Analytical treatment of nonlinear conformable time-fractional Boussinesq equations by three integration methods

Under investigation in this paper is a nonlinear conformable time-fractional Boussinesq equations as an important class of fractional differential equations in mathematical physics. The extended trial equation method, the $\exp (-\Omega (\eta \left)\right)$-expansion method and the $\tan \left(\varphi \right(\eta \left)/2\right)$-expansion method are used in examining the analytical solution of the nonlinear fractional equations. The proposed methods are based on the integration method and a wave transformation. The fractional derivative in the sense of conformable time-fractional derivative is defined. Fractional complex transform is implemented to change fractional differential equations into ordinary differential equations in this paper. In addition, explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of nonlinear conformable time-fractional Boussinesq equations.