Group formalism of Lie transformations, exact solutions and conservation laws of nonlinear time-fractional Kramers equation

In this paper, we will concentrate on a systematic investigation of finding Lie point symmetries of the nonlinear (2 + 1)-dimensional time-fractional Kramers equation via Riemann-Liouville and Caputo derivatives. By using the Lie group analysis method, the invariance properties and the symmetry reductions of the time-fractional Kramers equation are provided. It is shown that by using one of the symmetries of the underlying equation, it can be transformed into a nonlinear (1+1)-dimensional fractional differential equation with a new dependent variable and the derivative in Erdelyi-Kober sense. Furthermore, we construct some exact solutions for the time-fractional Kramers equation using the invariant subspace method. In addition, adapting Ibragimov's method, using Noether identity, Noether operators and formal Lagrangian, we construct conservation laws of this equation.