This article proposes some higher order splitting-up techniques based on the cubic B-spline Galerkin finite element method in analyzing the Burgers equation model. The strong form of both conservation and diffusion parts of the time-split Burgers equation have been considered in building the Galerkin approach. To integrate the corresponding ODE system, the Crank-Nicolson time discretization scheme is used. The proposed schemes are shown to be unconditionally stable. Three challenging examples have been considered that have changing values of the kinematic viscosity constant of the medium. Moreover, cases of shock waves of severe gradient are solved and compared with the exact solution and the literature. The qualitative and quantitative results demonstrate that our numerical approach has far higher accuracy than rival methods.