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

The concept of nonlinear self-adjointness is employed to construct the conservation laws for fractional evolution equations using its Lie point symmetries. The 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 obtained using appropriate formal Lagrangians. 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 conserved vectors have been constructed for the linear and nonlinear fractional subdiffusion and diffusion-wave equations corresponding to its Lie point symmetries.