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 [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.

Classification of static plane symmetric spacetime via Noether gauge symmetries

In this paper, general static plane symmetric spacetime is classified with respect to Noether operators. For this purpose, Noether theorem is used which yields a set of linear partial differential equations (PDEs) with unknown radial functions [Formula: see text], [Formula: see text] and [Formula: see text]. Further, these PDEs are solved by taking different possibilities of radial functions. In the first case, all radial functions are considered same, while two functions are taken proportional to each other in second case, which further discussed by taking four different relationships between [Formula: see text], [Formula: see text] and [Formula: see text]. For all cases, different forms of unknown functions of radial factor [Formula: see text] are reported for nontrivial Noether operators with non-zero gauge term. At the end, a list of conserved quantities for each Noether operator Tables 1–4 is presented.

On Conservation Forms and Invariant Solutions for Classical Mechanics Problems of Liénard Type

In this study we apply partial Noether andλ-symmetry approaches to a second-order nonlinear autonomous equation of the formy′′+fyy′+g(y)=0, called Liénard equation corresponding to some important problems in classical mechanics field with respect tof(y)andg(y)functions. As a first approach we utilize partial Lagrangians and partial Noether operators to obtain conserved forms of Liénard equation. Then, as a second approach, based on theλ-symmetry method, we analyzeλ-symmetries for the case thatλ-function is in the form ofλ(x,y,y′)=λ1(x,y)y′+λ2(x,y). Finally, a classification problem for the conservation forms and invariant solutions are considered.

First Integrals for Two Linearly Coupled Nonlinear Duffing Oscillators

We investigate Noether and partial Noether operators of point type corresponding to a Lagrangian and a partial Lagrangian for a system of two linearly coupled nonlinear Duffing oscillators. Then, the first integrals with respect to Noether and partial Noether operators of point type are obtained explicitly by utilizing Noether and partial Noether theorems for the system under consideration. Moreover, if the partial Euler-Lagrange equations are independent of derivatives, then the partial Noether operators become Noether point symmetry generators for such equations. The difference arises in the gauge terms due to Lagrangians being different for respective approaches. This study points to new ways of constructing first integrals for nonlinear equations without regard to a Lagrangian. We have illustrated it here for nonlinear Duffing oscillators.

CONSERVATION LAWS OF SOME NON-VARIATIONAL PERTURBED PDE'S VIA A PARTIAL VARIATIONAL APPROACH

We show how one can construct approximate conservation laws of systems of perturbed or approximate partial differential equations that are not derivable by the variational principle but are approximate Euler–Lagrange in part by using approximate partial Noether operators associated with partial Lagrangians. We investigate perturbed partial differential equations that arise in several important physical applications. Finally, we obtain new approximate conservation laws for these equations.