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On solutions of two coupled fractional time derivative Hirota equations

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Abstract

We consider the well-known nonlinear Hirota equation (NLH) with fractional time derivative and derive its periodic wave solution and approximate analytic solitary wave solution using the homotopy analysis method (HAM). We also apply HAM to two coupled time fractional NLHs and construct their periodic wave solution and approximate solitary wave solution. We observe that the obtained periodic wave solution in both cases can be written in terms of the Mittag–Leffler function when the convergence control parameter \({c}_0\) equals \(-1\). Convergence of the obtained solution is discussed. The derived approximate analytic solution and the effect of time-fractional order \(\alpha \) are shown graphically.

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Notes

  1. Liao originally used the symbol \(\hbar \) to denote the nonzero auxiliary parameter. But it is well known as Planck’s constant. To avoid misunderstanding Liao [16] replaced \(\hbar \) by \({c}_0\).

References

  1. Samko, S., Kilbas, A.A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science, Switzerland (1993)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  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. Debnath, L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 54, 3413–3442 (2003)

    Article  MathSciNet  Google Scholar 

  6. Sahadevan, R., Bakkyaraj, T.: Invariant analysis of time fractional generalised Burgers and Korteweg-de Vries equations. J. Math. Anal. Appl. 393, 341–347 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bakkyaraj, T., Sahadevan, R.: An approximate solution to some classes of fractional nonlinear partial differential- difference equation using Adomian decomposition method. J. Fract. Calc. Appl. 5(1), 37–52 (2014)

    Google Scholar 

  8. Liao, S.: The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. dissertation. Shanghai jiao Tong University (1992)

  9. Liao, S.: A kind of approximate solution technique which doesnot depend upon small parameters (II)—an application in fluid mechanics. Int. J. Nonlinear Mech. 32, 815–822 (1997)

    Article  MATH  Google Scholar 

  10. Liao, S.: On the analytic solution of magnetohydro dynamic flows of non Newtonian fluids over a stretching sheet. J. Fluid Mech. 488, 189–212 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liao, S.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall, CRC, Boca Raton (2003)

    Book  Google Scholar 

  12. Liao, S.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Liao, S.: A new branch of solutions of boundary layer flows over a permeable stretching plate. Int. J. Nonlinear Mech. 42, 819–930 (2007)

    Article  MATH  Google Scholar 

  14. Liao, S.: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14, 983–997 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Liao, S.: An optimal homotopy analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 2003–2016 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Liao, S.: Homotopy Analysis Method in Nonlinear Differential Equation. Springer, Berlin (2012)

    Book  Google Scholar 

  17. Liang, S.X., Jeffrey, D.J.: Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation. Commun. Nonlinear Sci. Numer. Simul. 14, 4057–4064 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A. 360, 109–113 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Abbasbandy, S.: Solitary wave solutions to the Kuramoto - Sivashinsky equation by means of the homotopy analysis method. Nonlinear Dyn. 52, 35–40 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ganjiani, M., Ganjiani, H.: Solution of coupled system of nonlinear differential equations using homotopy analysis method. Nonlinear Dyn. 56, 159–167 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Song, L., Zhang, H.: Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation. Phys. Lett. A 367, 88–94 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hashim, I., Abdulaziz, O., Momani, S.: Homotopy analysis method for fractional IVPs. Commun. Nonlinear Sci. Numer. Simul. 14, 674–684 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Xu, H., Liao, S., You, X.C.: Analysis of nonlinear fractional partial differential equations with the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14, 1152–1156 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Cang, J., Tan, Y., Xu, H., Liao, S.: Series solutions of nonlinear Riccati differential equations with fractional order. Chaos Solitons Fractals 40, 1–9 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ. 26, 448–479 (2009)

    MathSciNet  Google Scholar 

  26. Jafari, H., Seifi, S.: Homotopy analysis method for solving linear and nonlinear fractional diffusion—wave equation. Commun. Nonlinear Sci. Numer. Simul. 14, 2006–2012 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Ganjiani, M.: Solution of nonlinear fractional differential equations using homotopy analysis method. Appl. Math. Modell. 34, 1634–1641 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  28. Khan, N.A., Jamil, M., Ara, A.: Approximate solution to time fractional Schrödinger equation via homotopy analysis method. ISRN Math. Phys. 2012, 1–11 (2012)

    Article  Google Scholar 

  29. Jafari, H., Seifi, S.: Solving a system of nonlinear fractional partial differential equations using homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14, 1962–1969 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  30. Zurigat, M., Momani, S., Odibat, Z., Alawneh, A.: The homotopy analysis method for handling system of fractional differential equations. Appl. Math. Modell. 34, 24–35 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  31. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution and Inverse Scattering. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  32. Lakshmanan, M., Rajasekar, S.: Nonlinear Dynamics: Integrability, Chaos, and Patterns. Springer, India (2003)

    Book  Google Scholar 

  33. Radhakrishnan, R., Sahadevan, R., Lakshmanan, M.: Integrability and singularity structure of coupled nonlinear Schrödinger equations. Chaos Solitons Fractals 5, 2315–2327 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  34. Gilson, C., Hietarinta, J., Nimmo, J., Ohta, Y.: The Sasa-Satsuma higher order nonlinear Schrödinger equation and its bilinearization and multi-soliton solutions. http://arxiv.org/abs/nlin/0304025v2 (2003)

  35. Turkyilmazoglu, M.: Convergence of the homotopy analysis method. arXiv:1006:4460v1 (2010)

  36. El-Ajou, A., Odibat, Z., Momani, S., Alawneh, A.: Construction of analytical solutions to fractional differential equations using homotopy analysis method. Int. J. Appl. Math. 40, 43 (2010)

    Google Scholar 

  37. Abbasbandy, S., Hashemi, M.S., Hashim, I.: On convergence of homotopy analysis method and its application to fractional integro-differential equations. Quaest. Math. 36, 93–105 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to thank anonymous referees for their valuable suggestions. One of the authors (T.Bakkyaraj) would like to thank the Council of Scientific and Industrial Research (CSIR), Government of India, New Delhi, for providing Senior Research Fellowship.

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Authors and Affiliations

  1. Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, Chennai, 600 005, India

    T. Bakkyaraj & R. Sahadevan

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  1. T. Bakkyaraj
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  2. R. Sahadevan
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Correspondence to R. Sahadevan.

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Bakkyaraj, T., Sahadevan, R. On solutions of two coupled fractional time derivative Hirota equations. Nonlinear Dyn 77, 1309–1322 (2014). https://doi.org/10.1007/s11071-014-1380-7

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  • DOI: https://doi.org/10.1007/s11071-014-1380-7

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