Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Travelling wave solutions for fractional Korteweg-de Vries equations via an approximate-analytical method

1 Department of Mathematics, Savitribai Phule Pune University, Pune 411007, India
2 Computational Intelligence Laboratory, University of Manitoba, WPG, MB, R3T 5V6, Winnipeg, Canada
3 Department of Mathematics, Faculty of Arts and Science, Adiyaman University, 02040 Adiyaman, Turkey

Special Issues: Recent Advances in Fractional Calculus with Real World Applications

This paper introduces an approximate-analytical method (AAM) for solving nonlinear fractional partial differential equations (NFPDEs) in full general forms. The main advantage of the paper is to apply the proposed AAM to solve the fractional Korteweg-de Vries (KdV) equations. Moreover, the analytical travelling wave solutions for the fractional KdV equation and the modified fractional KdV equation are successfully obtained. The numerical solutions are also obtained in the forms of tables and graphs. The fractional partial derivatives are considered in Caputo sense.
  Article Metrics

Keywords approximate-analytical method; nonlinear fractional partial differential equations; fractional KdV equations; traveling wave solutions

Citation: Hayman Thabet, Subhash Kendre, James Peters. Travelling wave solutions for fractional Korteweg-de Vries equations via an approximate-analytical method. AIMS Mathematics, 2019, 4(4): 1203-1222. doi: 10.3934/math.2019.4.1203


  • 1.K. A. Touchent, Z. Hammouch, T. Mekkaoui, et al. Implementation and Convergence Analysis of Homotopy Perturbation Coupled With Sumudu Transform to Construct Solutions of Local-Fractional PDEs, Fractal and Fractional, 2 (2018), 22.
  • 2.A. Akbulut and F. Taşcan, Lie symmetries, symmetry reductions and conservation laws of time fractional modified korteweg–de vries (mkdv) equation, Chaos Soliton. Fract., 100 (2017), 1-6.    
  • 3.N. A. Asif, Z. Hammouch, M. B. Riaz, et al. Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, Eur. Phys. J. Plus, 133 (2018), 272.
  • 4.A. M. A. El-Sayed, A. Elsaid, I. L. El-Kalla, et al. A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains, Appl. Math. Comput., 218 (2012), 8329-8340.
  • 5.B. Guo, X. Pu and F. Huang, Fractional partial differential equations and their numerical solutions, World Scientific, 2015.
  • 6.P. K. Gupta and M. Singh, Homotopy perturbation method for fractional fornberg–whitham equation, Comput. Math. Appl., 61 (2011), 250-254.    
  • 7.Q. Huang and R. Zhdanov, Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann–Liouville derivative, Physica A, 409 (2014), 110-118.    
  • 8.A. Keten, M. Yavuz and D. Baleanu, Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces, Fractal and Fractional, 3 (2019), 27.
  • 9.A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Limited, 2006.
  • 10.W. X. Ma and Z. Zhu, Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Appl. Math. Comput., 218 (2012), 11871-11879.
  • 11.K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley-Interscience, 1993.
  • 12.S. Momani, Z. Odibat and V. S. Erturk, Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation, Phys. Lett. A, 370 (2007), 379-387.    
  • 13.G. Paquin and P. Winternitz, Group theoretical analysis of dispersive long wave equations in two space dimensions, Physica D: Nonlinear Phenomena, 46 (1990), 122-138.    
  • 14.I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic press, 1998.
  • 15.M. B. Riaz, M. A. Imran and K. Shabbir, Analytic solutions of Oldroyd-B fluid with fractional derivatives in a circular duct that applies a constant couple, Alex. Eng. J., 55 (2016), 3267-3275.    
  • 16.M. B. Riaz and A. A. Zafar, Exact solutions for the blood flow through a circular tube under the influence of a magnetic field using fractional Caputo-Fabrizio derivatives, Math. Model. Nat. Phenom., 13 (2018), 8.
  • 17.R. Sahadevan and T. Bakkyaraj, Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations, J. Math. Anal. Appl., 393 (2012), 341-347.    
  • 18.H. Thabet and S. Kendre, Analytical solutions for conformable space-time fractional partial differential equations via fractional differential transform, Chaos Soliton. Fract., 109 (2018), 238-245.    
  • 19.H. Thabet and S. Kendre, Modified least squares homotopy perturbation method for solving fractional partial differential equations, Malaya Journal of Matematik, 6 (2018), 420-427.    
  • 20.H. Thabet and S. Kendre, New modification of Adomian decomposition method for solving a system of nonlinear fractional partial differential equations, Int. J. Adv. Appl. Math. and Mech, 6 (2019), 1-13.
  • 21.H. Thabet, S. Kendre and D. Chalishajar, New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations, Mathematics, 5 (2017), 47.
  • 22.H. Thabet, S. Kendre and J. Peters, Analytical Solutions for Nonlinear Systems of Conformable Space-Time Fractional Partial Differential Equations via Generalized Fractional Differential Transform, Vietnam Journal of Mathematics, (2019), 1-21.
  • 23.M. Yavuz and E. Bonyah, New approaches to the fractional dynamics of schistosomiasis disease model, Physica A, 525 (2019), 373-393.    
  • 24.M. Yavuz and N. Özdemir, Comparing the new fractional derivative operators involving exponential and mittag-leffler kernel, Discrete & Continuous Dynamical Systems-S, (2019), 1098-1107.
  • 25.E. M. E. Zayed, Y. A. Amer and R. M. A. Shohib, The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics, Journal of the Association of Arab Universities for Basic and Applied Sciences, 19 (2016), 59-69.


This article has been cited by

  • 1. Hayman Thabet, Subhash Kendre, James Peters, Melike Kaplan, Solitary wave solutions and traveling wave solutions for systems of time-fractional nonlinear wave equations via an analytical approach, Computational and Applied Mathematics, 2020, 39, 3, 10.1007/s40314-020-01163-1

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved