Symmetry classification and exact solutions of the Kramers equation

1998 ◽  
Vol 39 (6) ◽  
pp. 3505-3510 ◽  
Author(s):  
Stanislav Spichak ◽  
Valerii Stognii
Keyword(s):  
Exact Solutions ◽  
Symmetry Classification ◽  
Kramers Equation

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

2020 ◽  
Vol 17 (12) ◽  
pp. 2050190
Author(s):  
Zahra Momennezhad ◽  
Mehdi Nadjafikhah
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Group Analysis ◽  
Systematic Investigation ◽  
Analysis Method ◽  
Invariance Properties ◽  
Kramers Equation ◽  
Fractional Differential ◽  
Lie Transformations ◽  
Noether Operators

In this paper, we will concentrate on a systematic investigation of finding Lie point symmetries of the nonlinear [Formula: see text]-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 [Formula: see text]-dimensional fractional differential equation with a new dependent variable and the derivative in Erdélyi–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.

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