For numerous fluids between elastic and viscous materials, the fractional derivative models have an advantage over the integer order models. On the basis of conformable fractional derivative and the respective useful properties, the bilinear form of time fractional Burgers equation and Boussinesq-Burgers equations are obtained using the generalized Bell polynomials and bilinear method. The kink soliton solution, anti-kink soliton solution, and the single-soliton solution for different fractional order are derived, respectively. The time fractional order system possesses property of time memory. Higher oscillation frequency appears as the time fractional order increasing. The fractional derivative increases the possibility of improving the control performance in complex systems with fluids between different elastic and viscous materials.