Search

In this paper, we employ the time fractional higher order nonlinear Boussinesq–Burgers system to investigate shallow water waves based on the Riemann–Liouville and Caputo fractional partial derivatives. The Lie algebra of infinitesimal generators associated with the symmetry groups that leave the studied system invariant is systematically constructed, and the similarity reductions are established. Furthermore, conserved quantities of the system under consideration are formulated. Next, the power series solution, including the convergence analysis, is obtained. Based on the fractional Sumudu–Adomian decomposition method, some approximate solutions are derived. Finally, in order to show the efficiency of the considered methods, the effect of the fractional order, as well as the dynamical behavior of solutions, numerical validation and graphical representations are designed. [ABSTRACT FROM AUTHOR]

The nonlocal residual symmetries of a generalized Broer–Kaup–Kupershmidt system are constructed using the truncated Painlevé expansion. By considering appropriate potential variables, these nonlocal symmetries are localized into Lie point symmetries by prolonging the generalized Broer–Kaup–Kupershmidt system to an enlarged system. Moreover, multiple residual symmetries of the studied system are derived, their localization is performed, and the N-th Bäcklund transformation theorem is presented. Furthermore, some exact interaction solutions between solitons and various kinds of nonlinear waves are formulated in view of the new consistent tanh expansion method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Lie group formalism for the time fractional generalized nonlinear Broer–Kaup system in the sense of Riemann–Liouville fractional partial derivative. The Lie algebra corresponding to the symmetry groups in which the studied equation remains invariant is established, and the similarity reductions are performed. Next, based on the invariant subspace method as well as the power series method, including the convergence analysis, some exact solutions of the time fractional generalized Broer–Kaup system and its standard form are derived. Moreover, in order to show the dynamical behavior and the impact of the fractional order on the profile of solutions, some figures in 2D and 3D have been depicted. Finally, in accordance with the nonlinear self-adjointness property, conservation laws are successfully formulated using infinitesimal symmetries. [ABSTRACT FROM AUTHOR]

In this paper, the Lie symmetry group method is employed to study the time fractional generalized advection–diffusion equation. The Lie symmetry algebra classification is established by considering three different cases. Next, the similarity reductions are performed, and some solutions including invariants are derived. Finally, conservation laws are successfully constructed using the symmetry generators. [ABSTRACT FROM AUTHOR]