New Analytical Approach for Fractional Cubic Nonlinear Schrödinger Equation Via Laplace Transform

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 236)


In this paper, a user-friendly algorithm based on new homotopy perturbation transform method (HPTM) is proposed to obtain approximate solution of a time-space fractional cubic nonlinear Schrödinger equation. The numerical solutions obtained by the HPTM indicate that the technique is easy to implement and computationally very attractive.


Fractional cubic nonlinear Schrödinger equation Laplace transform Homotopy perturbation transform method He’s polynomials Maple code 


  1. 1.
    Hilfer, R. (ed.): Applications of Fractional Calculus in Physics, pp. 87–130. World Scientific Publishing Company, Singapore, New Jersey, Hong Kong (2000)Google Scholar
  2. 2.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)MATHGoogle Scholar
  3. 3.
    Caputo, M.: Elasticita e Dissipazione. Zani-Chelli, Bologna (1969)Google Scholar
  4. 4.
    Miller, K.S., Ross, B.: An Introduction to the fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)MATHGoogle Scholar
  5. 5.
    Oldham, K.B., Spanier, J.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)MATHGoogle Scholar
  6. 6.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)MATHGoogle Scholar
  7. 7.
    He, J.H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999)Google Scholar
  8. 8.
    He, J.H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135, 73–79 (2003)Google Scholar
  9. 9.
    He, J.H.: New interpretation of homotopy perturbation method. Int. J. Mod. Phys. B 20, 2561–2568 (2006)CrossRefGoogle Scholar
  10. 10.
    Abbasbandy, S.: Application of He’s homotopy perturbation method to functional integral equations. Chaos, Solitons Fractals 31, 1243–1247 (2007)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Ganji, D.D., Sadighi, A.: Application of He’s homotopy perturbation method to nonlinear coupled system of reaction-diffusion equations. Int. J. Nonlinear Sci. Numer. Simul. 7, 411–418 (2006)Google Scholar
  12. 12.
    Rafei, M., Ganji, D.D.: Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method. Int. J. Nonlinear Sci. Numer. Simul. 7, 321–328 (2006)Google Scholar
  13. 13.
    Rafei, M., Ganji, D.D., Daniali, D.: Solution of epidemic model by homotopy perturbation method. Appl. Math. Comput. 187, 1056–1062 (2007)Google Scholar
  14. 14.
    Ozis, T., Yildirim, A.: Travelling wave solution of KdV equation using He’ homotopy perturbation method. Int. J. Nonlinear Sci. Numer. Simul. 8, 239–242 (2007)Google Scholar
  15. 15.
    Khuri, S.A.: A Laplace decomposition algorithm applied to a class of nonlinear differential equations. J. Appl. Math. 1, 141–155 (2001)Google Scholar
  16. 16.
    Khan, Y.: An effective modification of the Laplace decomposition method for nonlinear equations. Int. J. Nonlinear Sci. Numer. Simul. 10, 1373–1376 (2009)Google Scholar
  17. 17.
    Khan, Y., Wu, Q.: Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput. Math. Appl. 61(8), 1963–1967 (2011)Google Scholar
  18. 18.
    Singh, J., Kumar, D., Rathore, S.: Application of homotopy perturbation transform method for solving linear and nonlinear Klein-Gordon equations. J. Inf. Comput. Sci. 7(2), 131–139 (2012)Google Scholar
  19. 19.
    Herzallah Mohamed, A.E., Gepreel,K.A.: Approximate solution to time-space fractional cubic nonlinear Schrödinger equation. Appl. Math. Model. 36(11), 56–78 (2012)Google Scholar
  20. 20.
    Hemida, K.M., Gerpreel, K.A., Mohamed, M.S.: Analytical approximate solution to the time-space nonlinear partial fractional differential equations. Int. J. Pure Appl. Math. 78(2), 233–243 (2012)Google Scholar
  21. 21.
    Ghorbani, A.: Beyond adomian’s polynomials: He polynomials. Chaos, Solitons Fractals 39, 1486–1492 (2009)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Mohyud-Din, S.T., Noor, M.A., Noor, K.I.: Traveling wave solutions of seventh-order generalized KdV equation using He’s polynomials. Int. J. Nonlinear Sci. Numer. Simul. 10, 227–233 (2009)Google Scholar

Copyright information

© Springer India 2014

Authors and Affiliations

  1. 1.Department of MathematicsJagannath UniversityJaipurIndia
  2. 2.Department of MathematicsJagannath Gupta Institute of Engineering and Technology JaipurIndia

Personalised recommendations