Group analysis, invariance results, exact solutions and conservation laws of the perturbed fractional Boussinesq equation

The scope of this work is based on symmetry analysis of perturbed fractional Boussinesq equation. For beginning the group formalism, invariance properties and conservation laws of the nonlinear perturbed fractional Boussinesq equation have been explored generally. This method was first described by Lukashchuk [Commun. Nonlinear Sci. Numer. Simul. 68 (2019) 147-159]. The key subject is that when the order of fractional derivative in a fractional differential equation (FDE) is nearly integers, we can approximate it to a perturbed integer-order differential equation with a small perturbation parameter. For obtaining the results, perturbed and unperturbed symmetries are computed. Then, the methodology of reduction is applied for finding some new solutions by the symmetry operators of the equation. These solutions are obtained by the similarity transformations of the symmetries. Another exact solution will be found with constructing one-dimensional optimal system of the symmetries. Finally, the meaning of nonlinear self-adjointness concept is attended in order to find conservation laws with informal Lagrangians.