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New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions

Year 2017, Volume: 5 Issue: 4, 232 - 241, 01.10.2017

Abstract

 In this study, new families of analytical exact solutions of the space fractional Korteweg-de Vries (KdV) equation are presented. Here, the fractional derivative is considered in conformable sense. By utilizing the Jacobi elliptic function expansion method, the solutions are obtained in general form containing the hyperbolic, trigonometric, and rational functions. Also, the complex valued solutions are obtained and some solutions of this equation are demonstrated. 

References

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  • Z. M. Odibat, Exact solitary solutions for variants of the KdV equations with fractional time derivatives, Chaos, Solitons and Fractals 40 (2009) 1264-1270.
  • Q. Wang, Homotopy perturbation method for fractional KdV equation, Applied Mathematics and Computation 190 (2007) 1795-1802.
  • Z. M. Odibat, Compact and noncompact structures for nonlinear fractional evolution equations, Physics Letters A 372 (2008) 1219-1227.
  • Z. M. Odibat, Compact structures in a class of nonlinearly dispersive equations with time-fractional derivatives, Applied Mathematics and Computation 205 (2008) 273-280.
  • O. Abdulaziz, I. Hashim, E.S. Ismail, Approximate analytical solution to fractional modified KdV equations, Mathematical and Computer Modelling 49 (2009) 136-145.
  • Z. Odibat, S. Momani, The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers and Mathematics with Applications 58 (2009) 2199-2208.
  • M. Kurulay, M. Bayram, Approximate analytical solution for the fractional modified KdV by differential transform method, Commun Nonlinear Sci Numer Simulat 15 (2010) 1777-1782.
  • M. Younis, H. Rehman, M. Iftikhar, Travelling wave solutions to some time-space nonlinear evolution equations, Applied Mathematics and Computation 249 (2014) 81-88.
  • S. Sahoo, S. S. Ray, Solitary wave solutions for time fractional third order modified KdV equation using two reliable techniques expansion method and improved expansion method, Physica A 448 (2016) 265-282.
  • S. Arshad, A. Sohail, K. Maqbool, Nonlinear shallow water waves: A fractional order Approach, Alexandria Engineering Journal (2016) 55, 525-532.
  • W. Rui, Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs, Commun Nonlinear Sci Numer Simulat 47 (2017) 253-266.
  • L. Hua-Mei, New exact solutions of nonlinear Gross-Pitaevskii equation with weak bias magnetic and time-dependent laser fields, Chinese Physics 14 (2) (2005) 251-256.
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  • K.S. Miller, An Introduction to Fractional Calculus and Fractional Differential Equations, J. Wiley and Sons, New York, 1993.
  • A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, in: Math. Studies., North-Holland, New York, 2006.
  • I. Podlubny, Fractional Differential Equations, Academic Press, USA, 1999.
  • R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics 264 (2014) 65-70.
  • T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57-66.
Year 2017, Volume: 5 Issue: 4, 232 - 241, 01.10.2017

Abstract

References

  • S. Momani, An explicit and numerical solutions of the fractional KdV equation, Mathematics and Computers in Simulation 70 (2005) 110 - 118.
  • Z. M. Odibat, Exact solitary solutions for variants of the KdV equations with fractional time derivatives, Chaos, Solitons and Fractals 40 (2009) 1264-1270.
  • Q. Wang, Homotopy perturbation method for fractional KdV equation, Applied Mathematics and Computation 190 (2007) 1795-1802.
  • Z. M. Odibat, Compact and noncompact structures for nonlinear fractional evolution equations, Physics Letters A 372 (2008) 1219-1227.
  • Z. M. Odibat, Compact structures in a class of nonlinearly dispersive equations with time-fractional derivatives, Applied Mathematics and Computation 205 (2008) 273-280.
  • O. Abdulaziz, I. Hashim, E.S. Ismail, Approximate analytical solution to fractional modified KdV equations, Mathematical and Computer Modelling 49 (2009) 136-145.
  • Z. Odibat, S. Momani, The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers and Mathematics with Applications 58 (2009) 2199-2208.
  • M. Kurulay, M. Bayram, Approximate analytical solution for the fractional modified KdV by differential transform method, Commun Nonlinear Sci Numer Simulat 15 (2010) 1777-1782.
  • M. Younis, H. Rehman, M. Iftikhar, Travelling wave solutions to some time-space nonlinear evolution equations, Applied Mathematics and Computation 249 (2014) 81-88.
  • S. Sahoo, S. S. Ray, Solitary wave solutions for time fractional third order modified KdV equation using two reliable techniques expansion method and improved expansion method, Physica A 448 (2016) 265-282.
  • S. Arshad, A. Sohail, K. Maqbool, Nonlinear shallow water waves: A fractional order Approach, Alexandria Engineering Journal (2016) 55, 525-532.
  • W. Rui, Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs, Commun Nonlinear Sci Numer Simulat 47 (2017) 253-266.
  • L. Hua-Mei, New exact solutions of nonlinear Gross-Pitaevskii equation with weak bias magnetic and time-dependent laser fields, Chinese Physics 14 (2) (2005) 251-256.
  • J. V Armitage ve W. F. Eberlein, Elliptic Functions, New York: Cambridge University Press, 2006.
  • K.S. Miller, An Introduction to Fractional Calculus and Fractional Differential Equations, J. Wiley and Sons, New York, 1993.
  • A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, in: Math. Studies., North-Holland, New York, 2006.
  • I. Podlubny, Fractional Differential Equations, Academic Press, USA, 1999.
  • R. Khalil, M. A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics 264 (2014) 65-70.
  • T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57-66.
There are 19 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Aysegul Dascioglu This is me

Sevil Culha This is me

Dilek Varol Bayram This is me

Publication Date October 1, 2017
Published in Issue Year 2017 Volume: 5 Issue: 4

Cite

APA Dascioglu, A., Culha, S., & Bayram, D. V. (2017). New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions. New Trends in Mathematical Sciences, 5(4), 232-241.
AMA Dascioglu A, Culha S, Bayram DV. New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions. New Trends in Mathematical Sciences. October 2017;5(4):232-241.
Chicago Dascioglu, Aysegul, Sevil Culha, and Dilek Varol Bayram. “New Analytical Solutions of the Space Fractional KdV Equation in Terms of Jacobi Elliptic Functions”. New Trends in Mathematical Sciences 5, no. 4 (October 2017): 232-41.
EndNote Dascioglu A, Culha S, Bayram DV (October 1, 2017) New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions. New Trends in Mathematical Sciences 5 4 232–241.
IEEE A. Dascioglu, S. Culha, and D. V. Bayram, “New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions”, New Trends in Mathematical Sciences, vol. 5, no. 4, pp. 232–241, 2017.
ISNAD Dascioglu, Aysegul et al. “New Analytical Solutions of the Space Fractional KdV Equation in Terms of Jacobi Elliptic Functions”. New Trends in Mathematical Sciences 5/4 (October 2017), 232-241.
JAMA Dascioglu A, Culha S, Bayram DV. New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions. New Trends in Mathematical Sciences. 2017;5:232–241.
MLA Dascioglu, Aysegul et al. “New Analytical Solutions of the Space Fractional KdV Equation in Terms of Jacobi Elliptic Functions”. New Trends in Mathematical Sciences, vol. 5, no. 4, 2017, pp. 232-41.
Vancouver Dascioglu A, Culha S, Bayram DV. New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions. New Trends in Mathematical Sciences. 2017;5(4):232-41.