effect is very important in mathematical physics. In the present article, we examine a novel fractional extension of the non-linear Fornberg-Whitham equation occurring in wave breaking. We consider the most recent theory of differentiation involving the non-singular kernel based on the extended Mittag-Leffler-type function to modify the Fornberg-Whitham equation. We examine the existence of the solution of the non-linear Fornberg-Whitham equation of fractional order. Further, we show the uniqueness of the solution. We obtain the numerical solution of the new arbitrary order model of the non-linear Fornberg-Whitham equation with the aid of the Laplace decomposition technique. The numerical outcomes are displayed in the form of graphs and tables. The results indicate that the Laplace decomposition algorithm is a very user-friendly and reliable scheme for handling such type of non-linear problems of fractional order.

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. G.B. Whitham, Proc. R. Soc. A 299, 6 (1967)

    Article  ADS  Google Scholar 

  2. B. Fornberg, G.B. Whitham, Phil. Trans. R. Soc. London A 289, 373 (1978)

    Article  ADS  Google Scholar 

  3. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)

  4. L. Debnath, Frac. Calc. Appl. Anal. 6, 119 (2003)

    Google Scholar 

  5. M. Caputo, Elasticità e Dissipazione (Zanichelli, Bologna, 1969)

  6. J. Singh, D. Kumar, J.J. Nieto, Entropy 18, 206 (2016)

    Article  ADS  Google Scholar 

  7. H.M. Srivastava, D. Kumar, J. Singh, Appl. Math. Model 45, 192 (2017)

    Article  MathSciNet  Google Scholar 

  8. P.K. Gupta, M. Singh, Comput. Math. Appl. 61, 250 (2011)

    Article  MathSciNet  Google Scholar 

  9. M.G. Saker, F. Erdogan, A. Yildirim, Comput. Math. Appl. 63, 1382 (2012)

    Article  MathSciNet  Google Scholar 

  10. M. Merdan, A. Gökdoğan, A. Yildirim, S.T. Mohyud-Din, Abstr. Appl. Anal. 2012, 965367 (2012)

    Article  Google Scholar 

  11. J. Singh, D. Kumar, S. Kumar, Ain Shams Eng. J. 4, 557 (2013)

    Article  Google Scholar 

  12. M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)

    Google Scholar 

  13. A. Atangana, B.T. Alkahtani, Chaos, Solitons Fractals 89, 566 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  14. J. Singh, D. Kumar, J.J. Nieto, Chaos, Solitons Fractals 99, 109 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  15. J. Singh, D. Kumar, M.A. Qurashi, D. Baleanu, Adv. Differ. Equ. 2017, 88 (2017)

    Article  Google Scholar 

  16. A. Atangana, D. Baleanu, Therm. Sci. 20, 763 (2016)

    Article  Google Scholar 

  17. D. Kumar, J. Singh, D. Baleanu, Physica A 492, 155 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  18. J. Singh, D. Kumar, D. Baleanu, Chaos 27, 103113 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  19. D. Baleanu, A. Jajarmi, M. Hajipour, J. Optim. Theory Appl. 175, 718 (2017)

    Article  MathSciNet  Google Scholar 

  20. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method (Kluwer Academic Publishers, Boston, 1994)

  21. Z. Odibat, S. Momani, Appl. Math. Model. 32, 28 (2008)

    Article  Google Scholar 

  22. D. Kumar, J. Singh, A. Kilicman, Abstr. Appl. Anal. 2013, 608943 (2013)

    Article  Google Scholar 

  23. D. Kumar, J. Singh, D. Baleanu, J. Comput. Nonlinear Dyn. 11, 061004 (2016)

    Article  Google Scholar 

  24. S.A. Khuri, J. Appl. Math. 1, 141 (2001)

    Article  MathSciNet  Google Scholar 

  25. D. Baleanu, G.C. Wu, S.D. Zeng, Chaos, Solitons Fractals 102, 99 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  26. G.C. Wu, D. Baleanu, H.P. Xie, Chaos 26, 084308 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  27. J.S. Duan, Appl. Math. Comput. 217, 6337 (2011)

    Article  MathSciNet  Google Scholar 

  28. A. Atangana, I. Koca, Chaos, Solitons Fractals 89, 447 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  29. A. Atangana, Appl. Math. Model. 39, 2909 (2015)

    Article  MathSciNet  Google Scholar 

  30. A. Atangana, E.F.D. Goufo, J. Nonlinear Sci. Appl. 8, 763 (2015)

    Article  MathSciNet  Google Scholar 

  31. A. Atangana, J.F. Gómez-Aguilar, Phys. A 476, 1 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, JECRC University, Jaipur, 303905, Rajasthan, India

    Devendra Kumar & Jagdev Singh

  2. Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Eskisehir Yolu 29. Km, Yukarıyurtcu Mahallesi Mimar Sinan Caddesi No: 4 06790, Etimesgut, Turkey

    Dumitru Baleanu

  3. Institute of Space Sciences, Magurele, Bucharest, Romania

    Dumitru Baleanu

Authors
  1. Devendra Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jagdev Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dumitru Baleanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Devendra Kumar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, D., Singh, J. & Baleanu, D. A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler-type kernel. Eur. Phys. J. Plus 133, 70 (2018). https://doi.org/10.1140/epjp/i2018-11934-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2018-11934-y

Search

Navigation