Symmetry group classification for two-dimensional elastodynamics problems in nonlocal elasticity

First Integrals, Integrating Factors, and Invariant Solutions of the Path Equation Based on Noether and -Symmetries

“…Let ∈ A be a vector that satisfies ̸ = + , where is a constant. Then ( ) represents th prolongation of the generalized operator (7), and partial Noether operator corresponding to a partial Lagrangian is formulated as…”

“…The main purpose of the work is to study Noether and -symmetry classifications of the path equation for the different forms of arbitrary function of the governing equation [3][4][5][6][7]. Based on Noether's theorem, if Noether symmetries of an ordinary differential equation are known, then the conservation laws of this equation can be obtained directly by using Euler-Lagrange equations [8].…”

Symmetry properties of two-dimensional Ciarlet–Mooney–Rivlin constitutive models in nonlinear elastodynamics

“…Particular classes of exact solutions have been obtained for several models using continuous symmetry groups. In particular, the classification of Lie point symmetries for 1D and 2D nonlocal elastodynamics appears in [11,12]. Invariant solutions for the radial motions of compressible hyperelastic spheres and cylinders have been obtained in [13,14].…”

Invariant solutions of integro-differential Vlasov–Maxwell equations in Lagrangian variables by Lie group analysis

“…Indirect methods can be found in the works of Krasnoslobodtsev [12] and Taranov [13]. Using Lie-Bäcklund type operators is a direct method for investigating the solution of nonlocal determining equations which was introduced by Bobylev [14,15], Meleshko [16], Ibragimov et al [17], and Özer, [18][19][20]. There exist also direct methods in which instead of the Lie-Bäcklund type operator, a Lie point group is used.…”