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Symmetry properties and explicit solutions of some nonlinear differential and fractional equations

Author

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  • Zhang, Yufeng
  • Mei, Jianqin
  • Zhang, Xiangzhi

Abstract

One generalized Burgers hierarchy is derived by applying the Cole-Hopf transforation, whose dark-equation hierarchy is also generated by the dark-equation method, from which a generalized Burgers equation and a generalized Kupershmidt equation, respectively, are followed to obtain. Through Lie-group analysis method we produce similarity reductions, exact solutions of the generalized Burgers and the Kupershmidt equations. Specially, we investigate the similarity reductions of the fractional Kupershmidt equation and its exact solutions. In addition, we obtain the conservation laws of the Kupershmidt equation and its adjoint equation. Finally, we give rise to symmetries, primary branch solutions as well various recursion operators of degenerated equations from the Kupershmidt equation.

Suggested Citation

  • Zhang, Yufeng & Mei, Jianqin & Zhang, Xiangzhi, 2018. "Symmetry properties and explicit solutions of some nonlinear differential and fractional equations," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 408-418.
    • Handle: RePEc:eee:apmaco:v:337:y:2018:i:c:p:408-418
    DOI: 10.1016/j.amc.2018.05.030
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    References listed on IDEAS

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    1. Gang wei Wang & Tian zhou Xu & Tao Feng, 2014. "Lie Symmetry Analysis and Explicit Solutions of the Time Fractional Fifth-Order KdV Equation," PLOS ONE, Public Library of Science, vol. 9(2), pages 1-6, February.
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    Cited by:

    1. Amber Shaikh & M Asif Jamal & Fozia Hanif & M Sadiq Ali Khan & Syed Inayatullah, 2019. "Neural minimization methods (NMM) for solving variable order fractional delay differential equations (FDDEs) with simulated annealing (SA)," PLOS ONE, Public Library of Science, vol. 14(10), pages 1-22, October.
    2. Hengtai Wang & Zhiwei Zou & Xin Shen, 2021. "Lie Symmetry Analysis, Self-Adjointness and Conservation Law for a Type of Nonlinear Equation," Mathematics, MDPI, vol. 9(12), pages 1-10, June.

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