Modeling of viscoelastic flows in terms of fractional calculus is an inquisitive area in engineering and industry. The physical models with fractional operators in nonNewtonian fluid mechanics represent more realistic behavior of flows involving nonlinear complex dynamics. The efficiency is because of freedom to choose derivative of any order in the mathematical formulation of the flows. Non-integer derivatives are important for the systems having hereditary behavior as they depend on the past conditions along with the local conditions. Viscoelastic fluids keep memory of old deformations and their behavior is related to these deformations. The fractional derivatives are more adequate in predicting the characteristics of viscoelastic fluids than the ordinary derivatives. In literature linear flow problems, with non-integer derivatives are solved by classical transforms methods. Unfortunately, most of the viscoelastic fluids unlike Newtonian fluids, are not characterized by only one relation. Mathematical equations, for the viscoelastic fluid flows are highly nonlinear in nature. In most of situations, highly nonlinear PDEs cannot be solved exactly, by existing techniques. Literature survey indicates, that appropriate consideration is not given to the numerical solutions, of anomalous nonlinear flow problems with noninteger derivatives. In this study, we have modeled the viscoelastic flow problems via fractional calculus approach and considered numerical techniques using finite difference approximations along with ”L1 algorithm”to discretize non-integer derivatives of time and finite element, discretization is used for space variables, in order to solve the governing fractional viscoelastic models. Finally we have predicted the behavior of viscoelastic fluids that can be used directly for the simulations of industrial processes.