Construction of new exact solutions to time-fractional two-component evolutionary system of order 2 via different methods

  • Linjun Wang
  • Wei Shen
  • Yiping Meng
  • Xumei ChenEmail author


This paper is concerned with the applications of five different methods including the sub-equation method, the tanh method, the modified Kudryashov method, the \(\left( \frac{G'}{G}\right)\)-expansion method and the Exp-function method to construct exact solutions of time-fractional two-component evolutionary system of order 2. We first convert this type of fractional equations to the nonlinear ordinary differential equations by means of fractional complex transform. Then, the five methods are adopted to solve the nonlinear ordinary differential equations. As a result, some new exact solutions are obtained. It is also shown that each of the considered methods can be used as an alternative for solving fractional differential equations.


Exact solutions Modified Riemann–Liouville derivative The time-fractional two-component evolutionary system of order 2 



This work is supported by National Natural Science Foundation of China (No. 11601192), Natural Science Foundation of Jiangsu Province (No. BK20140522), Scientific Research Fund of Jiangsu University of Science and Technology.


  1. Ablowitz, M.J., Clarkson, P.A.: Soliton, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York (1991)CrossRefzbMATHGoogle Scholar
  2. Alquran, M.: Solitons and periodic solutions to nonlinear partial differential equations by the sine–cosine method. Appl. Math. Inf. Sci. 6(1), 85–88 (2012)MathSciNetzbMATHGoogle Scholar
  3. Alquran, M.: Analytical solution of time-fractional two-component evolutionary system of order 2 by residual power series method. J Appl. Anal. Comput. 5(4), 589–599 (2015)MathSciNetGoogle Scholar
  4. Alquran, M., Al-Khaled, K., Ananbeh, H.: New soliton solutions for systems of nonlinear evolution equations by the rational sine–cosine method. Stud. Math. Sci. 3(1), 1–9 (2011)Google Scholar
  5. Bekir, A., Aksoy, E., Cevikel, A.C.: Exact solutions of nonlinear time fractional partial differential equations by sub-equation method. Math. Method Appl. Sci. 38(13), 2779–2784 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. Bhrawy, A., Zaky, M.: An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations. Appl. Numer. Math. 111, 197–218 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Ege, S.M., Misirli, E.: The modified Kudryashov method for solving some fractional-order nonlinear equations. Adv. Differ. Equ. 2014(1), 135 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Eslami, M., Neirameh, A.: New exact solutions for higher order nonlinear Schrödinger equation in optical fibers. Opt. Quantum Electron. 50(1), 47 (2018)CrossRefGoogle Scholar
  9. Eslami, M., Vajargah, B.F., Mirzazadeh, M., Biswas, A.: Application of first integral method to fractional partial differential equations. Indian J. Phys. 88(2), 177–184 (2014)ADSCrossRefGoogle Scholar
  10. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  11. Foursov, M.V.: Classification of certain integrable coupled potential KdV and modified KdV-type equations. J. Math. Phys. 41(9), 6173–6185 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. Foursov, M.V., Maza, M.M.: On computer-assisted classification of coupled integrable equations. J. Symb. Comput. 33, 647–660 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Guner, O., Aksoy, E., Bekir, A., Cevikel, A.C.: Different methods for \((3+1)\)-dimensional space-time fractional modified Kdv–Zakharov–Kuznetsov equation. Comput. Math. Appl. 71(6), 1259–1269 (2016)MathSciNetCrossRefGoogle Scholar
  14. Guner, O., Atik, H., Kayyrzhanovich, A.A.: New exact solution for space-time fractional differential equations via \((G^{\prime }/G)\)-expansion method. Optik 130, 696–701 (2017)ADSCrossRefGoogle Scholar
  15. He, J.H., Elagan, S., Li, Z.: Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Phys. Lett. A 376(4), 257–259 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. Jumarie, G.: Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 51(9–10), 1367–1376 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Jumarie, G.: Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions. Appl. Math. Lett. 22(3), 378–385 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Korkmaz, A., Hepson, O.E.: Traveling waves in rational expressions of exponential functions to the conformable time fractional Jimbo–Miwa and Zakharov–Kuznetsov equations. Opt. Quantum Electron. 50(1), 42 (2018)CrossRefGoogle Scholar
  19. Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2248–2253 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. Lopes, A.M., Machado, J.T., Pinto, C.M., Galhano, A.M.: Fractional dynamics and MDS visualization of earthquake phenomena. Comput. Math. Appl. 66(5), 647–658 (2013)MathSciNetCrossRefGoogle Scholar
  21. Lu, D., Seadawy, A.R., Khater, M.M.: Bifurcations of new multi soliton solutions of the van der Waals normal form for fluidized granular matter via six different methods. Results Phys. 7, 2028–2035 (2017)ADSCrossRefGoogle Scholar
  22. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, NY (1993)zbMATHGoogle Scholar
  23. Obeidat, N.A., Rawashdeh, M.S., Alquran, M.: An improved approximate solutions to nonlinear partial differential equations using differential transform method and adomian decomposition method. Thai J. Math. 12(3), 569–589 (2014)MathSciNetzbMATHGoogle Scholar
  24. Perdikaris, P., Karniadakis, E.: Fractional-order viscoelasticity in one-dimensional blood flow models. Ann. Biomed. Eng. 42(5), 1012–1023 (2014)CrossRefGoogle Scholar
  25. Sahoo, S., Ray, S.S.: A new method for exact solutions of variant types of time-fractional Korteweg–de Vries equations in shallow water waves. Math. Methods Appl. Sci. 40(1), 106–114 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. Tariq, H., Akram, G.: New approach for exact solutions of time fractional Cahn–Allen equation and time fractional Phi-4 equation. Physica A 473, 352–362 (2017)ADSMathSciNetCrossRefGoogle Scholar
  27. Wang, L., Wang, F.: Approximate solutions for time-fractional two-component evolutionary system of order 2 using coupled fractional reduced differential transform method. J. Appl. Anal. Comput. 7(4), 1312–1322 (2017)MathSciNetGoogle Scholar
  28. Wang, M., Li, X., Zhang, J.: The \((G^{\prime }/G)\)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372(4), 417–423 (2008)ADSMathSciNetCrossRefGoogle Scholar
  29. Wang, W., Chen, X., Ding, D., Lei, S.L.: Circulant preconditioning technique for barrier options pricing under fractional diffusion models. Int. J. Comput. Math. 92(12), 2596–2614 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Yin, C., Cheng, Y., Zhong, S.M., Bai, Z.: Fractional-order switching type control law design for adaptive sliding mode technique of 3D fractional-order nonlinear systems. Complexity 21(6), 363–373 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of ScienceJiangsu UniversityZhenjiangChina
  2. 2.School of ScienceJiangsu University of Science and TechnologyZhenjiangChina

Personalised recommendations