Nonlinear Dynamics

, Volume 80, Issue 1–2, pp 791–802 | Cite as

Conservation laws for time-fractional subdiffusion and diffusion-wave equations

Original Paper


A new technique for constructing conservation laws for fractional differential equations not having a Lagrangian is proposed. The technique is based on the methods of Lie group analysis and employs the concept of nonlinear self-adjointness which is enhanced to the certain class of fractional evolution equations. The proposed approach is demonstrated on subdiffusion and diffusion-wave equations with the Riemann–Liouville and Caputo time-fractional derivatives. It is shown that these equations are nonlinearly self-adjoint, and therefore, desired conservation laws can be calculated using the appropriate formal Lagrangians. The explicit forms of fractional generalizations of the Noether operators are also proposed for the equations with the Riemann–Liouville and Caputo time-fractional derivatives of order \(\alpha \in (0,2)\). Using these operators and formal Lagrangians, new conservation laws are constructed for the linear and nonlinear time-fractional subdiffusion and diffusion-wave equations by their Lie point symmetries.


Time-fractional diffusion equation Conservation law  Nonlinear self-adjointness Symmetry 



This work was supported by the grant of the Ministry of Education and Science of the Russian Federation (contract No. 11.G34.31.0042 with Ufa State Aviation Technical University and leading scientist Professor N. H. Ibragimov). The author is also grateful to Professor Rafail K. Gazizov for helpful discussion of the manuscript.


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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Laboratory “Group Analysis of Mathematical Models in Natural and Engineering Sciences”Ufa State Aviation Technical UniversityUfaRussia

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