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Symmetry analysis and conservation laws to the space-fractional Prandtl equation

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Abstract

This paper studies the Lie point symmetries, conservation laws and invariant solutions to the space-fractional Prandtl equation with the Riemann–Liouville derivative. Exploiting the classical Lie symmetry analysis method extended to fractional equations, three vector fields are calculated, which were used to reduce the fractional partial differential equation into an ordinary differential equation through similarity transformation. Further, it is found that the fractional Prandtl equation is nonlinearly self-adjoint. Therefore, the nonlinear self-adjointness method not requiring the existence of a Lagrangian is utilized to explore the conservation laws for this equation. Three conservation laws are constructed, and one of them is a trivial conservation law. Finally, due to the difficulties of obtaining analytical solution for the ordinary equation, similarity solutions are presented numerically.

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Notes

  1. This result was pointed out by Prof. Lukashchuk.

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Acknowledgements

Mingyang Pan would like to thank Prof. S. Yu. Lukashchuk(Ufa State Aviation Technical University) for his timely, pertinent and valuable guidance. This work was supported by the National Natural Science Foundation of China (No. 51276014).

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Authors and Affiliations

  1. School of Energy and Environmental Engineering, University of Science and Technology Beijing, Beijing, 100083, China

    Mingyang Pan

  2. School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    Liancun Zheng & Chunyan Liu

  3. School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, QLD, 4001, Australia

    Fawang Liu

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  1. Mingyang Pan
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  2. Liancun Zheng
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  3. Chunyan Liu
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  4. Fawang Liu
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Correspondence to Liancun Zheng.

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Pan, M., Zheng, L., Liu, C. et al. Symmetry analysis and conservation laws to the space-fractional Prandtl equation. Nonlinear Dyn 90, 1343–1351 (2017). https://doi.org/10.1007/s11071-017-3730-8

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